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A $2.00 \mathrm{~kg}$ particle moves along an $x$ axis in one-dimensional motion while a conservative force along that axis acts on it. The potential energy $U(x)$ associated with the force is plotted in Fig. 8-10a. That is, if the particle were placed at any position between $x=0$ and $x=7.00 \mathrm{~m}$, it would have the plotted value of $U$. At $x=6.5 \mathrm{~m}$, the particle has velocity $\vec{v}_0=(-4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$ From Figure, determine the particle's speed at $x_1=4.5 \mathrm{~m}$.

If a force is conservative, it is possible to assign a numerical value for the potential at any point and conversely, when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken, contributing to the mechanical energy and the overall conservation of energy. The work done by a conservative force is equal to the negative of change in potential energy during that process. Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. \begin{align} v & = at+v_0 & [1]\\\ r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\\ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\\ r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\\ \end{align} where: * is the particle's initial position * is the particle's final position * is the particle's initial velocity * is the particle's final velocity * is the particle's acceleration * is the time interval Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; \begin{align} \mathbf{v} & = \int \mathbf{a} dt = \mathbf{a}t+\mathbf{v}_0 \,, & [1] \\\ \mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) dt = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \,, & [2] \\\ \end{align} in magnitudes, \begin{align} v & = at+v_0 \,, & [1] \\\ r & = \frac{{a}t^2}{2}+v_0t +r_0 \,. A conservative force depends only on the position of the object. Graph of the Lennard-Jones potential function: Intermolecular potential energy as a function of the distance of a pair of particles. (Velocity is on the y-axis and time on the x-axis. If the force is not conservative, then defining a scalar potential is not possible, because taking different paths would lead to conflicting potential differences between the start and end points. If the velocity or positions change non-linearly over time, such as in the example shown in the figure, then differentiation provides the correct solution. thumb|360px|v vs t graph for a moving particle under a non-uniform acceleration a. In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken.HyperPhysics - Conservative force Equivalently, if a particle travels in a closed loop, the total work done (the sum of the force acting along the path multiplied by the displacement) by a conservative force is zero. The position of the particle is \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r where and are the polar unit vectors. For example, if a child slides down a frictionless slide, the work done by the gravitational force on the child from the start of the slide to the end is independent of the shape of the slide; it only depends on the vertical displacement of the child. ==Mathematical description== A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions: # The curl of F is the zero vector: \vec{ abla} \times \vec{F} = \vec{0}. where in two dimensions this reduces to: \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0 # There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place: W \equiv \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0. The Lennard-Jones model describes the potential intermolecular energy V between two particles based on the outlined principles. The mean potential energy per particle is negative. Accordingly, some authors classify the magnetic force as conservative,For example, : "In general, a force which depends explicitly upon the velocity of the particle is not conservative. Specific potential energy is potential energy of an object per unit of mass of that object. Suppose a particle starts at point A, and there is a force F acting on it. In mechanics, the derivative of the position vs. time graph of an object is equal to the velocity of the object. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B. Using equation [4] in the set above, we have: s= \frac{v^2 - u^2}{-2g}. Therefore, the slope of the curve gives the change in position divided by the change in time, which is the definition of the average velocity for that interval of time on the graph. Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object.

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A playful astronaut releases a bowling ball, of mass $m=$ $7.20 \mathrm{~kg}$, into circular orbit about Earth at an altitude $h$ of $350 \mathrm{~km}$. What is the mechanical energy $E$ of the ball in its orbit?

Hence, mechanical energy E_\text{mechanical} of the satellite-Earth system is given by E_\text{mechanical} = U + K E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2 If the satellite is in circular orbit, the energy conservation equation can be further simplified into E_\text{mechanical} = - G \frac{M m}{2r} since in circular motion, Newton's 2nd Law of motion can be taken to be G \frac{M m}{r^2}\ = \frac{m v^2}{r} ==Conversion== Today, many technological devices convert mechanical energy into other forms of energy or vice versa. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced. ===Satellite=== thumb|plot of kinetic energy K, gravitational potential energy, U and mechanical energy E_\text{mechanical} versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius A satellite of mass m at a distance r from the centre of Earth possesses both kinetic energy, K, (by virtue of its motion) and gravitational potential energy, U, (by virtue of its position within the Earth's gravitational field; Earth's mass is M). Specific mechanical energy is the mechanical energy of an object per unit of mass. The relations are used. p= \frac{h^2}{\mu} = a(1-{e^2}) = r_{p}(1+e) where * p\,\\! is the conic section semi-latus rectum. * r_p\,\\! is distance at periastron of the body from the center of mass. v = \sqrt{\mu\left({2\over{r}} - {1\over{a}}\right)} where *\mu\, is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other. *r\, is the distance between the orbiting body and center of mass. *a\,\\! is the length of the semi-major axis. === Orbital Mechanics === When calculating the specific mechanical energy of a satellite in orbit around a celestial body, the mass of the satellite is assumed to be negligible: \mu = G(M + m) \approx GM where M is the mass of the celestial body. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant. Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is defined as: \epsilon= \epsilonk+\epsilonp where * \epsilonk is the specific kinetic energy * \epsilonp it the specific potential energy == Astrodynamics == In the gravitational two-body problem, the specific mechanical energy of one body \epsilon is given as: \begin{align} \epsilon &= \frac{v^2}{2} - \frac{\mu}{r} = -\frac{1}{2} \frac{\mu^2}{h^2} \left(1 - e^2\right) = -\frac{\mu}{2a} \end{align} where * v\,\\! is the orbital speed of the body; relative to center of mass. * r\,\\! is the orbital distance between the body and center of mass; * \mu = {G}(m_1 + m_2)\,\\! is the standard gravitational parameter of the bodies; * h\,\\! is the specific relative angular momentum of the same body referenced to the center of mass. The gravitational potential energy of an object is equal to the weight W of the object multiplied by the height h of the object's center of gravity relative to an arbitrary datum: U = W h The potential energy of an object can be defined as the object's ability to do work and is increased as the object is moved in the opposite direction of the direction of the force. Specific potential energy is potential energy of an object per unit of mass of that object. Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. Though energy cannot be created or destroyed, it can be converted to another form of energy. ===Swinging pendulum=== 200px|thumb|A swinging pendulum with the velocity vector (green) and acceleration vector (blue). The change in potential energy moving from the surface (a distance R from the center) to a height h above the surface is \begin{align} \Delta U &= \frac{GMm}{R}-\frac{GMm}{R+h} \\\ &= \frac{GMm}{R}\left(1-\frac{1}{1+h/R}\right). \end{align} If h/R is small, as it must be close to the surface where g is constant, then this expression can be simplified using the binomial approximation \frac{1}{1+h/r} \approx 1-\frac{h}{R} to \begin{align} \Delta U &\approx \frac{GMm}{R}\left[1-\left(1-\frac{h}{R}\right)\right] \\\ \Delta U &\approx \frac{GMmh}{R^2}\\\ \Delta U &\approx m\left(\frac{GM}{R^2}\right)h. \end{align} As the gravitational field is g = GM / R^2, this reduces to \Delta U \approx mgh. The gravitational potential energy is the potential energy an object has because it is within a gravitational field. thumb|250px|An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; its mechanical energy is therefore conserved. In a mechanical system like a swinging pendulum subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible, energy passes back and forth between kinetic and potential energy but never leaves the system. The force between a point mass, M, and another point mass, m, is given by Newton's law of gravitation: Extract of page 10 F = \frac {GMm}{r^2} To get the total work done by an external force to bring point mass m from infinity to the final distance R (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement: W = \int_\infty^R \frac {GMm}{r^2}dr = -\left . \frac{G M m}{r} \right|_{\infty}^{R} Because \lim_{r\to \infty} \frac{1}{r} = 0, the total work done on the object can be written as: Extract of page 143 In the common situation where a much smaller mass m is moving near the surface of a much larger object with mass M, the gravitational field is nearly constant and so the expression for gravitational energy can be considerably simplified. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity. In an elastic collision, mechanical energy is conserved – the sum of the mechanical energies of the colliding objects is the same before and after the collision. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. In physical sciences, mechanical energy is the sum of potential energy and kinetic energy.

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Let the disk in Figure start from rest at time $t=0$ and also let the tension in the massless cord be $6.0 \mathrm{~N}$ and the angular acceleration of the disk be $-24 \mathrm{rad} / \mathrm{s}^2$. What is its rotational kinetic energy $K$ at $t=2.5 \mathrm{~s}$ ?

The rotational energy depends on the moment of inertia for the system, I . Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Knowledge of the Euler angles as function of time t and the initial coordinates \mathbf{r}(0) determine the kinematics of the rigid rotor. === Classical kinetic energy === The following text forms a generalization of the well- known special case of the rotational energy of an object that rotates around one axis. The rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. The classical kinetic energy T of the rigid rotor can be expressed in different ways: * as a function of angular velocity * in Lagrangian form * as a function of angular momentum * in Hamiltonian form. The kinetic energy T of the linear rigid rotor is given by 2T = \mu R^2 \left[\dot{\theta}^2 + (\dot\varphi\,\sin\theta)^2\right] = \mu R^2 \begin{pmatrix}\dot{\theta} & \dot{\varphi}\end{pmatrix} \begin{pmatrix} 1 & 0 \\\ 0 & \sin^2\theta \\\ \end{pmatrix} \begin{pmatrix}\dot{\theta} \\\ \dot{\varphi}\end{pmatrix} = \mu \begin{pmatrix}\dot{\theta} & \dot{\varphi}\end{pmatrix} \begin{pmatrix} h_\theta^2 & 0 \\\ 0 & h_\varphi^2 \\\ \end{pmatrix} \begin{pmatrix}\dot{\theta} \\\ \dot{\varphi}\end{pmatrix}, where h_\theta = R\, and h_\varphi= R\sin\theta\, are scale (or Lamé) factors. Note the close relationship between the result for rotational energy and the energy held by linear (or translational) motion: :E_\mathrm{translational} = \tfrac{1}{2} m v^2 In the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity, \omega , takes the role of the linear velocity, v. As the Earth has a sidereal rotation period of 23.93 hours, it has an angular velocity of .Launching From Florida: Life in the Fast Lane!, NASA The Earth has a moment of inertia, I = .Moment of inertia--Earth, Wolfram Therefore, it has a rotational kinetic energy of . Looking at rotational energy separately around an object's axis of rotation, the following dependence on the object's moment of inertia is observed:Resnick, R. and Halliday, D. (1966) PHYSICS, Equation 12-11 :E_\mathrm{rotational} = \tfrac{1}{2} I \omega^2 where : \omega \ is the angular velocity : I \ is the moment of inertia around the axis of rotation : E \ is the kinetic energy The mechanical work required for or applied during rotation is the torque times the rotation angle. – Page 37 of 45 (graphic) For a miniature disc with a diameter of 8 cm (radius of 4 cm), the speed ratio of outer to inner data edge is 1.6. The rotor is rigid if R is independent of time. For a stationary open rotor with no outer duct, such as a helicopter in hover, the power required to produce a given thrust is: :P = \sqrt{\frac{T^3}{2 \rho A}} where: * T is the thrust * \rho is the density of air (or other medium) * A is the area of the rotor disc * P is power A device which converts the translational energy of the fluid into rotational energy of the axis or vice versa is called a Rankine disk actuator. Note that a different rotation matrix would result from a different choice of Euler angle convention used. ==== Lagrange form ==== Backsubstitution of the expression of \boldsymbol{\omega} into T gives the kinetic energy in Lagrange form (as a function of the time derivatives of the Euler angles). The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow). A linear (data reading and writing) speeds of 2.4 times higher can be reached at the outer disc edge with the same angular (rotation) speed. Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. This is because, although the model does account for bond stretching due to rotation, it ignores any bond stretching due to vibrational energy in the bond (anharmonicity in the potential). == Arbitrarily shaped rigid rotor == An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear motion) in field-free space R3, so that its energy consists only of rotational kinetic energy (and possibly constant translational energy that can be ignored). In matrix-vector notation, 2 T = \begin{pmatrix} \dot{\alpha} & \dot{\beta} & \dot{\gamma} \end{pmatrix} \; \mathbf{g} \; \begin{pmatrix} \dot{\alpha} \\\ \dot{\beta} \\\ \dot{\gamma}\\\ \end{pmatrix}, where \mathbf{g} is the metric tensor expressed in Euler angles--a non-orthogonal system of curvilinear coordinates-- \mathbf{g}= \begin{pmatrix} I_1 \sin^2\beta \cos^2\gamma+I_2\sin^2\beta\sin^2\gamma+I_3\cos^2\beta & (I_2-I_1) \sin\beta\sin\gamma\cos\gamma & I_3\cos\beta \\\ (I_2-I_1) \sin\beta\sin\gamma\cos\gamma & I_1\sin^2\gamma+I_2\cos^2\gamma & 0 \\\ I_3\cos\beta & 0 & I_3 \\\ \end{pmatrix}. ==== Angular momentum form ==== Often the kinetic energy is written as a function of the angular momentum \mathbf{L} of the rigid rotor. (Note: The corresponding eigenvalue equation gives the Schrödinger equation for the rigid rotor in the form that it was solved for the first time by Kronig and Rabi (for the special case of the symmetric rotor). thumb|An actuator disk accelerating a fluid flow from right to left In fluid dynamics, momentum theory or disk actuator theory is a theory describing a mathematical model of an ideal actuator disk, such as a propeller or helicopter rotor, by W.J.M. Rankine (1865), Alfred George Greenhill (1888) and (1889). This disc creates a flow around the rotor. The instantaneous power of an angularly accelerating body is the torque times the angular velocity.

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A food shipper pushes a wood crate of cabbage heads (total mass $m=14 \mathrm{~kg}$ ) across a concrete floor with a constant horizontal force $\vec{F}$ of magnitude $40 \mathrm{~N}$. In a straight-line displacement of magnitude $d=0.50 \mathrm{~m}$, the speed of the crate decreases from $v_0=0.60 \mathrm{~m} / \mathrm{s}$ to $v=0.20 \mathrm{~m} / \mathrm{s}$. What is the increase $\Delta E_{\text {th }}$ in the thermal energy of the crate and floor?

When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material. The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. The Ergun equation, derived by the Turkish chemical engineer Sabri Ergun in 1952, expresses the friction factor in a packed column as a function of the modified Reynolds number. ==Equation== f_p = \frac {150}{Gr_p}+1.75 where f_p and Gr_p are defined as f_p = \frac{\Delta p}{L} \frac{D_p}{\rho v_s^2} \left(\frac{\epsilon^3}{1-\epsilon}\right) and Gr_p = \frac{\rho v_s D_p}{(1-\epsilon)\mu} = \frac{Re}{(1-\epsilon)}; where: Gr_p is the modified Reynolds number, f_p is the packed bed friction factor \Delta p is the pressure drop across the bed, L is the length of the bed (not the column), D_p is the equivalent spherical diameter of the packing, \rho is the density of fluid, \mu is the dynamic viscosity of the fluid, v_s is the superficial velocity (i.e. the velocity that the fluid would have through the empty tube at the same volumetric flow rate) \epsilon is the void fraction (porosity) of the bed. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. Specific mechanical energy is the mechanical energy of an object per unit of mass. The heat loss due to linear thermal bridging (H_{TB}) is a physical quantity used when calculating the energy performance of buildings. This is an efficient way of increasing the rate, since the alternative way of doing so is by increasing either the heat transfer coefficient (which depends on the nature of materials being used and the conditions of use) or the temperature gradient (which depends on the conditions of use). In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, \partial Q/\partial t, is proportional to the rate of change of its temperature, \partial u/\partial t. One of the experimentally obtained equations for heat transfer coefficient for the fin surface for low wind velocities is: k=2.11 v^{0.71} \theta^{0.44} a^{-0.14} where k= Fin surface heat transfer coefficient [W/m2K ] a=fin length [mm] v=wind velocity [km/hr] θ=fin pitch [mm] Another equation for high fluid velocities, obtained from experiments conducted by Gibson, is k= 241.7[0.0247-0.00148(a^{0.8}/\theta^{0.4})] v^{0.73} where k=Fin surface heat transfer coefficient[W/m2K ] a=Fin length[mm] θ=Fin pitch[mm] v=Wind velocity[km/hr] A more accurate equation for fin surface heat transfer coefficient is: k_{avg} = (2.47-2.55/\theta^{0.4}) v^{0.9} 0.0872 \theta + 4.31 where k (avg)= Fin surface heat transfer coefficient[W/m2K ] θ=Fin pitch[mm] v=Wind velocity[km/hr] All these equations can be used to evaluate average heat transfer coefficient for various fin designs. == Design == The momentum conservation equation for this case is given as follows: {\partial(\rho v)\over\partial t} + v abla . (\rho v) = - abla P + abla . \tau + F + \rho g This is used in combination with the continuity equation. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. Applying the law of conservation of energy to a small element of the medium centered at x, one concludes that the rate at which heat accumulates at a given point x is equal to the derivative of the heat flow at that point, negated. The energy equation is also needed, which is: {\partial (\rho E)\over\partial t} + abla.[v(\rho E + p)] = abla.[k_{eff} abla T- \Sigma_j h_j J_j +(\tau.v)]. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. The heat equation is a consequence of Fourier's law of conduction (see heat conduction). Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. It appears in both United Kingdom and Irish methodologies. ==Calculation== The calculation of the heat loss due to linear thermal bridging is relatively simple, given by the formula below: :H_{TB} = y \sum A_{exp} In the formula, y = 0.08 if Accredited Construction details used, and y = 0.15 otherwise, and \sum A_{exp} is the sum of all the exposed areas of the building envelope, ==References== Category:Energy economics Category:Thermodynamic properties This results in velocity profiles and temperature profiles for various surfaces and this knowledge can be used to design the fin. == References == Category:Unit operations Category:Transport phenomena Category:Heat transfer Thus the rate of heat flow into V is also given by the surface integral q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS where n(x) is the outward pointing normal vector at x. When solved as a scalar equation, it can be used to calculate the temperatures at the fin and cylinder surfaces, by reducing to: abla^2 T + {\overset{.}{q}\over k} = {1 \over \alpha} {\partial T\over\partial t} Where: q = internal heat generation = 0 (in this case). * The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x: \kappa = 1/(\rho c_p).

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While you are operating a Rotor (a large, vertical, rotating cylinder found in amusement parks), you spot a passenger in acute distress and decrease the angular velocity of the cylinder from $3.40 \mathrm{rad} / \mathrm{s}$ to $2.00 \mathrm{rad} / \mathrm{s}$ in $20.0 \mathrm{rev}$, at constant angular acceleration. (The passenger is obviously more of a "translation person" than a "rotation person.") What is the constant angular acceleration during this decrease in angular speed?

In physics, angular acceleration refers to the time rate of change of angular velocity. Therefore, the instantaneous angular acceleration α of the particle is given by : \alpha = \frac{d}{dt} \left(\frac{v_{\perp}}{r}\right). The angular velocity satisfies equations of motion known as Euler's equations (with zero applied torque, since by assumption the rotor is in field-free space). The angular velocity ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as: : \omega = \frac{d\phi}{dt} = \frac{v_\perp}{r}. Spin angular acceleration refers to the angular acceleration of a rigid body about its centre of rotation, and orbital angular acceleration refers to the angular acceleration of a point particle about a fixed origin. The vector \boldsymbol{\omega} = (\omega_x, \omega_y, \omega_z) on the left hand side contains the components of the angular velocity of the rotor expressed with respect to the body-fixed frame. Angular acceleration is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (α). Thus the formula for Advance ratio is \mu = \frac {V}{u} = \frac {V}{\Omega\cdot R} where Omega (Ω) is the rotor's angular velocity, and R is the rotor radius (about the length of one rotor blade)Jackson, Dave. The SI unit of angular velocity is radians per second, Extract of page 27 with the radian being a dimensionless quantity, thus the SI units of angular velocity may be listed as s−1. The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The same equations for the angular speed can be obtained reasoning over a rotating rigid body. Therefore, the orbital angular acceleration is the vector \boldsymbol\alpha defined by : \boldsymbol\alpha = \frac{d}{dt} \left(\frac{\mathbf r \times \mathbf v}{r^2}\right). Angular acceleration then may be termed a pseudoscalar, a numerical quantity which changes sign under a parity inversion, such as inverting one axis or switching the two axes. === Particle in three dimensions === In three dimensions, the orbital angular acceleration is the rate at which three-dimensional orbital angular velocity vector changes with time. : Quantity correspondence Angular frequency \omega Frequency u = \omega/{2\pi} 2π rad/s 1 Hz 1 rad/s ≈ 0.159155 Hz 1 rad/s ≈ 9.5493 rpm 0.1047 rad/s ≈ 1 rpm == Coherent units == A use of the unit radian per second is in calculation of the power transmitted by a shaft. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceleration, called spin angular acceleration and orbital angular acceleration respectively. The radian per second is defined as the angular frequency that results in the angular displacement increasing by one radian every second. * Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotation and is independent of the choice of origin, in contrast to orbital angular velocity. 300px|right|thumb|Rotation of a rigid body P about a fixed axis O. Angular displacement of a body is the angle (in radians, degrees or turns) through which a point revolves around a centre or a specified axis in a specified sense. The instantaneous angular velocity ω at any point in time is given by : \omega = \frac{v_{\perp}}{r}, where r is the distance from the origin and v_{\perp} is the cross-radial component of the instantaneous velocity (i.e. the component perpendicular to the position vector), which by convention is positive for counter-clockwise motion and negative for clockwise motion. Angular velocity is usually represented by the symbol omega (, sometimes ). In two dimensions, angular acceleration is a number with plus or minus sign indicating orientation, but not pointing in a direction. For rigid bodies, angular acceleration must be caused by a net external torque.

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A living room has floor dimensions of $3.5 \mathrm{~m}$ and $4.2 \mathrm{~m}$ and a height of $2.4 \mathrm{~m}$. What does the air in the room weigh when the air pressure is $1.0 \mathrm{~atm}$ ?

This can be seen by using the ideal gas law as an approximation. ==Dry air== The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure: \begin{align} \rho &= \frac{p}{R_\text{specific} T}\\\ R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\\ \rho &= \frac{pM}{RT} = \frac{pm}{k_{\rm B}T}\\\ \end{align} where: *\rho, air density (kg/m3)In the SI unit system. Air returns from the room at ceiling level or the maximum allowable height above the occupied zone. Depending on the measuring instruments used, different sets of equations for the calculation of the density of air can be applied. The partial pressure of dry air p_\text{d} is found considering partial pressure, resulting in: p_\text{d} = p - p_\text{v} where p simply denotes the observed absolute pressure. ==Variation with altitude== thumb|upright=2.0|Standard atmosphere: , , ===Troposphere=== To calculate the density of air as a function of altitude, one requires additional parameters. At 101.325kPa (abs) and , air has a density of approximately , which is about that of water, according to the International Standard Atmosphere (ISA). * At 70°F and 14.696psi, dry air has a density of 0.074887lb/ft3. Air density is a property used in many branches of science, engineering, and industry, including aeronautics;Olson, Wayne M. (2000) AFFTC-TIH-99-01, Aircraft Performance FlightICAO, Manual of the ICAO Standard Atmosphere (extended to 80 kilometres (262 500 feet)), Doc 7488-CD, Third Edition, 1993, .Grigorie, T.L., Dinca, L., Corcau J-I. and Grigorie, O. (2010) Aircrafts' Altitude Measurement Using Pressure Information:Barometric Altitude and Density Altitude gravimetric analysis;A., Picard, R.S., Davis, M., Gläser and K., Fujii (CIPM-2007) Revised formula for the density of moist air the air-conditioningS. * At 20°C and 101.325kPa, dry air has a density of 1.2041 kg/m3. The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa: ==Humid air== thumb|right|400px|Effect of temperature and relative humidity on air density The addition of water vapor to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. thumb|350px|alt=Diagram of underfloor air distribution showing cool, fresh air moving through the underfloor plenum and supplied via floor diffusers and desktop vents. Therefore: * At IUPAC standard temperature and pressure (0°C and 100kPa), dry air has a density of approximately 1.2754kg/m3. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. At 101.325 kPa (abs) and 20 °C (68 °F), air has a density of approximately , according to the International Standard Atmosphere (ISA). thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. Both the pressure and density obey this law, so, denoting the height of the border between the troposphere and the tropopause as U: \begin{align} p &= p(U) e^{-\frac{h - U}{H_\text{TP}}} = p_0 \left(1 - \frac{L U}{T_0}\right)^\frac{g M}{R L} e^{-\frac{h - U}{H_\text{TP}}} \\\ \rho &= \rho(U) e^{-\frac{h - U}{H_\text{TP}}} = \rho_0 \left(1 - \frac{L U}{T_0}\right)^{\frac{g M}{R L} - 1} e^{-\frac{h - U}{H_\text{TP}}} \end{align} ==Composition== ==See also== *Air *Atmospheric drag *Lighter than air *Density *Atmosphere of Earth *International Standard Atmosphere *U.S. Standard Atmosphere *NRLMSISE-00 ==Notes== ==References== ==External links== *Conversions of density units ρ by Sengpielaudio *Air density and density altitude calculations and by Richard Shelquist *Air density calculations by Sengpielaudio (section under Speed of sound in humid air) *Air density calculator by Engineering design encyclopedia *Atmospheric pressure calculator by wolfdynamics *Air iTools - Air density calculator for mobile by JSyA *Revised formula for the density of moist air (CIPM-2007) by NIST Category:Atmospheric thermodynamics A The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Also, the investigation of energy saving has shown that this amount varies for buildings located in different climates, suggesting further studies should investigate this factor prior to designing a suitable HVAC system. ==Applications== Underfloor air distribution is frequently used in office buildings, particularly highly-reconfigurable and open plan offices where raised floors are desirable for cable management. Room air distribution is characterizing how air is introduced to, flows through, and is removed from spaces.Fundamentals volume of the ASHRAE Handbook, Atlanta, GA, USA, 2005 HVAC airflow in spaces generally can be classified by two different types: mixing (or dilution) and displacement. ==Mixing systems== Mixing systems generally supply air such that the supply air mixes with the room air so that the mixed air is at the room design temperature and humidity. Displacement room airflow presents an opportunity to improve both the thermal comfort and indoor air quality (IAQ) of the occupied space. In architecture, construction, and real estate, floor area, floor space, or floorspace is the area (measured as square feet or square metres) taken up by a building or part of it. At a certain plane in the room, the airflow rate returned to the UZ is equal to the supply air. thumb|upright=1.25|Different air masses which affect North America as well as other continents, tend to be separated by frontal boundaries In meteorology, an air mass is a volume of air defined by its temperature and humidity.

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An astronaut whose height $h$ is $1.70 \mathrm{~m}$ floats "feet down" in an orbiting space shuttle at distance $r=6.77 \times 10^6 \mathrm{~m}$ away from the center of Earth. What is the difference between the gravitational acceleration at her feet and at her head?

It is calculated as the distance between the centre of gravity of a ship and its metacentre. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from the rotation of the Earth (but the latter is small enough to be negligible for most purposes); the total (the apparent gravity) is about 0.5% greater at the poles than at the Equator."Curious About Astronomy?", Cornell University, retrieved June 2007 Although the symbol is sometimes used for standard gravity, (without a suffix) can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity). Acceleration due to gravity, acceleration of gravity or gravity acceleration may refer to: *Gravitational acceleration, the acceleration caused by the gravitational attraction of massive bodies in general *Gravity of Earth, the acceleration caused by the combination of gravitational attraction and centrifugal force of the Earth *Standard gravity, or g, the standard value of gravitational acceleration at sea level on Earth ==See also== *g-force, the acceleration of a body relative to free-fall The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by or , is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. The metacentric height (GM) is a measurement of the initial static stability of a floating body. As the ship heels over, the centre of gravity generally remains fixed with respect to the ship because it just depends on the position of the ship's weight and cargo, but the surface area increases, increasing BMφ. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. Hence, a sufficiently, but not excessively, high metacentric height is considered ideal for passenger ships. ==Metacentre== When a ship heels (rolls sideways), the centre of buoyancy of the ship moves laterally. Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level (assumed zero potential) that represents the work done by lifting one unit mass one unit distance through a region in which the acceleration of gravity is uniformly 9.80665 m/s2. thumb|upright=1.8|A simplified spacecraft system. thumb|upright=1.6|Ship stability diagram showing centre of gravity (G), centre of buoyancy (B), and metacentre (M) with ship upright and heeled over to one side. * Cardiovascular events and changes occurring during spaceflight: these are due to body fluids shift and redistribution, heart rhythm disturbances and decrease in maximal exercise capacity in the micro gravity environment. Geopotential height may be obtained from normalizing geopotential by the acceleration of gravity: :{H} = \frac{\Phi}{g_{0}}\ = \frac{1}{g_{0}}\int_0^Z\ g(\phi,Z)\,dZ where g_0 = 9.80665 m/s2, the standard gravity at mean sea level. This position meant that a person's legs experienced only one sixth of their weight, which was the equivalent of being on the lunar surface. The value of defined above is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. thumb|250px|A test subject being suited up for studies on the Reduced Gravity Walking Simulator. This illusion will alter the astronaut's perception of the orienting force of gravity and then lose spatial direction. "G", is the center of gravity. While objects are weightless in space, an astronaut has to be familiar with an object's forces of inertia and understand how the object will respond to simple motions to avoid losing it in space. Geopotential height (altitude) differs from geometric (tapeline) height but remains a historical convention in aeronautics as the altitude used for calibration of aircraft barometric altimeters. ==Definition== Geopotential is the gravitational potential energy per unit mass at elevation Z: :\Phi(Z) = \int_0^Z\ g(\phi,Z)\,dZ where g(\phi,Z) is the acceleration due to gravity, \phi is latitude, and Z is the geometric elevation. Because of this any future medical criteria for commercial spaceflight participants needs to focus specifically on the detrimental effects of rapidly changing gravitational levels, and which individuals will be capable of tolerating this. The centre of gravity of the ship is commonly denoted as point G or CG.

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If the particles in a system all move together, the com moves with them-no trouble there. But what happens when they move in different directions with different accelerations? Here is an example. The three particles in Figure are initially at rest. Each experiences an external force due to bodies outside the three-particle system. The directions are indicated, and the magnitudes are $F_1=6.0 \mathrm{~N}, F_2=12 \mathrm{~N}$, and $F_3=14 \mathrm{~N}$. What is the acceleration of the center of mass of the system?

And any system of particles that move under Newtonian gravitation as if they are a rigid body must do so in a central configuration. In Euler's three-body problem we assume that the two centres of attraction are stationary. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem. In celestial mechanics and the mathematics of the -body problem, a central configuration is a system of point masses with the property that each mass is pulled by the combined gravitational force of the system directly towards the center of mass, with acceleration proportional to its distance from the center. The central-force problem concerns an ideal situation (a "one-body problem") in which a single particle is attracted or repelled from an immovable point O, the center of force.Goldstein, p. 71; Landau and Lifshitz, p. 30; Whittaker, p. Thus, the equation of motion for r can be written in the form \mu \ddot{\mathbf{r}} = \mathbf{F} where \mu is the reduced mass \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} = \frac{m_1 m_2}{m_1 + m_2} As a special case, the problem of two bodies interacting by a central force can be reduced to a central-force problem of one body. ==Qualitative properties== ===Planar motion=== thumb|right|alt=The image shows a yellow disc with three vectors. In an extended modern sense, a three- body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles. ==Mathematical description== The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions \mathbf{r_i} = (x_i, y_i, z_i) of three gravitationally interacting bodies with masses m_i: :\begin{align} \ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\\ \ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\\ \ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}. \end{align} where G is the gravitational constant. In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. This is the only central configuration for these masses that does not lie in a lower- dimensional subspace. ==Dynamics== Under Newton's law of universal gravitation, bodies placed at rest in a central configuration will maintain the configuration as they collapse to a collision at their center of mass. An additional mass (which may be zero) is placed at the center of the system. Systems of bodies in a two-dimensional central configuration can orbit stably around their center of mass, maintaining their relative positions, with circular orbits around the center of mass or in elliptical orbits with the center of mass at a focus of the ellipse. Those equations are an accurate description of a particular form of the three-body problem. Therefore, both bodies are accelerated if a force is present between them; there is no perfectly immovable center of force. If either center were absent, the particle would move on one of these ellipses, as a solution of the Kepler problem. Special cases of these generalized problems include Darboux's problemDarboux JG, Archives Néerlandaises des Sciences (ser. 2), 6, 371-376 and Velde's problem.Velde (1889) Programm der ersten Höheren Bürgerschule zu Berlin ==Overview and history== Euler's three-body problem is to describe the motion of a particle under the influence of two centers that attract the particle with central forces that decrease with distance as an inverse-square law, such as Newtonian gravity or Coulomb's law. The simplest generalization of Euler's three-body problem is to add a third center of force midway between the original two centers, that exerts only a linear Hooke force (confer Hooke's law). The simplest bodies or elements of a multibody system were treated by Newton (free particle) and Euler (rigid body). Each multibody system formulation may lead to a different mathematical appearance of the equations of motion while the physics behind is the same. In other words, a central force must act along the line joining O with the present position of the particle. Moreover, the motion of three bodies is generally non-repeating, except in special cases. Magnus, Dynamics of multibody systems, Springer Verlag, Berlin (1978). In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space.

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An asteroid, headed directly toward Earth, has a speed of $12 \mathrm{~km} / \mathrm{s}$ relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed $v_f$ when it reaches Earth's surface.

Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 is an Aten near-Earth asteroid less than 20 meters in diameter crudely estimated to have passed roughly 6500 km above the surface of Earth on 31 March 2004. The estimated 4 to 6 meter sized body made one of the closest known approaches to Earth. == Description == On 31 March 2004, around 15:35 UTC, the asteroid is crudely estimated to have passed within approximately 1 Earth radius () or 6,400 kilometers of the surface of the Earth (or 2.02 from Earth's center). Due to this elongated orbit, the Aten asteroid and near-Earth asteroid also classifies as Earth-crosser, Venus-crosser and Mercury-grazer. 1989 VA was the first asteroid discovered with such a small semi-major axis (0.728 AU, about the same as Venus), breaking 2100 Ra-Shalom's distance record (0.832 AU), which had held for over a decade. This asteroid orbits the Sun with a short orbital period at a distance of 0.3–1.2 AU once every 227 days. is a very eccentric, stony asteroid and near-Earth object, approximately 1 kilometer in diameter. is a sub-kilometer asteroid that orbits near Mars's Lagrangian point, on average trailing 60° behind it. It passed closest approach to Earth on 3 March 2016 05:17 UT at a distance of and was quickly approaching the glare of the Sun thus preventing further optical observations. == 2021 approach == It was recovered on 17 February 2021 by Pan-STARRS when the uncertainty in the asteroid's sky position covered about 1.2° of the sky. is an asteroid and near-Earth object approximately in diameter. Another, larger near-Earth asteroid, 2004 FH passed just two weeks prior to . is a near-Earth asteroid estimated to be roughly in diameter. It remained the asteroid with the smallest known semi-major axis for five years until the discovery of (0.683 AU), which was the first asteroid discovered closer to the Sun than Venus. Due to its eccentric orbit, is also a Mars-crosser, crossing the orbit of the Red Planet at 1.66 AU. == 2016 discovery == It was first observed by the Mount Lemmon Survey on 28 February 2016, when the asteroid was about from Earth and had a solar elongation of 174°. By early February 2021 the asteroid was brighter than apparent magnitude 24, which still placed it near the limiting magnitude of even the best automated astronomical surveys. With an exceptionally high eccentricity of 0.59, it was the most eccentric Aten asteroid known at the time of discovery, more eccentric than previously discovered Aten, 3753 Cruithne. The formerly poorly known trajectory of this asteroid was further complicated by close approaches to Venus and Mercury. It was not until (0.277 AU) was discovered that an Aten asteroid with a lower perihelion was found. 's eccentric orbit takes it out past the Earth, where it has encounters of about 0.15 to 0.20 AU about every 3 to 5 years around October–November. While listed on the Sentry Risk Table, virtual clones of the asteroid that fit the uncertainty in the known trajectory showed 116 potential impacts between 2054 and 2109. On 26 March 2010, it may have come within 0.0825 AU (12.3 million km) of Earth, but with an uncertainty parameter of 9, the orbit is poorly determined. It was the eighth Aten asteroid discovered. The combination of a small semi- major axis and high eccentricity made the first Aten asteroid discovered to get closer to the Sun (0.295 AU) than Mercury ever does. 2340 Hathor (the second Aten discovered, in 1976) had the smallest perihelion (0.464 AU) earlier, which was about the same distance as Mercury's aphelion (0.467 AU). Its orbit has an eccentricity of 0.40 and an inclination of 7° with respect to the ecliptic.

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The huge advantage of using the conservation of energy instead of Newton's laws of motion is that we can jump from the initial state to the final state without considering all the intermediate motion. Here is an example. In Figure, a child of mass $m$ is released from rest at the top of a water slide, at height $h=8.5 \mathrm{~m}$ above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child's speed at the bottom of the slide.

The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Classically, conservation of energy was distinct from conservation of mass. The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. The kinetic energy, as determined by the velocity, is converted to potential energy as it reaches the same height as the initial ball and the cycle repeats. thumb|An idealized Newton's cradle with five balls when there are no energy losses and there is always a small separation between the balls, except for when a pair is colliding thumb|Newton's cradle three-ball swing in a five-ball system. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. Many physicists at that time, such as Newton, held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum: :\sum_{i} m_i v_i was the conserved vis viva. All enforce the conservation of energy and momentum. The caloric theory maintained that heat could neither be created nor destroyed, whereas conservation of energy entails the contrary principle that heat and mechanical work are interchangeable. Thus, conservation of energy (total, including material or rest energy) and conservation of mass (total, not just rest) are one (equivalent) law. The Newton's cradle is a device that demonstrates the conservation of momentum and the conservation of energy with swinging spheres. He showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle. Energy conservation has been a foundational physical principle for about two hundred years. For systems that include large gravitational fields, general relativity has to be taken into account; thus mass–energy conservation becomes a more complex concept, subject to different definitions, and neither mass nor energy is as strictly and simply conserved as is the case in special relativity. == Formulation and examples == The law of conservation of mass can only be formulated in classical mechanics, in which the energy scales associated with an isolated system are much smaller than mc^2, where m is the mass of a typical object in the system, measured in the frame of reference where the object is at rest, and c is the speed of light. Given the stationary- action principle, conservation of energy can be rigorously proven by Noether's theorem as a consequence of continuous time translation symmetry; that is, from the fact that the laws of physics do not change over time. Some say that this behavior demonstrates the conservation of momentum and kinetic energy in elastic collisions. ===Mechanical equivalent of heat=== A key stage in the development of the modern conservation principle was the demonstration of the mechanical equivalent of heat. A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind cannot exist; that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings.Planck, M. (1923/1927). The law of conservation of mass and the analogous law of conservation of energy were finally generalized and unified into the principle of mass–energy equivalence, described by Albert Einstein's famous formula E = mc^2. The energy conservation law is a consequence of the shift symmetry of time; energy conservation is implied by the empirical fact that the laws of physics do not change with time itself. Engineers such as John Smeaton, Peter Ewart, , Gustave-Adolphe Hirn, and Marc Seguin recognized that conservation of momentum alone was not adequate for practical calculation and made use of Leibniz's principle. In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass cannot change, so the quantity can neither be added nor be removed. In reality, the conservation of mass only holds approximately and is considered part of a series of assumptions in classical mechanics.

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Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.

* If one has $1000 invested for 1 year at a 7-day SEC yield of 2%, then: :(0.02 × $1000 ) / 365 ~= $0.05479 per day. * If one has $1000 invested for 30 days at a 7-day SEC yield of 5%, then: :(0.05 × $1000 ) / 365 ~= $0.137 per day. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. :Multiply by 365/7 to give the 7-day SEC yield. It is also referred to as the 7-day Annualized Yield. The calculation is performed as follows: :Take the net interest income earned by the fund over the last 7 days and subtract 7 days of management fees. It is important to note that the 7-day SEC yield is only an estimate of the fund's actual yield, and may not necessarily reflect the yield that an investor would receive if they held the fund for a longer period of time. ==Examples== The examples assume interest is withdrawn as it is earned and not allowed to compound. To calculate approximately how much interest one might earn in a money fund account, take the 7-day SEC yield, multiply by the amount invested, divide by the number of days in the year, and then multiply by the number of days in question. The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. Multiply by 30 days to yield $4.11 in interest. *For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years. The Summa de arithmetica of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The E-M rule could thus be written also as : t \approx \frac{70}{r} \times \frac{198}{200-r} or t \approx \frac{72}{r} \times \frac{192}{200-r} In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate. ===Padé approximant=== The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula: : t \approx \frac{69.3}{r} \times \frac{600+4r}{600+r} which simplifies to: : t \approx \frac{207900+1386r}{3000r+5r^2} ==Derivation== ===Periodic compounding === For periodic compounding, future value is given by: :FV = PV \cdot (1+r)^t where PV is the present value, t is the number of time periods, and r stands for the interest rate per time period. :Divide that dollar amount by the average size of the fund's investments over the same 7 days. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5. == Using the rule to estimate compounding periods == To estimate the number of periods required to double an original investment, divide the most convenient "rule-quantity" by the expected growth rate, expressed as a percentage. Similarly, replacing the "R" in R/200 on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2%. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. The effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum. The 7-day SEC Yield is a measure of performance in the interest rates of money market mutual funds offered by US mutual fund companies. The function f(r) is shown to be accurate in the approximation of t for a small, positive interest rate when r=.08 (see derivation below). f(.08)\approx1.03949, and we therefore approximate time t as: : t=\bigg(\frac{\ln2}{r}\bigg)f(.08) \approx \frac{.72}{r} Written as a percentage: : \frac{.72}{r}=\frac{72}{100r} This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). 100 r is r written as a percentage.

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Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \mathrm{~L}$ of a dye solution with a concentration of $1 \mathrm{~g} / \mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \mathrm{~L} / \mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \%$ of its original value.

Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. If one adds 1 litre of water to this solution, the salt concentration is reduced. Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. D_t=\left [ \frac{V}{Q} \right ] \cdot \ln \left [ \frac{C_\text{initial}}{C_\text{ending}}\right ] Sometimes the equation is also written as: \ln \left [ \frac{C_\text{ending}}{C_\text{initial}}\right ] \quad = {-}\frac{Q}{V} \cdot (t_\text{ending} - t_\text{initial}) where t_\text{initial} = 0 *Dt = time required; the unit of time used is the same as is used for Q *V = air or gas volume of the closed space or room in cubic feet, cubic metres or litres *Q = ventilation rate into or out of the room in cubic feet per minute, cubic metres per hour or litres per second *Cinitial = initial concentration of a vapor inside the room measured in ppm *Cfinal = final reduced concentration of the vapor inside the room in ppm ==Dilution ventilation equation== The basic room purge equation can be used only for purge scenarios. The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). Mathematically this relationship can be shown by equation: c_1 V_1 = c_2 V_2 where *c1 = initial concentration or molarity *V1 = initial volume *c2 = final concentration or molarity *V2 = final volume .... ==Basic room purge equation== The basic room purge equation is used in industrial hygiene. The mean free time for a molecule in a fluid is the average time between collisions. Time of concentration is useful in predicting flow rates that would result from hypothetical storms, which are based on statistically derived return periods through IDF curves.Sherman, C. (1931): Frequency and intensity of excessive rainfall at Boston, Massachusetts, Transactions, American Society of Civil Engineers, 95, 951–960. (pdf) For many (often economic) reasons, it is important for engineers and hydrologists to be able to accurately predict the response of a watershed to a given rain event. The concentration of this admixture should be small and the gradient of this concentration should be also small. Lake retention time (also called the residence time of lake water, or the water age or flushing time) is a calculated quantity expressing the mean time that water (or some dissolved substance) spends in a particular lake. In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). upright=1.35|thumb|Diluting a solution by adding more solvent Dilution is the process of decreasing the concentration of a solute in a solution, usually simply by mixing with more solvent like adding more water to the solution. In a scenario where a liquid continuously evaporates from a container in a ventilated room, a differential equation has to be used: \frac{dC}{dt} = \frac{G - Q' C}{V} where the ventilation rate has been adjusted by a mixing factor K: Q' = \frac{Q}{K} *C = concentration of a gas *G = generation rate *V = room volume *Q′ = adjusted ventilation rate of the volume ==Welding== The dilution in welding terms is defined as the weight of the base metal melted divided by the total weight of the weld metal. The solutions on the left are more dilute, compared to the more concentrated solutions on the right. The equation can only be applied when the purged volume of vapor or gas is replaced with "clean" air or gas. He introduced several mechanisms of diffusion and found rate constants from experimental data.

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A certain vibrating system satisfies the equation $u^{\prime \prime}+\gamma u^{\prime}+u=0$. Find the value of the damping coefficient $\gamma$ for which the quasi period of the damped motion is $50 \%$ greater than the period of the corresponding undamped motion.

For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac{c}{c_c} = \frac {\text{actual damping}} {\text{critical damping}}, where the system's equation of motion is : m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 and the corresponding critical damping coefficient is : c_c = 2 \sqrt{k m} or : c_c = 2 m \sqrt{\frac{k}{m}} = 2m \omega_n where : \omega_n = \sqrt{\frac{k}{m}} is the natural frequency of the system. The damping ratio is a system parameter, denoted by (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped (). The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. The general equation for an exponentially damped sinusoid may be represented as: y(t) = A e^{-\lambda t} \cos(\omega t - \varphi) where: *y(t) is the instantaneous amplitude at time ; *A is the initial amplitude of the envelope; *\lambda is the decay rate, in the reciprocal of the time units of the independent variable ; *\varphi is the phase angle at ; *\omega is the angular frequency. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. == Derivation == Using the natural frequency of a harmonic oscillator \omega_n = \sqrt{{k}/{m}} and the definition of the damping ratio above, we can rewrite this as: : \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. ==Damped sine wave== thumb|350px|Plot of a damped sinusoidal wave represented as the function y(t) = e^{- t} \cos(2 \pi t) A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases. * Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. If the system has n degrees of freedom un and is under application of m damping forces. * Q factor: Q = 1 / (2 \zeta) is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. == Damping ratio definition == thumb|400px|upright=1.3|The effect of varying damping ratio on a second-order system. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). == Q factor and decay rate == The Q factor, damping ratio ζ, and exponential decay rate α are related such that : \zeta = \frac{1}{2 Q} = { \alpha \over \omega_n }. Two such solutions, for the two values of s satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: thumb|Phase portrait of damped oscillator, with increasing damping strength. *Damped harmonic motion, see animation (right). The equation of motion is \ddot x + 2\gamma \dot x + \omega^2 x = 0. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. It corresponds to the underdamped case of damped second-order systems, or underdamped second-order differential equations.Douglas C. Giancoli (2000). Critically damped systems have a damping ratio of exactly 1, or at least very close to it. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Low damping materials may be utilized in musical instruments where sustained mechanical vibration and acoustic wave propagation is desired.

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Find the value of $y_0$ for which the solution of the initial value problem $$ y^{\prime}-y=1+3 \sin t, \quad y(0)=y_0 $$ remains finite as $t \rightarrow \infty$

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. We are trying to find a formula for y(t) that satisfies these two equations. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Another solution is given by : y_s(x) = 0 . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. Starting with \varphi_0(t)=0, we iterate :\varphi_{k+1}(t)=\int_0^t (1+(\varphi_k(s))^2)\,ds so that \varphi_n(t) \to y(t): :\varphi_1(t)=\int_0^t (1+0^2)\,ds = t :\varphi_2(t)=\int_0^t (1+s^2)\,ds = t + \frac{t^3}{3} :\varphi_3(t)=\int_0^t \left(1+\left(s + \frac{s^3}{3}\right)^2\right)\,ds = t + \frac{t^3}{3} + \frac{2t^5}{15} + \frac{t^7}{63} and so on. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem.

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A certain spring-mass system satisfies the initial value problem $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0 $$ where $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter. Suppose $k=2$. Find the time $\tau$ after which $|u(t)|<0.1$ for all $t>\tau$.

The modified KdV–Burgers equation is a nonlinear partial differential equationAndrei D. Polyanin, Valentin F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, second edition, p 1041 CRC PRESS :u_t+u_{xxx}-\alpha u^2\,u_x - \beta u_{xx}=0. ==See also== *Burgers' equation *Korteweg–de Vries equation *modified KdV equation ==References== #Graham W. Griffiths William E. Shiesser Traveling Wave Analysis of Partial Differential Equations Academy Press # Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 #Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation :u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x = \frac16 (6u_{xxxx}+60uu_{xx}+45u_x^2+40u^3)_x. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. Unnormalized KdV equation is a nonlinear partial differential equationAndrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS u_{t}+\alpha*u_{xxx}+\beta*u*u_{x}=0 ==References== #Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press # Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 #Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. The case f(u) = 3u2 is the original Korteweg–De Vries equation. ==References== * Category:Partial differential equations The term for 0 < k < u, k even, may be simplified using the properties of the gamma function to :\operatorname E(T^k)= u^{\frac{k}{2}} \, \prod_{i=1}^{k/2} \frac{2i-1}{ u - 2i} \qquad k\text{ even},\quad 0 For a t-distribution with u degrees of freedom, the expected value is 0 if u>1, and its variance is \frac{ u}{ u-2} if u>2. Also, \omega(u)-e^{-\gamma} oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. In mathematics, a generalized Korteweg–De Vries equation is the nonlinear partial differential equation :\partial_t u + \partial_x^3 u + \partial_x f(u) = 0.\, The function f is sometimes taken to be f(u) = uk+1/(k+1) + u for some positive integer k (where the extra u is a "drift term" that makes the analysis a little easier). This may also be written as :f(t) = \frac{1}{\sqrt{ u}\,\mathrm{B} (\frac{1}{2}, \frac{ u}{2})} \left(1+\frac{t^2} u \right)^{-( u+1)/2}, where B is the Beta function. For t > 0, :F(t) = \int_{-\infty}^t f(u)\,du = 1 - \tfrac{1}{2} I_{x(t)}\left(\tfrac{ u}{2}, \tfrac{1}{2}\right), where :x(t) = \frac{ u}. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). Here a, b, and k are parameters. An alternative formula, valid for t^2 < u, is :\int_{-\infty}^t f(u)\,du = \tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}( u+1) \right)} {\sqrt{\pi u}\,\Gamma \left(\tfrac{ u}{2}\right)} \, {}_2F_1 \left( \tfrac{1}{2}, \tfrac{1}{2}( u+1); \tfrac{3}{2}; -\tfrac{t^2}{ u} \right), where 2F1 is a particular case of the hypergeometric function. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. 300px|thumbnail|Graph of the Buchstab function ω(u) from u = 1 to u = 4\. The Buchstab function (or Buchstab's function) is the unique continuous function \omega: \R_{\ge 1}\rightarrow \R_{>0} defined by the delay differential equation :\omega(u)=\frac 1 u, \qquad\qquad\qquad 1\le u\le 2, :{\frac{d}{du}} (u\omega(u))=\omega(u-1), \qquad u\ge 2. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. #Eryk Infeld and George Rowlands, Nonlinear Waves, Solitons and Chaos,Cambridge 2000 #Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 #Dongming Wang, Elimination Practice, Imperial College Press 2004 # David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 # George Articolo Partial Differential Equations and Boundary Value Problems with Maple V Academic Press 1998 Category:Nonlinear partial differential equations #Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000 #Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 #Dongming Wang, Elimination Practice,Imperial College Press 2004 # David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 # George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 Category:Nonlinear partial differential equations It is the first equation in a hierarchy of integrable equations with the Lax operator : \partial_x^3 + 2u\partial_x + u_x, . The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.p. 131, Cheer and Goldston 1990. ==Applications== The Buchstab function is used to count rough numbers.

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Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously. Determine $T$ if $r=7 \%$.

The function f(r) is shown to be accurate in the approximation of t for a small, positive interest rate when r=.08 (see derivation below). f(.08)\approx1.03949, and we therefore approximate time t as: : t=\bigg(\frac{\ln2}{r}\bigg)f(.08) \approx \frac{.72}{r} Written as a percentage: : \frac{.72}{r}=\frac{72}{100r} This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). 100 r is r written as a percentage. *For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. The E-M rule could thus be written also as : t \approx \frac{70}{r} \times \frac{198}{200-r} or t \approx \frac{72}{r} \times \frac{192}{200-r} In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate. ===Padé approximant=== The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula: : t \approx \frac{69.3}{r} \times \frac{600+4r}{600+r} which simplifies to: : t \approx \frac{207900+1386r}{3000r+5r^2} ==Derivation== ===Periodic compounding === For periodic compounding, future value is given by: :FV = PV \cdot (1+r)^t where PV is the present value, t is the number of time periods, and r stands for the interest rate per time period. Similarly, replacing the "R" in R/200 on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2%. * P = principal deposit * r = rate of return (monthly) * M = monthly deposit, and * t = time, in months The compound interest for each deposit is: M'=M(1+r)^{t} and adding all recurring deposits over the total period t (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month) : M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}} recognizing the geometric series: M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}} and applying the closed-form formula (common ratio :1/(1+r)) we obtain: P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as \text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y} where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. Let RSt be the simple rate of return on the security from t − 1 to t. Accumulation functions for simple and compound interest are a(t)=1 + r t a(t) = \left(1 + \frac {r} {n}\right) ^ {nt} If n t = 1, then these two functions are the same. ===Continuous compounding=== As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The continuously compounded rate of return or instantaneous rate of return RCt obtained during that period is : RC_{t}=\ln\left (\frac{P_{t}}{P_{t-1}}\right ). For continuous compounding, 69 gives accurate results for any rate, since ln(2) is about 69.3%; see derivation below. A practical estimate for reverse calculation of the rate of return when the exact date and amount of each recurring deposit is not known, a formula that assumes a uniform recurring monthly deposit over the period, is:http://moneychimp.com/features/portfolio_performance_calculator.htm "recommended by The Four Pillars of Investing and The Motley Fool" r=\left(\frac{P'-P-\sum{M}}{P+\sum{M}/2}\right)^{1/t} or r=\left(\frac{P'-\sum{M}/2}{P+\sum{M}/2}\right)^{1/t}-1 ==See also== * Credit card interest * Exponential growth * Fisher equation * Interest * Interest rate * Rate of return * Rate of return on investment * Real versus nominal value (economics) * Yield curve ==References== Category:Interest Category:Exponentials Category:Mathematical finance Category:Actuarial science it:Anatocismo When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5. == Using the rule to estimate compounding periods == To estimate the number of periods required to double an original investment, divide the most convenient "rule-quantity" by the expected growth rate, expressed as a percentage. Compound interest is standard in finance and economics. For every three percentage points away from 8%, the value of 72 could be adjusted by 1: : t \approx \frac{72 + (r - 8)/3}{r} or, for the same result: : t \approx \frac{70 + (r - 2)/3}{r} Both of these equations simplify to: : t \approx \frac{208}{3r} + \frac{1}{3} Note that \frac{208}{3} is quite close to 69.3. ===E-M rule=== The Eckart–McHale second- order rule (the E-M rule) provides a multiplicative correction for the rule of 69.3 that is very accurate for rates from 0% to 20%, whereas the rule is normally only accurate at the lowest end of interest rates, from 0% to about 5%. Thus, continuing the above nominal example, the final value of the investment expressed in real terms is :P_t^{real} = P_t \cdot \frac{PL_{t-1}}{PL_t}. For periodic compounding, the exact doubling time for an interest rate of r percent per period is :t = \frac{\ln(2)}{\ln(1+r/100)}\approx \frac{72}{r}, where t is the number of periods required.

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A mass weighing $2 \mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, determine the position $u$ of the mass at any time $t$. Find the frequency of the motion.

The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. Since the inertia of the beam can be found from its mass, the spring constant can be calculated. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. In the case of =1, the considered problem has a closed solution: y(\tau )=\left[\frac{4}{3}\tau (1-\tau) -\frac{4}{3}\tau \left( 1+2 \tau\ln (1-\tau )+2\ln (1-\tau )\right)\right]\ . ==References== Category:Mechanical vibrations thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. We assume dimensionless displacements of the string and dimensionless time : thumb|240px|Massless string and a moving mass - mass trajectory. : y(\tau)=\frac{w(vt,t)}{w_{st}}\ ,\ \ \ \ \tau\ =\ \frac{vt}{l}\ , where st is the static deflection in the middle of the string. More complex problems can be solved by the finite element method or space-time finite element method. massless load inertial load thumb|321px|Vibrations of a string under a moving massless force (v=0.1c); c is the wave speed. thumb|321px|Vibrations of a string under a moving massless force (v=0.5c); c is the wave speed. thumb|321px|Vibrations of a string under a moving inertial force (v=0.1c); c is the wave speed. thumb|321px|Vibrations of a string under a moving inertial force (v=0.5c); c is the wave speed. It has been the basis of all the most significant > experiments on gravitation ever since. ==Torsional harmonic oscillators== Definition of terms Term Unit Definition \theta\, rad Angle of deflection from rest position I\, kg m2 Moment of inertia C\, joule s rad−1 Angular damping constant \kappa\, N m rad−1 Torsion spring constant \tau\, \mathrm{N\,m}\, Drive torque f_n\, Hz Undamped (or natural) resonant frequency T_n\, s Undamped (or natural) period of oscillation \omega_n\, \mathrm{rad\,s^{-1}}\, Undamped resonant frequency in radians f\, Hz Damped resonant frequency \omega\, \mathrm{rad\,s^{-1}}\, Damped resonant frequency in radians \alpha\, \mathrm{s^{-1}}\, Reciprocal of damping time constant \phi\, rad Phase angle of oscillation L\, m Distance from axis to where force is applied Torsion balances, torsion pendulums and balance wheels are examples of torsional harmonic oscillators that can oscillate with a rotational motion about the axis of the torsion spring, clockwise and counterclockwise, in harmonic motion. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. If the damping is low, this can be obtained by measuring the natural resonant frequency of the balance, since the moment of inertia of the balance can usually be calculated from its geometry, so: :\kappa = (2\pi f_n)^2 I\, In measuring instruments, such as the D'Arsonval ammeter movement, it is often desired that the oscillatory motion die out quickly so the steady state result can be read off. *The torsion pendulum used in torsion pendulum clocks is a wheel-shaped weight suspended from its center by a wire torsion spring. Inertial load in numerical models is described in Unexpected property of differential equations that govern the motion of the mass particle travelling on the string, Timoshenko beam, and Mindlin plate is described in. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. In the case of inertial moving load, the analytical solutions are unknown. Made in the USA of cherry wood and power coated steel, it simulates a walking motion, using see-saw oscillations from 0-10mm, with a frequency of 0 to 15.5 Hz. The equation of motion is increased by the term related to the inertia of the moving load. We will use the word "torsion" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. The kinetic energy is given by: :T=\frac{1}{2}mv^2 where v is the velocity of the mass.

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If $\mathbf{x}=\left(\begin{array}{c}2 \\ 3 i \\ 1-i\end{array}\right)$ and $\mathbf{y}=\left(\begin{array}{c}-1+i \\ 2 \\ 3-i\end{array}\right)$, find $(\mathbf{y}, \mathbf{y})$.

More precisely, given two sets of variables represented as coordinate vectors and y, then each equation of the system can be written y^TA_ix=g_i, where, is an integer whose value ranges from 1 to the number of equations, each A_i is a matrix, and each g_i is a real number. There are several possible ways to compute these quantities for a given implicit curve. XHJTA-FM is a radio station on 94.3 FM in Irapuato, Guanajuato. 300px|thumb|Cassini ovals: (1) a=1.1, c=1 (above), (2) a=c=1 (middle), (3) a=1, c=1.05 (below) 300px|thumb|Implicit curve: \sin(x+y)-\cos(xy)+1=0 In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa. is a passenger railway station located in the city of Himeji, Hyōgo Prefecture, Japan, operated by West Japan Railway Company (JR West). ==Lines== Yobe Station is served by the Kishin Line, and is located 6.1 kilometers from the terminus of the line at . ==Station layout== The station consists of two ground-level opposed side platforms connected by a level crossing. Finally, we calculate the value c via linear interpolation of c_{0} and c_{1} In practice, a trilinear interpolation is identical to two bilinear interpolation combined with a linear interpolation: :c \approx l\left( b(c_{000}, c_{010}, c_{100}, c_{110}),\, b(c_{001}, c_{011}, c_{101}, c_{111})\right) ===Alternative algorithm=== An alternative way to write the solution to the interpolation problem is :f(x, y, z) \approx a_0 + a_1 x + a_2 y + a_3 z + a_4 x y + a_5 x z + a_6 y z + a_7 x y z where the coefficients are found by solving the linear system :\begin{align} \begin{bmatrix} 1 & x_0 & y_0 & z_0 & x_0 y_0 & x_0 z_0 & y_0 z_0 & x_0 y_0 z_0 \\\ 1 & x_1 & y_0 & z_0 & x_1 y_0 & x_1 z_0 & y_0 z_0 & x_1 y_0 z_0 \\\ 1 & x_0 & y_1 & z_0 & x_0 y_1 & x_0 z_0 & y_1 z_0 & x_0 y_1 z_0 \\\ 1 & x_1 & y_1 & z_0 & x_1 y_1 & x_1 z_0 & y_1 z_0 & x_1 y_1 z_0 \\\ 1 & x_0 & y_0 & z_1 & x_0 y_0 & x_0 z_1 & y_0 z_1 & x_0 y_0 z_1 \\\ 1 & x_1 & y_0 & z_1 & x_1 y_0 & x_1 z_1 & y_0 z_1 & x_1 y_0 z_1 \\\ 1 & x_0 & y_1 & z_1 & x_0 y_1 & x_0 z_1 & y_1 z_1 & x_0 y_1 z_1 \\\ 1 & x_1 & y_1 & z_1 & x_1 y_1 & x_1 z_1 & y_1 z_1 & x_1 y_1 z_1 \end{bmatrix}\begin{bmatrix} a_0 \\\ a_1 \\\ a_2 \\\ a_3 \\\ a_4 \\\ a_5 \\\ a_6 \\\ a_7 \end{bmatrix} = \begin{bmatrix} c_{000} \\\ c_{100} \\\ c_{010} \\\ c_{110} \\\ c_{001} \\\ c_{101} \\\ c_{011} \\\ c_{111} \end{bmatrix}, \end{align} yielding the result :\begin{align} a_0 ={} &\frac{-c_{000} x_1 y_1 z_1 + c_{001} x_1 y_1 z_0 + c_{010} x_1 y_0 z_1 - c_{011} x_1 y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{ c_{100} x_0 y_1 z_1 - c_{101} x_0 y_1 z_0 - c_{110} x_0 y_0 z_1 + c_{111} x_0 y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_1 ={} &\frac{ c_{000} y_1 z_1 - c_{001} y_1 z_0 - c_{010} y_0 z_1 + c_{011} y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} y_1 z_1 + c_{101} y_1 z_0 + c_{110} y_0 z_1 - c_{111} y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_2 ={} &\frac{ c_{000} x_1 z_1 - c_{001} x_1 z_0 - c_{010} x_1 z_1 + c_{011} x_1 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} x_0 z_1 + c_{101} x_0 z_0 + c_{110} x_0 z_1 - c_{111} x_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_3 ={} &\frac{ c_{000} x_1 y_1 - c_{001} x_1 y_1 - c_{010} x_1 y_0 + c_{011} x_1 y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} x_0 y_1 + c_{101} x_0 y_1 + c_{110} x_0 y_0 - c_{111} x_0 y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_4 ={} &\frac{-c_{000} z_1 + c_{001} z_0 + c_{010} z_1 - c_{011} z_0 + c_{100} z_1 - c_{101} z_0 - c_{110} z_1 + c_{111} z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_5 = &\frac{-c_{000} y_1 + c_{001} y_1 + c_{010} y_0 - c_{011} y_0 + c_{100} y_1 - c_{101} y_1 - c_{110} y_0 + c_{111} y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_6 ={} &\frac{-c_{000} x_1 + c_{001} x_1 + c_{010} x_1 - c_{011} x_1 + c_{100} x_0 - c_{101} x_0 - c_{110} x_0 + c_{111} x_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_7 ={} &\frac{ c_{000} - c_{001} - c_{010} + c_{011} - c_{100} + c_{101} + c_{110} - c_{111}}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}. \end{align} ==See also== * Linear interpolation * Bilinear interpolation * Tricubic interpolation * Radial interpolation * Tetrahedral interpolation * Spherical Linear Interpolation ==External links== *pseudo-code from NASA, describes an iterative inverse trilinear interpolation (given the vertices and the value of C find Xd, Yd and Zd). Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. 'Systems of bilinear equations'. 250px|thumb|right|Scalar multiplication of a vector by a factor of 3 stretches the vector out. 250px|thumb|right|The scalar multiplications −a and 2a of a vector a In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). Vector \mathbf t(x_0,y_0,z_0) is a tangent vector of the curve at point (x_0,y_0,z_0). 300px|thumb|Intersection curve between a sphere and a cylinder Examples: (1)\quad x+y+z-1=0 \ ,\ x-y+z-2=0 ::is a line. (2)\quad x^2+y^2+z^2-4=0 \ , \ x+y+z-1=0 ::is a plane section of a sphere, hence a circle. (3)\quad x^2+y^2-1=0 \ , \ x+y+z-1=0 ::is an ellipse (plane section of a cylinder). (4)\quad x^2+y^2+z^2-16=0 \ , \ (y-y_0)^2+z^2-9=0 ::is the intersection curve between a sphere and a cylinder. One method is to use implicit differentiation to compute the derivatives of y with respect to x. For a real scalar and matrix: : \lambda = 2, \quad \mathbf{A} =\begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix} : 2 \mathbf{A} = 2 \begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix} = \begin{pmatrix} 2 \\!\cdot\\! a & 2 \\!\cdot\\! b \\\ 2 \\!\cdot\\! c & 2 \\!\cdot\\! d \\\ \end{pmatrix} = \begin{pmatrix} a \\!\cdot\\! 2 & b \\!\cdot\\! 2 \\\ c \\!\cdot\\! 2 & d \\!\cdot\\! 2 \\\ \end{pmatrix} = \begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix}2= \mathbf{A}2. For quaternion scalars and matrices: : \lambda = i, \quad \mathbf{A} = \begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix} : i\begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix} = \begin{pmatrix} i^2 & 0 \\\ 0 & ij \\\ \end{pmatrix} = \begin{pmatrix} -1 & 0 \\\ 0 & k \\\ \end{pmatrix} e \begin{pmatrix} -1 & 0 \\\ 0 & -k \\\ \end{pmatrix} = \begin{pmatrix} i^2 & 0 \\\ 0 & ji \\\ \end{pmatrix} = \begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix}i\,, where are the quaternion units. The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along x, then along y, and finally along z, produces the same value. ) :(P3) until the distance between the points (x_{j+1},y_{j+1}),\, (x_j,y_j) is small enough. Next, we perform linear interpolation between c_{000} and c_{100} to find c_{00}, c_{001} and c_{101} to find c_{01}, c_{011} and c_{111} to find c_{11}, c_{010} and c_{110} to find c_{10}. First we interpolate along x (imagine we are "pushing" the face of the cube defined by C_{0jk} to the opposing face, defined by C_{1jk}), giving: : \begin{align} c_{00} &= c_{000} (1 - x_\text{d}) + c_{100} x_\text{d} \\\ c_{01} &= c_{001} (1 - x_\text{d}) + c_{101} x_\text{d} \\\ c_{10} &= c_{010} (1 - x_\text{d}) + c_{110} x_\text{d} \\\ c_{11} &= c_{011} (1 - x_\text{d}) + c_{111} x_\text{d} \end{align} Where c_{000} means the function value of (x_0, y_0, z_0). There are several ways to arrive at trilinear interpolation, which is equivalent to 3-dimensional tensor B-spline interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators. ==Method== right|thumb|Eight corner points on a cube surrounding the interpolation point C right|thumb|Depiction of 3D interpolation thumb|A geometric visualisation of trilinear interpolation. Contains a very clever and simple method to find trilinear interpolation that is based on binary logic and can be extended to any dimension (Tetralinear, Pentalinear, ...). A normal vector to the curve at the point is given by : \mathbf{n}(x_0,y_0) = (F_x(x_0,y_0), F_y(x_0,y_0)) (here written as a row vector). === Curvature === For readability of the formulas, the arguments (x_0,y_0) are omitted.

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15. Consider the initial value problem $$ 4 y^{\prime \prime}+12 y^{\prime}+9 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-4 . $$ Determine where the solution has the value zero.

Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. Furthermore, for a given x ot=0, this is the unique solution going through (x,y(x)). ==Failure of uniqueness== Consider the differential equation : y'(x)^2 = 4y(x) . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. If p' = 0 it means that y' = p = c = constant, and the general solution of this new equation is: : y_c(x) = c \cdot x + c^2 where c is determined by the initial value. Hence, : y_s(x) = -\tfrac{1}{4} \cdot x^2 \,\\! is tangent to every member of the one- parameter family of solutions : y_c(x) = c \cdot x + c^2 \,\\! of this Clairaut equation: : y(x) = x \cdot y' + (y')^2. \,\\! ==See also== * Chandrasekhar equation * Chrystal's equation * Caustic (mathematics) * Envelope (mathematics) * Initial value problem * Picard–Lindelöf theorem ==Bibliography== * Category:Differential equations Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Another solution is given by : y_s(x) = 0 . Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Now we shall check when these solutions are singular solutions. Uniqueness fails for these solutions on the interval c_1\leq x\leq c_2, and the solutions are singular, in the sense that the second derivative fails to exist, at x=c_1 and x=c_2. ==Further example of failure of uniqueness== The previous example might give the erroneous impression that failure of uniqueness is directly related to y(x)=0. The general solution to this equation is : y(x)= C x^{-2} . If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. # Check that the solution is consistent with step 2.

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A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?

If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. So the present value of the drawdowns is equal to the present value of the repayments, given the APR as the interest rate. For example, consider a 30-year loan of $200,000 with a stated APR of 10.00%, i.e., 10.0049% APR or the EAR equivalent of 10.4767%. There are two primary methods of borrowing money to buy a car: direct and indirect. thumb|279px|Parts of total cost and effective APR for a 12-month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. If the $1000 one-time fees are taken into account then the yearly interest rate paid is effectively equal to 10.31%. The monthly payments, using APR, would be $1755.87. If the fee is not considered, this loan has an effective APR of approximately 80% (1.0512 = 1.7959, which is approximately an 80% increase). If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase). * APR is also an abbreviation for "Annual Principal Rate" which is sometimes used in the auto sales in some countries where the interest is calculated based on the "Original Principal" not the "Current Principal Due", so as the Current Principal Due decreases, the interest due does not. ==Rate format== An effective annual interest rate of 10% can also be expressed in several ways: * 0.7974% effective monthly interest rate, because 1.00797412=1.1 * 9.569% annual interest rate compounded monthly, because 12×0.7974=9.569 * 9.091% annual rate in advance, because (1.1-1)÷1.1=0.09091 These rates are all equivalent, but to a consumer who is not trained in the mathematics of finance, this can be confusing. In that case the formula becomes: :: S -A = R (1 + \mathrm{APR}/100)^{-t_N} + \sum_{k=1}^N A_k (1 + \mathrm{APR}/100)^{-t_k} :where: :: S is the borrowed amount or principal amount. Over 85% of new cars and half of used cars are financed (as opposed to being paid for in a lump sum with cash). If the consumer pays the loan off early, the effective interest rate achieved will be significantly higher than the APR initially calculated. Typically, the indirect auto lender will set an interest rate, known as the "buy rate". Using the improved notation of directive 2008/48/EC, the basic equation for calculation of APR in the EU is: :: \sum_{i=1}^M C_i (1 + \mathrm{APR}/100)^{-t_i} = \sum_{j=1}^N D_j (1 + \mathrm{APR}/100)^{-s_j} :where: :: M is the total number of drawdowns paid by the lender :: N is the total number of repayments paid by the borrower :: i is the sequence number of a drawdown paid by the lender :: j is the sequence number of a repayment paid by the borrower :: Ci is the cash flow amount for drawdown number i :: Dj is the cash flow amount for repayment number j :: ti is the interval, expressed in years and fractions of a year, between the date of the first drawdown* and the date of drawdown i :: sj is the interval, expressed in years and fractions of a year, between the date of the first drawdown* and the date of repayment j. In this equation the left side is the present value of the drawdowns made by the lender and the right side is the present value of the repayments made by the borrower. Suppose that the complete amount including the interest is withdrawn after exactly one year. Buy Here Pay Here financing accounts for 6% of the total financing market. Consumers can, of course, use the nominal interest rate and any costs on the loan (or savings account) and compute the APR themselves, for instance using one of the calculators on the internet. An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid.

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Consider the initial value problem $$ y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0 $$ where $k$ is the magnitude of an impulse at $t=1$ and $\gamma$ is the damping coefficient (or resistance). Let $\gamma=\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.

* Determine the system steady-state gain k=A_0with k=\lim_{t\to\infty} \dfrac{y(t)}{x(t)} * Calculate r=\dfrac{t_{25}}{t_{75}} P=-18.56075\,r+\dfrac{0.57311}{r-0.20747}+4.16423 X=14.2797\,r^3-9.3891\,r^2+0.25437\,r+1.32148 * Determine the two time constants \tau_2=T_2=\dfrac{t_{75}-t_{25}}{X\,(1+1/P)} \tau_1=T_1=\dfrac{T_2}{P} * Calculate the transfer function of the identified system within the Laplace-domain G(s) = \dfrac{k}{(1+s\,T_1)\cdot(1+s\,T_2)} ====Phase margin==== thumbnail|280px|Figure 5: Bode gain plot to find phase margin; scales are logarithmic, so labeled separations are multiplicative factors. Here damping ratio is greater than one. ==Properties== thumb|Typical second order transient system properties Transient response can be quantified with the following properties. In particular, the unit step response of the system is: :S(t) = \left(\frac {A_0} {1+ \beta A_0}\right)\left(1 - e^{- \rho t} \ \frac { \sin \left( \mu t + \phi \right)}{ \sin \phi}\right)\ , which simplifies to :S(t) = 1 - e^{- \rho t} \ \frac { \sin \left( \mu t + \phi \right)}{ \sin \phi} when A0 tends to infinity and the feedback factor β is one. * Determine the time-spans t_{25}and t_{75}where the step response reaches 25% and 75% of the steady state output value. : \Delta = e^{- \rho t_S }\text{ or }t_S = \frac { \ln \frac{1}{\Delta} } { \rho } = \tau_2 \frac {2 \ln \frac{1} { \Delta} } { 1 + \frac { \tau_2 } { \tau_1} } \approx 2 \tau_2 \ln \frac{1} { \Delta}, where the τ1 ≫ τ2 is applicable because of the overshoot control condition, which makes τ1 = αβAOL τ2. The final value of the step response is 1, so the exponential is the actual overshoot itself. Using the equations above, the amount of overshoot can be found by differentiating the step response and finding its maximum value. Its step response is of the same form: an exponential decay toward the new equilibrium value. Notice that the damping of the response is set by ρ, that is, by the time constants of the open-loop amplifier. The equation reads as :\frac{d^2y}{d\zeta^2} =(y^2-\zeta^2)e^{-\delta^{-1/3}(y+\gamma \zeta)} subjected to the boundary conditions : \begin{align} \zeta\rightarrow -\infty : &\quad \frac{dy}{d\zeta}=-1,\\\ \zeta\rightarrow \infty : &\quad \frac{dy}{d\zeta}=1 \end{align} where \delta is the reduced or rescaled Damköhler number and \gamma is the ratio of excess heat conducted to one side of the reaction sheet to the total heat generated in the reaction zone. In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. For \delta>\delta_E with |\gamma|<1, the equation possess two solutions, of which one is an unstable solution. The impulse response and step response are transient responses to a specific input (an impulse and a step, respectively). This forward amplifier has unit step response :S_{OL}(t) = A_0(1 - e^{-t / \tau}), an exponential approach from 0 toward the new equilibrium value of A0. The time dependence of the amplifier is easy to discover by switching variables to s = jω, whereupon the gain becomes: : A_{FB} = \frac {A_0} { \tau_1 \tau_2 } \; \cdot \; \frac {1} {s^2 +s \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) + \frac {1+ \beta A_0} {\tau_1 \tau_2}} The poles of this expression (that is, the zeros of the denominator) occur at: :2s = - \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) \pm \sqrt { \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right) ^2 -\frac {4 \beta A_0 } {\tau_1 \tau_2 } }, which shows for large enough values of βA0 the square root becomes the square root of a negative number, that is the square root becomes imaginary, and the pole positions are complex conjugate numbers, either s+ or s−; see Figure 2: : s_{\pm} = -\rho \pm j \mu, with : \rho = \frac {1}{2} \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right ), and : \mu = \frac {1} {2} \sqrt { \frac {4 \beta A_0} { \tau_1 \tau_2} - \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2 }. The result for maximum step response Smax is: :S_\max= 1 + \exp \left( - \pi \frac { \rho }{ \mu } \right). It is clear the overshoot is zero if μ = 0, which is the condition: : \frac {4 \beta A_0} { \tau_1 \tau_2} = \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2. Comparing the top panel (α = 4) with the lower panel (α = 0.5) shows lower values for α increase the rate of response, but increase overshoot. As an example of this formula, if the settling time condition is tS = 8 τ2. The step response can be described by the following quantities related to its time behavior, *overshoot *rise time *settling time *ringing In the case of linear dynamic systems, much can be inferred about the system from these characteristics. The transient response is not necessarily tied to abrupt events but to any event that affects the equilibrium of the system. If there is a second pole, then as the closed-loop time constant approaches the time constant of the second pole, a two-pole analysis is needed. ===Two-pole amplifiers=== In the case that the open-loop gain has two poles (two time constants, τ1, τ2), the step response is a bit more complicated.

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If a series circuit has a capacitor of $C=0.8 \times 10^{-6} \mathrm{~F}$ and an inductor of $L=0.2 \mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.

In the case of the series RLC circuit, the damping factor is given by :\zeta = \frac{\, R \,}{2} \sqrt{ \frac{C}{\, L \,} \,} = \frac{1}{\ 2 Q\ } ~. For the parallel circuit, the attenuation is given byNilsson and Riedel, p. 286. : \alpha = \frac{1}{\,2\,R\,C\,} and the damping factor is consequently :\zeta = \frac{1}{\,2\,R\,} \sqrt{\frac{L}{C}~}\,~. The value of the damping factor determines the type of transient that the circuit will exhibit.Irwin, pp. 217–220. ===Transient response=== thumb|350px|Plot showing underdamped and overdamped responses of a series RLC circuit to a voltage input step of 1 V. Rearranging for the case where is known – capacitance: : C = \frac{~\alpha + \beta~}{R\,\alpha\,\beta} \,, inductance (total of circuit and load): : L = \frac{R}{\,\alpha + \beta~} \,, initial terminal voltage of capacitor: : V_0 = \frac{\,-I_0 R\,\alpha\,\beta~}{\alpha + \beta} \left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,. ==See also== *RC circuit *RL circuit *Linear circuit == Footnotes == ==References== ==Bibliography== * * * * * Category:Analog circuits Category:Electronic filter topology For the case of the series RLC circuit these two parameters are given by:Agarwal and Lang, p. 641. :\begin{align} \alpha &= \frac{R}{\, 2L \,} \\\ \omega_0 &= \frac{1}{\, \sqrt{L\,C\,} \,} \;. \end{align} A useful parameter is the damping factor, , which is defined as the ratio of these two; although, sometimes is not used, and is referred to as damping factor instead; hence requiring careful specification of one's use of that term.Agarwal and Lang, p. 646. : \zeta \equiv \frac{\alpha}{\, \omega_0 \,} \;. These equations show that a series RC circuit has a time constant, usually denoted being the time it takes the voltage across the component to either rise (across the capacitor) or fall (across the resistor) to within of its final value. If the inductance is known, then the remaining parameters are given by the following – capacitance: : C = \frac{1}{~L\,\alpha\,\beta\,~} \,, resistance (total of circuit and load): : R = L\,(\,\alpha + \beta\,) \,, initial terminal voltage of capacitor: : V_0 = -I_0 L\,\alpha\,\beta\,\left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,. Parallel RC, series L circuit with resistance in parallel with the capacitor In the same vein, a resistor in parallel with the capacitor in a series LC circuit can be used to represent a capacitor with a lossy dielectric. This results in the linear differential equation :C\frac{dV}{dt} + \frac{V}{R}=0 \,, where is the capacitance of the capacitor. Series RL, parallel C circuit with resistance in series with the inductor is the standard model for a self-resonant inductor A series resistor with the inductor in a parallel LC circuit as shown in Figure 4 is a topology commonly encountered where there is a need to take into account the resistance of the coil winding and its self- capacitance. thumb|350px|A series RLC network (in order): a resistor, an inductor, and a capacitor An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. With complex impedances: :\begin{align} I_R &= \frac{V_\mathrm{in}}{R} \\\ I_C &= j\omega C V_\mathrm{in}\,. \end{align} This shows that the capacitor current is 90° out of phase with the resistor (and source) current. The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. As a result, : \omega_\mathrm{d} \approx \omega_0 \,. === Voltage multiplier === In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit : I = \frac{V}{R}\,. As a result, \sigma = 0 and the impedance becomes :Z_C = \frac{1}{j\omega C} = - \frac{j}{\omega C} \,. ==Series circuit== By viewing the circuit as a voltage divider, the voltage across the capacitor is: :V_C(s) = \frac{\frac{1}{Cs}}{R + \frac{1}{Cs}}V_\mathrm{in}(s) = \frac{1}{1 + RCs}V_\mathrm{in}(s) and the voltage across the resistor is: :V_R(s) = \frac{R}{R + \frac{1}{Cs}}V_\mathrm{in}(s) = \frac{RCs}{1 + RCs}V_\mathrm{in}(s)\,. ===Transfer functions=== The transfer function from the input voltage to the voltage across the capacitor is :H_C(s) = \frac{ V_C(s) }{ V_\mathrm{in}(s) } = \frac{ 1 }{ 1 + RCs } \,. The current through the resistor must be equal in magnitude (but opposite in sign) to the time derivative of the accumulated charge on the capacitor. Solving this equation for yields the formula for exponential decay: :V(t)=V_0 e^{-\frac{t}{RC}} \,, where is the capacitor voltage at time . A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. The article next gives the analysis for the series RLC circuit in detail. Considering the expression for again, when :R \ll \frac{1}{\omega C}\,, so :\begin{align} I &\approx \frac{V_\mathrm{in}}\frac{1}{j\omega C} \\\ V_\mathrm{in} &\approx \frac{I}{j\omega C} = V_C \,.\end{align} Now, :\begin{align} V_R &= IR = C\frac{dV_C}{dt}R \\\ V_R &\approx RC\frac{dV_{in}}{dt}\,, \end{align} which is a differentiator across the resistor. RLC parallel circuit – the voltage source powering the circuit – the current admitted through the circuit – the equivalent resistance of the combined source, load, and components – the inductance of the inductor component – the capacitance of the capacitor component The properties of the parallel RLC circuit can be obtained from the duality relationship of electrical circuits and considering that the parallel RLC is the dual impedance of a series RLC. This article considers the RC circuit, in both series and parallel forms, as shown in the diagrams below. == Natural response == 200px|thumb|right| RC circuit The simplest RC circuit consists of a resistor and a charged capacitor connected to one another in a single loop, without an external voltage source.

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If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\prime \prime}+2 y^{\prime}+t e^t y=0$ and if $W\left(y_1, y_2\right)(1)=2$, find the value of $W\left(y_1, y_2\right)(5)$.

In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. The Battle of the Ypres–Comines Canal was a battle of the Second World War fought between the British Expeditionary Force (BEF) and German Army Group B during the BEF's retreat to Dunkirk in 1940. In scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis function. Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. The Second Anglo-Sikh War was a military conflict between the Sikh Empire and the East India Company that took place in 1848 and 1849. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. Later developments indicated that the MFS can be used to solve partial differential equations with variable coefficients.C.M. Fan, C.S. Chen, J. Monroe, The method of fundamental solutions for solving convection- diffusion equations with variable coefficients, Advances in Applied Mathematics and Mechanics. 1 (2009) 215–230 The MFS has proved particularly effective for certain classes of problems such as inverse,Y.C. Hon, T. Wei, The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. A differential equation can be homogeneous in either of two respects. Introducing elliptic coordinates, : x = a \cosh \xi \cos \eta, : y = a \sinh \xi \sin \eta, the potential energy can be written as : V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)}, and the kinetic energy as : T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right). Terso Solutions, Inc., located in Madison, Wisconsin, USA, is the developer and distributor of an automated system for storage and distribution of high value research reagents and medical supplies. It follows that, if is a solution, so is , for any (non-zero) constant . Denoting the sum of the χ functions by Y, : Y = \chi_{1}(\varphi_{1}) + \chi_{2}(\varphi_{2}) + \cdots + \chi_{s}(\varphi_{s}), the kinetic energy can be written as : T = \frac{1}{2} Y F. Sci. 7 (2005) 119–132 unbounded domain, and free-boundary problems.A.K. G. Fairweather, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics. 9 (1998) 69–95. Retrieved on July 7, 2008. thumb|right|300px|Terso Solutions, Madison, WI, USA. == History == Developed initially as an on-site inventory supplier for Promega products, the privately held Terso Solutions, Inc. was spun off from Promega Corporation in 2005. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs. However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s,R. Mathon, R.L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM Journal on Numerical Analysis. (1977) 638–650. followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications. The MFS then gradually became a useful tool for the solution of a large variety of physical and engineering problems.Z. Fu, W. Chen, W. Yang, Winkler plate bending problems by a truly boundary-only boundary particle method, Computational Mechanics. 44 (2009) 757–763.W. Chen, J. Lin, F. Wang, Regularized meshless method for nonhomogeneous problems , Engineering Analysis with Boundary Elements. 35 (2011) 253–257.W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary , Engineering Analysis with Boundary Elements. 34 (2010) 530–532.JIANG Xin-rong, CHEN Wen, Method of fundamental solution and boundary knot method for helmholtz equations: a comparative study, Chinese Journal of Computational Mechanics, 28:3(2011) 338–344 (in Chinese) In the 1990s, M. A. Golberg and C. S. Chen extended the MFS to deal with inhomogeneous equations and time-dependent problems, greatly expanding its applicability.M.A. Golberg, C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Communications. 5 (1994) 57–61.M. a. Golberg, C.S. Chen, H. Bowman, H. Power, Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method, Computational Mechanics. 21 (1998) 141–148.

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Consider the initial value problem $$ 5 u^{\prime \prime}+2 u^{\prime}+7 u=0, \quad u(0)=2, \quad u^{\prime}(0)=1 $$ Find the smallest $T$ such that $|u(t)| \leq 0.1$ for all $t>T$.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Assuming W_c is nonsingular (if and only if the system is controllable), the minimum energy control is then : u(t) = -B^*e^{A^*(t_1-t)}W_c^{-1}(t_1)[e^{A(t_1-t_0)}x_0-x_1]. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Also, \omega(u)-e^{-\gamma} oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. Here the number of initial conditions necessary for obtaining a closed form solution is the dimension n = 1 times the order k, or simply k. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. In control theory, the minimum energy control is the control u(t) that will bring a linear time invariant system to a desired state with a minimum expenditure of energy. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.p. 131, Cheer and Goldston 1990. ==Applications== The Buchstab function is used to count rough numbers. Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. 300px|thumbnail|Graph of the Buchstab function ω(u) from u = 1 to u = 4\. We are trying to find a formula for y(t) that satisfies these two equations. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. One seeks an input u(t) so that the system will be in the state x_1 at time t_1, and for any other input \bar{u}(t), which also drives the system from x_0 to x_1 at time t_1, the energy expenditure would be larger, i.e., : \int_{t_0}^{t_1} \bar{u}^*(t) \bar{u}(t) dt \ \geq \ \int_{t_0}^{t_1} u^*(t) u(t) dt. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem.

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Consider the initial value problem $$ y^{\prime}=t y(4-y) / 3, \quad y(0)=y_0 $$ Suppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem.

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25. Consider the initial value problem $$ 2 y^{\prime \prime}+3 y^{\prime}-2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-\beta, $$ where $\beta>0$. Find the smallest value of $\beta$ for which the solution has no minimum point.

Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) \sqrt[x]{x} Unique global maximum at x = e. (See figure at top of page.) x3 \+ 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2] Local maximum at x = −1−/3, local minimum at x = −1+/3, global maximum at x = 2 and global minimum at x = −4. This is illustrated by the function :f(x,y)= x^2+y^2(1-x)^3,\qquad x,y \in \R, whose only critical point is at (0,0), which is a local minimum with f(0,0) = 0\. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction). ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem A function need not have a least fixed point, but if it does then the least fixed point is unique. ==Examples== With the usual order on the real numbers, the least fixed point of the real function f(x) = x2 is x = 0 (since the only other fixed point is 1 and 0 < 1). From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum. Then the second derivative test provides a sufficient condition for the point to be a local maximum or local minimum. ==Search techniques== Local search or hill climbing methods for solving optimization problems start from an initial configuration and repeatedly move to an improving neighboring configuration. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. If the first derivative exists everywhere, it can be equated to zero; if the function has an unbounded domain, for a point to be a local optimum it is necessary that it satisfy this equation. The definition of global minimum point also proceeds similarly. This function has no global maximum or minimum. x| Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0. cos(x) Infinitely many global maxima at 0, ±2, ±4, ..., and infinitely many global minima at ±, ±3, ±5, .... 2 cos(x) − x Infinitely many local maxima and minima, but no global maximum or minimum. cos(3x)/x with Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. thumb|right|220px|Attraction basins around locally optimal points thumb|right|233px|Polynomial of degree 4: the trough on the right is a local minimum and the one on the left is the global minimum. (See figure at right) x−x Unique global maximum over the positive real numbers at x = 1/e. x3/3 − x First derivative x2 − 1 and second derivative 2x. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability. thumb|150px|The function f(x) = x2 − 4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2 − /2. However, not all critical points are extrema.

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Consider the initial value problem $$ y^{\prime}=t y(4-y) /(1+t), \quad y(0)=y_0>0 . $$ If $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. We are trying to find a formula for y(t) that satisfies these two equations. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons. ==Linear system== ===Discrete time=== A linear matrix difference equation of the homogeneous (having no constant term) form X_{t+1}=AX_t has closed form solution X_t=A^tX_0 predicated on the vector X_0 of initial conditions on the individual variables that are stacked into the vector; X_0 is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being the number of time lags in the system. We use their notation, and assume that the unknown function is u, and that we have a known solution u_n at time t_n. Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. Here the constants c_1, \dots , c_k are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition x_t Is known. ===Continuous time=== A differential equation system of the first order with n variables stacked in a vector X is :\frac{dX}{dt}=AX. These problems can come from a more typical initial value problem :u'(t) = f(u(t)), \qquad u(t_0)=u_0, after linearizing locally about a fixed or local state u^*: : L = \frac{\partial f}{\partial u}(u^*); \qquad N = f(u) - L u. The characteristic equation of this dynamic equation is \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, whose solutions are the characteristic values \lambda_1,\dots , \lambda_k; these are used in the solution equation :x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}.

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28. A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$. If the mass is to attain a velocity of no more than $10 \mathrm{~m} / \mathrm{s}$, find the maximum height from which it can be dropped.

If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). Some heights are difficult to verify due to lack of documentation and are approximated. ==List of the highest falls survived without a parachute== Name Image Distance of fall Date Notes and References Feet Meters Vesna Vulović 33,330 10160 1972 Flight attendant from Serbia who was the sole survivor of an airplane bombing mid-air. The mass m0 used in the fall is 80 kg. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. thumb|250px|The climber will fall about the same height h in both cases, but they will be subjected to a greater force at position 1, due to the greater fall factor. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. This article attempts to list of the highest falls survived without a parachute. We first state an equation for this quantity and describe its interpretation, and then show its derivation and how it can be put into a more convenient form. ===Equation for the impact force and its interpretation=== When modeling the rope as an undamped harmonic oscillator (HO) the impact force Fmax in the rope is given by: :F_{max} = mg + \sqrt{(mg)^2 + 2mghk}, where mg is the climber's weight, h is the fall height and k is the spring constant of the portion of the rope that is in play. thumb|The aftermath of a hypervelocity impact, with a projectile the same size as the one that impacted for scale Hypervelocity is very high velocity, approximately over 3,000 meters per second (6,700 mph, 11,000 km/h, 10,000 ft/s, or Mach 8.8). When climbing from the ground up, the maximum possible fall factor is 1, since any greater fall would mean that the climber hit the ground. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. The amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. The mass of the displaced fluid can be expressed in terms of the density and its volume, . We will see below that when varying the height of the fall while keeping the fall factor fixed, the quantity hk stays constant. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). As a numerical example, consider a fall of 20 feet that occurs with 10 feet of rope out (i.e., the climber has placed no protection and falls from 10 feet above the belayer to 10 feet below—a factor 2 fall). This fall produces far more force on the climber and the gear than if a similar 20 foot fall had occurred 100 feet above the belayer. Therefore, the weight of the displaced fluid can be expressed as .

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A home buyer can afford to spend no more than $\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. Determine the maximum amount that this buyer can afford to borrow.

Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? "A capital budgeting model of the supply and demand of loanable funds", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. For a 24-month loan, the denominator is 300. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. The expansion P\approx P_0 \left(1 + X + \frac{X^2}{3}\right) is valid to better than 1% provided X\le 1 . ====Example of mortgage payment==== For a $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find: T=30 I=0.045 which gives X=\frac{1}{2}IT=.675 so that P\approx P_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96 The exact payment amount is P=\$608.02 so the approximation is an overestimate of about a sixth of a percent. ===Investing: monthly deposits=== Given a principal (initial) deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. For example, for interest rate of 6% (0.06/12), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives: = PMT(0.06/12, 25 * 12, -150000, 0, 0) = $966.45 ====Approximate formula for monthly payment==== A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (I<8\% and terms T=10–30 years), the monthly note rate is small compared to 1: r << 1 so that the \ln(1+r)\approx r which yields a simplification so that c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}} which suggests defining auxiliary variables Y\equiv n r = IT c_0\equiv \frac{P}{n} . Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. During the second month the borrower has use of two $1000 (2/3) amounts and so the payment should be $1000 plus two $10 interest fees.

1.61

21

0.36

89,034.79

1.25

D

A spring is stretched 6 in by a mass that weighs $8 \mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$ and is acted on by an external force of $4 \cos 2 t \mathrm{lb}$. If the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.

As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *"The Effective Mass of an Oscillating Spring" Am. J. Phys., 38, 98 (1970) *"Effective Mass of an Oscillating Spring" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the "effective total mass" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon "external" factors such as the acceleration of gravity along it. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. The effective mass of the spring can be determined by finding its kinetic energy. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. thumb|Phase portrait of damped oscillator, with increasing damping strength. The forcing amplitude increases from \gamma=0.20 to \gamma=0.65. The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. For the case with linearly distributed load of maximum intensity q_0, :M_{\mathrm{right}}^{\mathrm{fixed}} = \int_{0}^{L} q_0 \frac{x}{L} dx \frac{ x^2 (L-x)}{L^2} = \frac{q_0 L^2}{20} :M_{\mathrm{left}}^{\mathrm{fixed}} = \int_{0}^{L} \left \\{ - q_0 \frac{x}{L} dx \frac{x (L-x)^2}{L^2} \right \\} = - \frac{q_0 L^2}{30} == See also == * Moment distribution method * Statically Indeterminate * Slope deflection method * Matrix method == References == * Category:Structural analysis *Damped harmonic motion, see animation (right). The lower overhanging side is unstable – i.e. the dashed-line parts in the figures of the frequency response – and cannot be realized for a sustained time. The restoring force provided by the nonlinear spring is then \alpha x + \beta x^3. The dashed parts of the frequency response are unstable. In a real spring–mass system, the spring has a non-negligible mass m.

0.00539

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35.2

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C

A recent college graduate borrows $\$ 100,000$ at an interest rate of $9 \%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made. Assuming that this payment schedule can be maintained, when will the loan be fully paid?

If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). Dividing by loan time period t will then give the equivalent simple interest rate. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. A graduated payment loan typically involves negative amortization, and is intended for students in the case of student loans, and homebuyers in the case of real estate, who currently have moderate incomes and anticipate their income will increase over the next 5–10 years. However this would contradict the primary assumption upon which the "continuous payment" model is based: namely that the annual payment rate is defined as: :M_a=\lim_{N\to\infty}N\cdot x(N) \, Since it is of course impossible for an investor to make an infinitely small payment infinite times per annum, a bank or other lending institution wishing to offer "continuous payment" annuities or mortgages would in practice have to choose a large but finite value of N (annual frequency of payments) such that the continuous time formula will always be correct to within some minimal pre-specified error margin. For a theoretical continuous payment savings annuity we can only calculate an annual rate of payment: :M_a=\frac{500000 \times 12\%}{e^{0.12\cdot 10}-1}=25860.77 At this point there is a temptation to simply divide by 12 to obtain a monthly payment. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. In a (theoretical) continuous-repayment mortgage the payment interval is narrowed indefinitely until the discrete interval process becomes continuous and the fixed interval payments become—in effect—a literal cash "flow" at a fixed annual rate. As with many similar examples the discrete interval problem and its solution is closely approximated by calculations based on the continuous repayment model - Dr Hahn's solution for interest rate is 40.8% as compared to the 41.6% calculated above. ==Period of a loan== If a borrower can afford an annual repayment rate Ma, then we can re-arrange the formula for calculating Ma to obtain an expression for the time period T of a given loan P0: : \begin{align} & M_a = \frac{P_0 r}{1-e^{-rT}} \\\\[8pt] \Rightarrow & T = \frac{1}{r}\ln\frac{M_a}{M_a-P_0 r} = -\frac{1}{r}\ln\left(1 - \frac{P_0 r}{M_a} \right) \end{align} ==Minimum payment ratio== The minimum payment ratio of a loan is the ratio of minimum possible payment rate to actual payment rate. For example, consider a $100 loan which must be repaid after one month, plus 5%, plus a $10 fee. \, Another way to calculate balance due P(t) on a continuous-repayment loan is to subtract the future value (at time t) of the payment stream from the future value of the loan (also at time t): :P(t)=P_0 e^{rt}-\frac{M_a}{r}(e^{rt}-1). Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. Beware of extremely long repayment periods, as generally speaking, the longer the term, the more you will owe because the interest accrues over a long period of time. The minimum possible payment rate is that which just covers the loan interest – a borrower would in theory pay this amount forever because there is never any decrease in loan capital. This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. If the consumer pays the loan off early, the effective interest rate achieved will be significantly higher than the APR initially calculated. The sum of these interest and principal payments must equal the cumulative fixed payments at time t i.e. Mat. Evaluating the first integral on the right we obtain an expression for I(t), the interest paid: :I(t)=M_at-\frac{M_a(e^{rt}-1)}{re^{rT}} Unsurprisingly the second integral evaluates to P0 − P(t) and therefore: :I(t)=M_at-P_0+P(t) \, The reader may easily verify that this expression is algebraically identical to the one above. ==Loan cost factor== The cost of a loan is simply the annual rate multiplied by loan period: : C = M_aT= \frac{P_0 rT}{1-e^{-rT}} Let s = rT. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity.

-59.24

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Consider the initial value problem $$ y^{\prime}+\frac{1}{4} y=3+2 \cos 2 t, \quad y(0)=0 $$ Determine the value of $t$ for which the solution first intersects the line $y=12$.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. We are trying to find a formula for y(t) that satisfies these two equations. The condition of intersection is : ys(x) = yc(x). In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Hence, : y_s(x) = -\tfrac{1}{4} \cdot x^2 \,\\! is tangent to every member of the one- parameter family of solutions : y_c(x) = c \cdot x + c^2 \,\\! of this Clairaut equation: : y(x) = x \cdot y' + (y')^2. \,\\! ==See also== * Chandrasekhar equation * Chrystal's equation * Caustic (mathematics) * Envelope (mathematics) * Initial value problem * Picard–Lindelöf theorem ==Bibliography== * Category:Differential equations Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem.

0.396

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650000.0

10.065778

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D

An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \$25 per month. Find the balance in the account after 3 years.

Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. Over one month, :\frac{0.1299 \times \$2500}{12} = \$27.06 interest is due (rounded to the nearest cent). Simple interest applied over 3 months would be :\frac{0.1299 \times \$2500 \times 3}{12} = \$81.19 If the card holder pays off only interest at the end of each of the 3 months, the total amount of interest paid would be :\frac{0.1299 \times \$2500}{12} \times 3 = \$27.06\text{ per month} \times 3\text{ months} =\$81.18 which is the simple interest applied over 3 months, as calculated above. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. He realized that if an account that starts with $1.00 and pays say 100% interest per year, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.52 = $2.25. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. At the end of one year, 1,000 × 20% = 200 BRL interest is credited to the account. * P = principal deposit * r = rate of return (monthly) * M = monthly deposit, and * t = time, in months The compound interest for each deposit is: M'=M(1+r)^{t} and adding all recurring deposits over the total period t (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month) : M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}} recognizing the geometric series: M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}} and applying the closed-form formula (common ratio :1/(1+r)) we obtain: P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as \text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y} where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling. thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. Suppose that the complete amount including the interest is withdrawn after exactly one year. The amount of interest paid (every six months) is the disclosed interest rate divided by two and multiplied by the principal. The principal remaining after the first month is P_1=(1+r)P - c, that is, the initial amount plus interest less the payment. With monthly payments, the monthly interest is paid out of each payment and so should not be compounded, and an annual rate of 12·r would make more sense. For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple annual interest rate is 12.99% per annum, applied monthly, so the frequency of applying interest is 12 per year. Compounding quarterly yields $1.00×1.254 = $2.4414..., and so on. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). If the whole loan is repaid after one month then P_1=0, so P=\frac{c}{1+r} After the second month P_2=(1+r) P_1 - c is left, so P_2=(1+r)((1+r)P-c)-c If the whole loan was repaid after two months, P_2 = 0, so P = \frac{c}{1+r}+\frac{c}{(1+r)^2} This equation generalizes for a term of n months, P = c \sum\limits_{j=1}^n \frac{1}{(1+r)^j} . The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. In total, the investor therefore now holds: :\$10\,000 + \$300 = \left(1 + \frac{r}{n}\right) \cdot B = \left(1 + \frac{6\%}{2}\right) \times \$10\,000 and so earns a coupon at the end of the next 6 months of: :\begin{align}\frac {r \cdot B \cdot m}{n} &= \frac {6\% \times \left(\$10\,000 + \$300\right)}{2}\\\ &= \frac {6\% \times \left(1 + \frac{6\%}{2}\right) \times \$10\,000}{2}\\\ &=\$309\end{align} Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of: :\begin{align}\$10,000 + \$300 + \$309 &= \$10\,000 + \frac {6\% \times \$10,000}{2} + \frac {6\% \times \left( 1 + \frac {6\%}{2}\right) \times \$10\,000}{2}\\\ &= \$10\,000 \times \left(1 + \frac{6\%}{2}\right)^2\end{align} and the investor earned in total: :\begin{align}\$10\,000 \times \left(1 + \frac {6\%}{2}\right)^2 - \$10\,000\\\ = \$10\,000 \times \left( \left( 1 + \frac {6\%}{2}\right)^2 - 1\right)\end{align} The formula for the annual equivalent compound interest rate is: :\left(1 + \frac{r}{n}\right)^n - 1 where :r is the simple annual rate of interest :n is the frequency of applying interest For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is: :\left(1 + \frac{6\%}{2}\right)^2 - 1 = 1.03^2 - 1 = 6.09\% ===Other formulations=== The outstanding balance Bn of a loan after n regular payments increases each period by a growth factor according to the periodic interest, and then decreases by the amount paid p at the end of each period: :B_{n} = \big( 1 + r \big) B_{n - 1} - p, where :i = simple annual loan rate in decimal form (for example, 10% = 0.10. A compounding instrument adds the previously accrued interest to the principal each period, applying compound interest. == External links == * What is Accrued Interest ?

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A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$. If the mass is dropped from a height of $30 \mathrm{~m}$, find its velocity when it hits the ground.

If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. The resulting outcome depends on the properties of the drop, the surface, and the surrounding fluid, which is most commonly a gas. == On a dry solid surface == When a liquid drop strikes a dry solid surface, it generally spreads on the surface, and then will retract if the impact is energetic enough to cause the drop to spread out more than it would generally spread due to its static receding contact angle. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. This outcome is representative of impact of small, low-velocity drops onto smooth wetting surfaces. The specific outcome of the impact depends mostly upon the drop size, velocity, surface tension, viscosity, and also upon the surface roughness and the contact angle between the drop and the surface.Rioboo, Romain, Cameron Tropea, and Marco Marengo. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. "Surface phenomena: Contact time of a bouncing drop." In fluid dynamics, drop impact occurs when a drop of liquid strikes a solid or liquid surface. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). The formula for terminal velocity (V)] appears on p. [52], equation (127). Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). "Phenomena of liquid drop impact on solid and liquid surfaces." The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. So instead of m use the reduced mass m_r = m-\rho V in this and subsequent formulas. Dividing both sides by m gives \frac{\mathrm{d}v}{\mathrm{d}t} = g \left( 1 - \alpha^2 v^2 \right). If the impact energy is high enough, the jet rises to the point where it pinches off, sending one or more droplets upward out of the surface. ==See also== *Splash (fluid mechanics) == References == Category:Fluid dynamics "Outcomes from a drop impact on solid surfaces." At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0.

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A mass of $100 \mathrm{~g}$ stretches a spring $5 \mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \mathrm{~cm} / \mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.

The effective mass of the spring can be determined by finding its kinetic energy. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the "effective total mass" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon "external" factors such as the acceleration of gravity along it. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *"The Effective Mass of an Oscillating Spring" Am. J. Phys., 38, 98 (1970) *"Effective Mass of an Oscillating Spring" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the system. Pulling the spring to a greater length causes it to exert a force that brings the spring back toward its equilibrium length. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. In a real spring–mass system, the spring has a non-negligible mass m. alt=mass connected to the ground with a spring and damper in parallel|thumb|Classic model used for deriving the equations of a mass spring damper model The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. The amount of force can be determined by multiplying the spring constant, characteristic of the spring, by the amount of stretch, also known as Hooke's Law. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. The stiffness of the spring, its spring coefficient, \kappa\, in N·m/radian^2, along with the balance wheel's moment of inertia, I\, in kg·m2, determines the wheel's oscillation period T\,. An idealized spring exerts a force proportional to the amount of deformation of the spring from its equilibrium length, exerted in a direction oppose the deformation. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. The balance spring provides a restoring torque that limits and reverses the motion of the balance so it oscillates back and forth.

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Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$. Consider a water tank in the form of a right circular cylinder that is $3 \mathrm{~m}$ high above the outlet. The radius of the tank is $1 \mathrm{~m}$ and the radius of the circular outlet is $0.1 \mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.

The results confirm the correctness of Torricelli's law very well. ==Discharge and time to empty a cylindrical vessel== Assuming that a vessel is cylindrical with fixed cross-sectional area A, with orifice of area A_A at the bottom, then rate of change of water level height dh/dt is not constant. From Torricelli's law, the rate of outflow is :\frac{dV}{dt} = A_A v = A_A \sqrt{2gh}, From these two equations, : \begin{align} A_A \sqrt{2gh} &= \pi r^2 c \\\ \Rightarrow \quad h &= \frac{\pi^2 c^2}{2g A_A^2} r^4. \end{align} Thus, the radius of the container should change in proportion to the quartic root of its height, r \propto \sqrt[4]{h}. More precisely, : \Delta t = \frac{A}{A_A} \sqrt{\frac{2}{g}} (\sqrt{h_1} - \sqrt{h_2}) where \Delta t is the time taken by the water level to fall from the height of h_1 to height of h_2. ==Torricelli's original derivation== thumb|Figures from Evangelista Torricelli's Opera Geometrica (1644) describing the derivation of his famous outflow formula: (a) One tube filled up with water from A to B. (b) In two connected tubes the water lift up to the same height. (c) When the tube C is removed, the water should rise up to the height D. Due to friction effects the water only rises to the point C. Evangelista Torricelli's original derivation can be found in the second book 'De motu aquarum' of his 'Opera Geometrica' (see A. Malcherek: History of the Torricelli Principle and a New Outflow Theory,Journal of Hydraulic Engineering 142(11),1-7,2016,https://doi.org/10.1061/(ASCE)HY.1943-7900.0001232)): He starts a tube AB (Figure (a)) filled up with water to the level A. Torricelli's law can only be applied when viscous effects can be neglected which is the case for water flowing out through orifices in vessels. ===Experimental verification: Spouting can experiment=== thumb|Experiment to determine the trajectory of an outflowing jet: Vertical rods are adjusted so they are nearly touching the jet. Since the water level is H-h above the orifice, the horizontal efflux velocity v = \sqrt{2g(H-h)}, as given by Torricelli's law. Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening. The water volume in the vessel is changing due to the discharge \dot{V} out of the vessel: :\frac{dV}{dt} = A \frac{dh}{dt} = \dot{V} = A_A v_A = A_A \sqrt{2gh} \quad \Rightarrow \quad A \frac{dh}{\sqrt{h}} = A_A \sqrt{2g} \; dt Integrating both sides and re- arranging, we obtain : T = \frac{A}{A_A} \sqrt{\frac{2H}{g}}, where H is the initial height of the water level and T is the total time taken to drain all the water and hence empty the vessel. If a tank with volume V with cross section A and height H, so that V = AH, is fully filled, then the time to drain all the water is : T = \frac{V}{A_A} \sqrt{\frac{2}{gH}}. Furthermore y_1 - y_2 is equal to the height h of the liquid's surface over the opening: :\frac{{v_1}^2}{2} + g h = \frac{{v_2}^2}{2} The velocity of the surface v_1 can by related to the outflow velocity v_2 by the continuity equation v_1 A = v_2 A_A, where A_A is the orifice's cross section and A is the (cylindrical) vessel's cross section. Lastly, we can re-arrange the above equation to determine the height of the water level h(t) as a function of time t as : h(t) = H \left(1 - \frac{t}{T} \right)^2, where H is the height of the container while T is the discharge time as given above. ===Discharge experiment, coefficient of discharge=== The dicharge theory can be tested by measuring the emptying time T or time series of the water level h(t) within the cylindrical vessel. Torricelli's law is obtained as a special case when the opening A_A is very small relative to the horizontal cross-section of the container A_1: :v_A = \sqrt{2gh}. This is normally done by introducing a discharge coefficient which relates the discharge to the orifice's cross- section and Torricelli's law: : {\dot {V}}_{\text{real}}=\mu A_A v_A \quad \text{with} \quad \mu = \frac{A_C}{A_A} For low viscosity liquids (such as water) flowing out of a round hole in a tank, the discharge coefficient is in the order of 0.65. The law states that the speed v of efflux of a fluid through a sharp-edged hole at the bottom of the tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. v = \sqrt{2gh}, where g is the acceleration due to gravity. The low pressure on sides on the cylinder is needed to provide the centripetal acceleration of the flow: :\frac{\partial p}{\partial r}=\frac{\rho V^2}{L} \,, where is the radius of curvature of the flow. Thus we find the maximum speed in the flow, , in the low pressure on the sides of the cylinder. Here it was also shown that the outflow velocity is predicted extremeliy well by Torricelli's law and that no velocity correction (like a "coefficient of velocity") is needed. In order to derive Torricelli's formula the first point with no index is taken at the liquid's surface, and the second just outside the opening. As given by the Torricelli's law, the rate of efflux through the hole depends on the height of the water; and as the water level diminishes, the discharge is not uniform. The fluid exit velocity is greater further down the tube.Spouting cylinder fluid flow. The instantaneous rate of change in water volume is :\frac{dV}{dt} = A \frac{dh}{dt} = \pi r^2 c. We want to find the radius such that the water level has a constant rate of decrease, i.e. dh/dt = c.

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Solve the initial value problem $y^{\prime \prime}-y^{\prime}-2 y=0, y(0)=\alpha, y^{\prime}(0)=2$. Then find $\alpha$ so that the solution approaches zero as $t \rightarrow \infty$.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. For example, for the equation (a<0), the stationary solution is , which is obtained for the initial condition . Suppose we have an ordinary differential equation in the complex domain. It can also be shown that J_\alpha^2(x)+Y_{\alpha}^2(x)=\frac{8\cos(\alpha\pi)}{\pi^2} \int_0^\infty K_{2\alpha}(2x\sinh t)\, dt, only when || < and but not when . Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):Abramowitz and Stegun, p. 375, 9.6.3, 9.6.5. \begin{align} J_\alpha(iz) &= e^{\frac{\alpha\pi i}{2}} I_\alpha(z), \\\\[5pt] Y_\alpha(iz) &= e^{\frac{(\alpha+1)\pi i}{2}}I_\alpha(z) - \frac{2}{\pi}e^{-\frac{\alpha\pi i}{2}}K_\alpha(z). \end{align} and are the two linearly independent solutions to the modified Bessel's equation:Abramowitz and Stegun, p. 374, 9.6.1. x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - \left(x^2 + \alpha^2 \right)y = 0. Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the order of the Bessel function.

−2

-242.6

0.23333333333

20.2

6

A

If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ and if $W\left(y_1, y_2\right)(2)=3$, find the value of $W\left(y_1, y_2\right)(4)$.

Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. The Third Solution (, also known as Russicum) is a 1988 Italian crime-thriller film written and directed by Pasquale Squitieri and starring Treat Williams.VV.AA. Variety Film Reviews, Volume 18. In mathematical physics, the Degasperis–Procesi equation : \displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx} is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs: :\displaystyle u_t - u_{xxt} + 2\kappa u_x + (b+1)u u_x = b u_x u_{xx} + u u_{xxx}, where \kappa and b are real parameters (b=3 for the Degasperis–Procesi equation). Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation : u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. thumb|300px|Time-Temperature-Transformation diagram for two steels: one with 0.4% wt. C (red line) and one with 0.4% wt. C and 2% weight Mn (green line). Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics In the case \kappa=0 the solution splits into an infinite linear combination of peakons (as previously conjectured). ==Geometric formulation== In the spatially periodic case, the Camassa–Holm equation can be given the following geometric interpretation. Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs. The fact that the equations admits solutions of this type was discovered by Camassa and Holm and these considerations were subsequently put on a firm mathematical basis. Then u is a solution to the Camassa–Holm equation with \kappa=0, if and only if the path t\mapsto\varphi_t\in\mathrm{Diff}(S^1) is a geodesic on \mathrm{Diff}(S^1) with respect to the right-invariant H^1 metric. As a result, κ is the phase speed for the long-wave limit of k approaching zero, and the Camassa–Holm equation is (if κ is non-zero) a model for one-directional wave propagation like the Korteweg–de Vries equation. ==Hamiltonian structure== Introducing the momentum m as :m = u - u_{xx} + \kappa, \, then two compatible Hamiltonian descriptions of the Camassa–Holm equation are: : \begin{align} m_t &= -\mathcal{D}_1 \frac{\delta \mathcal{H}_1}{\delta m} & & \text{ with }& \mathcal{D}_1 &= m \frac{\partial}{\partial x} + \frac{\partial}{\partial x} m & \text{ and } \mathcal{H}_1 &= \frac{1}{2} \int u^2 + \left(u_x\right)^2\; \text{d}x, \\\ m_t &= -\mathcal{D}_2 \frac{\delta \mathcal{H}_2}{\delta m} & & \text{ with }& \mathcal{D}_2 &= \frac{\partial}{\partial x} - \frac{\partial^3}{\partial x^3} & \text{ and } \mathcal{H}_2 &= \frac{1}{2} \int u^3 + u \left(u_{x}\right)^2 - \kappa u^2\; \text{d}x. \end{align} ==Integrability== The Camassa–Holm equation is an integrable system. These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.Lundmark & Szmigielski 2003, 2005 When \kappa > 0 the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as \kappa tends to zero.Matsuno 2005a, 2005b == Discontinuous solutions == The Degasperis–Procesi equation (with \kappa=0) is formally equivalent to the (nonlocal) hyperbolic conservation law : \partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \frac{3 u^2}{2} \right] = 0, where G(x) = \exp(-|x|), and where the star denotes convolution with respect to x. By the Sobolev embedding theorem, this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to x. == Notes == == References == * * * * * * * * * * * * * * * * * * == Further reading == * * * * * * * * * * * * * * * * * Category:Mathematical physics Category:Solitons Category:Partial differential equations Category:Equations of fluid dynamics For general \kappa, the Camassa–Holm equation corresponds to the geodesic equation of a similar right- invariant metric on the universal central extension of \mathrm{Diff}(S^1), the Virasoro group. ==See also== *Degasperis–Procesi equation *Hunter–Saxton equation ==Notes== ==References== * * * * * * * * * * * * * * * * * * * * * * * ==Further reading== ; Peakon solutions * ; Water wave theory * * ; Existence, uniqueness, wellposedness, stability, propagation speed, etc. * * * * ; Travelling waves * ; Integrability structure (symmetries, hierarchy of soliton equations, conservations laws) and differential-geometric formulation * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Category:Partial differential equations Category:Equations of fluid dynamics Category:Integrable systems Category:Solitons \, The equation was introduced by Roberto Camassa and Darryl Holm as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter κ is positive and the solitary wave solutions are smooth solitons. In the soliton region c>2 the solutions splits into a finite linear combination solitons. In the region 0 the solution is asymptotically given by a modulated sine function whose amplitude decays like t^{-1/2}.

0.3359

0.2307692308

1.602

0.65625

4.946

E

Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.

University Library of Tromso, Ravnetrykk No. 29. , pp. 195–202. by exposing natural radium-226 to neutrons to produce radium-227, which decays with a 42-minute half-life to actinium-227. Actinium-227 (half-life 21.8 years) in turn decays via thorium-227 (half-life 18.7 days) to radium-223. Radium-223 (223Ra, Ra-223) is an isotope of radium with an 11.4-day half-life. 209P/LINEAR is a periodic comet with an orbital period of 5.1 years. Plutonium-244 (244Pu) is an isotope of plutonium that has a half-life of 80 million years. This article concerns the period 229 BC – 220 BC. ==References== __NOTOC__ Year 226 BC was a year of the pre-Julian Roman calendar. Its half-life of 80 million years ensured its circulation across the solar system before its extinction, and indeed, 244Pu has not yet been found in matter other than meteorites. Scandium-44 (44Sc) is a radioactive isotope of scandium that decays by positron emission to stable 44Ca with a half-life of 4.042 hours. 44Sc can be obtained as a daughter radionuclide of long-lived 44Ti (t1/2 60.4 a) from 44Ti /44Sc generator or can be produced by nuclear reaction 44Ca ( p, n)44Sc in small cyclotrons. Although radium-223 and its decay products also emit beta and gamma radiation, over 95% of the decay energy is in the form of alpha radiation. The amount of 244Pu in the pre-Solar nebula (4.57×109 years ago) was estimated as 0.8% the amount of 238U..As the age of the Earth is about 57 half-lives of 244Pu, the amount of plutonium-244 left should be very small; Hoffman et al. estimated its content in the rare-earth mineral bastnasite as = 1.0×10−18 g/g, which corresponded to the content in the Earth crust as low as 3×10−25 g/g (i.e. the total mass of plutonium-244 in Earth's crust is about 9 g). This is longer than any of the other isotopes of plutonium and longer than any other actinide isotope except for the three naturally abundant ones: uranium-235 (704 million years), uranium-238 (4.468 billion years), and thorium-232 (14.05 billion years). The principal use of radium-223, as a radiopharmaceutical to treat metastatic cancers in bone, takes advantage of its chemical similarity to calcium, and the short range of the alpha radiation it emits. ==Origin and preparation== Although radium-223 is naturally formed in trace amounts by the decay of uranium-235, it is generally made artificially,Bruland O.S., Larsen R.H. (2003). The recommended regimen is six treatments of 55 kBq/kg (1.5 μCi/kg), repeated at 4-week intervals. === Mechanism of action === The use of radium-223 to treat metastatic bone cancer relies on the ability of alpha radiation from radium-223 and its short-lived decay products to kill cancer cells. The 44Ti /44Sc generator represents a secular equilibrium system with a half-life ratio between parent and daughter of ca. 130 000. Radium revisited. This decay path makes it convenient to prepare radium-223 by "milking" it from an actinium-227 containing generator or "cow", similar to the moly cows widely used to prepare the medically important isotope technetium-99m. 223Ra itself decays to 219Rn (half-life 3.96 s), a short-lived gaseous radon isotope, by emitting an alpha particle of 5.979 MeV. ==Medical use== The pharmaceutical product and medical use of radium-223 against skeletal metastases was invented by Roy H. Larsen, Gjermund Henriksen and Øyvind S. Bruland"Preparation and use of radium-223 to target calcified tissues for pain palliation, bone cancer therapy, and bone surface conditioning" US 6635234 and has been developed by the former Norwegian company Algeta ASA, in a partnership with Bayer, under the trade name Xofigo (formerly Alpharadin), and is distributed as a solution containing radium-223 chloride (1100 kBq/ml), sodium chloride, and other ingredients for intravenous injection. M.Pruszynski, A.Majkowska-Pilip, N.Loktionova, E.Eppard, F.Roesch Category:Isotopes of scandium Presence of 244Pu fission tracks can be established by using the initial ratio of 244Pu to 238U (Pu/U)0 at a time T0 = years, when Xe formation first began in meteorites, and by considering how the ratio of Pu/U fission tracks varies over time. In fact, by analyzing data from Earth's mantle which indicates that about 30% of the existing fissiogenic xenon is attributable to 244Pu decay, the timing of Earth's formation can be inferred to have occurred nearly 50–70 million years following the formation of the Solar System. Radium is preferentially absorbed by bone by virtue of its chemical similarity to calcium, with most radium-223 that is not taken up by the bone being cleared, primarily via the gut, and excreted. Radionuclides such as 244Pu undergo decay to produce fissiogenic (i.e., arising from fission) xenon isotopes that can then be used to time the events of the early solar system.

479

672.4

2500.0

67

-3.5

B

A tank originally contains $100 \mathrm{gal}$ of fresh water. Then water containing $\frac{1}{2} \mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \mathrm{~min}$.

The solution is 9 grams of sodium chloride (NaCl) dissolved in water, to a total volume of 1000 ml (weight per unit volume). The state (1, 2), for example, is impossible to reach from an initial state of (0, 0), since (1, 2) has both jugs partially full, and no reversible action is possible from this state. === Jug with initial water === thumb|Starting with 9 liters in the 12-liter jug, the solution for 5 liters is plotted in red on the left, and the solution for 4 liters is plotted in blue on the right. Other assumptions of these problems may include that no water can be spilled, and that each step pouring water from a source jug to a destination jug stops when either the source jug is empty or the destination jug is full, whichever happens first. ==Standard example== The standard puzzle of this kind works with three jugs of capacity 8, 5 and 3 liters. The solver must pour the water so that the first and second jugs both contain 4 units, and the third is empty. Since normal saline contains 9 grams of NaCl, the concentration is 9 grams per litre divided by 58.4 grams per mole, or 0.154 mole per litre. It is most commonly used as a sterile 9 g of salt per litre (0.9%) solution, known as normal saline. In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. Saline (also known as saline solution) is a mixture of sodium chloride (salt) and water. Puzzles of this type ask how many steps of pouring water from one jug to another (until either one jug becomes empty or the other becomes full) are needed to reach a goal state, specified in terms of the volume of liquid that must be present in some jug or jugs. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal. Therefore, there are only two solutions to the problem: :(5, 0) ↔ (0, 5) ↔ (12, 5) ↔ (9, 8) ↔ (9, 0) :(5, 0) ↔ (5, 8) ↔ (12, 1) ↔ (0, 1) ↔ (1, 0) ↔ (1, 8) ↔ (9, 0) For the 4 liter question, since 4\equiv 0 \\! \mod \\!4, one irreversible action is necessary at the start of the solution; It could be simply pouring the whole 9 liters of water from the 12-liter jug to the sink (0,0), or fully fill it to 12 liters from the tap (12,0). Salt water chlorination is a process that uses dissolved salt (1000–36,000 ppm or 1–36 g/L) for the chlorination of swimming pools and hot tubs. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. For example, if one jug that holds 8 liters is empty and the other jug that hold 12 liters has 9 liters of water in it to begin with, then with a source (tap) and a drain (sink), these two jugs can measure volumes of 9 liters, 5 liters, 1 liter, as well as 12 liters, 8 liters, 4 liters and 0 liters. The answer is that the mixtures will be of equal purity. At its simplest, this figure is the result of dividing the lake volume by the flow in or out of the lake. The graph shows two ways to obtain 4 liters using 3-liter and 5-liter jugs, and a water source and sink on a Cartesian grid with diagonal lines of slope −1 (such that x+y=const. on these diagonal lines, which represent pouring water from one jug to the other jug). However, if the osmotic coefficient (a correction for non-ideal solutions) is taken into account, then the saline solution is much closer to isotonic. The question is then posed—which of the two mixtures is purer? The solution still applies no matter how many cups of any sizes and compositions are exchanged, or how little or much stirring at any point in time is done to any barrel, as long as at the end each barrel has the same amount of liquid. However, as the ideal saline concentration of a salt-chlorinated pool is very low (<3,500ppm, the threshold for human perception of salt by taste; seawater is about ten times this concentration), damage usually occurs due to improperly-maintained pool chemistry or improper maintenance of the electrolytic cell.

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0.2553

7.42

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C

A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously. If $r=7.5 \%$, determine $k$ so that $\$ 1$ million will be available for retirement in 40 years.

The return over the five-year period for such an investor would be ($19.90 + $5.78) / $14.21 − 1 = 80.72%, and the arithmetic average rate of return would be 80.72%/5 = 16.14% per year. ==See also== * Annual percentage yield * Average for a discussion of annualization of returns * Capital budgeting * Compound annual growth rate * Compound interest * Dollar cost averaging * Economic value added * Effective annual rate * Effective interest rate * Expected return * Holding period return * Internal rate of return * Modified Dietz method * Net present value * Rate of profit * Return of capital * Return on assets * Return on capital * Returns (economics) * Simple Dietz method * Time value of money * Time-weighted return * Value investing * Yield ==Notes== ==References== ==Further reading== * A. A. Groppelli and Ehsan Nikbakht. Assuming returns are reinvested however, due to the effect of compounding, the relationship between a rate of return r, and a return R over a length of time t is: :1 + R = (1 + r)^t which can be used to convert the return R to a compound rate of return r: :r = (1 + R)^\frac {1}{t} - 1 = \sqrt[t]{1 + R} - 1 For example, a 33.1% return over 3 months is equivalent to a rate of: :\sqrt[3]{1.331} - 1 = 10\% per month with reinvestment. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). If the value of the investment at the end of the second period is C, the holding period return in the second period is: :R_2 = \frac {C-B}{B} Multiplying together the growth factors in each period 1 + R_1 and 1 + R_2: :(1 + R_1)(1 + R_2) = \left( 1 + \frac {B-A}{A} \right) \left( 1 + \frac {C-B}{B} \right) = \left( \frac {B}{A}\right) \left( \frac {C}{B} \right) = \frac {C}{A} :(1 + R_1)(1 + R_2) - 1 = \frac {C}{A} - 1 = \frac {C-A}{A} is the holding period return over the two successive periods. As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1)(12/24) − 1), assuming reinvestment at the end of the first year. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. The return on the deposit over the year in yen terms is therefore: :\frac{1,346,400 - 1,200,000}{1,200,000} = 12.2\% This is the rate of return experienced either by an investor who starts with yen, converts to dollars, invests in the USD deposit, and converts the eventual proceeds back to yen; or for any investor, who wishes to measure the return in Japanese yen terms, for comparison purposes. ====Annualization==== Without any reinvestment, a return R over a period of time t corresponds to a rate of return r: :r = \frac {R}{t} For example, let us suppose that 20,000 USD is returned on an initial investment of 100,000 USD. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. * The final investment value of $103.02 compared with the initial investment of $100 means the return is $3.02 or 3.02%. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. There may also be more than one real solution to the equation, requiring some interpretation to determine the most appropriate one. ====Money-weighted return over multiple sub-periods==== Note that the money- weighted return over multiple sub-periods is generally not equal to the result of combining together the money-weighted returns within the sub-periods using the method described above, unlike time-weighted returns. ===Comparing ordinary return with logarithmic return=== The value of an investment is doubled if the return r = +100%, that is, if r_{\mathrm{log}} = ln($200 / $100) = ln(2) = 69.3%. Over 4 years, this translates into an overall return of: :1.05^4-1=21.55\% Example #2 Volatile rates of return, including losses Year 1 Year 2 Year 3 Year 4 Rate of return 50% −20% 30% −40% Geometric average at end of year 50% 9.5% 16% −1.6% Capital at end of year $150.00 $120.00 $156.00 $93.60 Dollar profit/(loss) ($6.40) The geometric average return over the 4-year period was −1.64%. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). The return over the five-year period is $19.90 × 91.314 / $1,000 − 1 = 81.71% * Geometric average annual total return with reinvestment = ($19.90 × 91.314 / $1,000) ^ (1 / 5) − 1 = 12.69% * An investor who did not reinvest would have received total distributions (cash payments) of $5.78 per share. Accumulation functions for simple and compound interest are a(t)=1 + r t a(t) = \left(1 + \frac {r} {n}\right) ^ {nt} If n t = 1, then these two functions are the same. ===Continuous compounding=== As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. The Summa de arithmetica of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72. * The continuously compounded rate of return in this example is: :\ln\left(\frac{103.02}{100}\right) = 2.98\%. Suppose the value of the investment at the beginning is A, and at the end of the first period is B. The rate of return is 4,000 / 100,000 = 4% per year. Cash flow example on $1,000 investment Year 1 Year 2 Year 3 Year 4 Dollar return $100 $55 $60 $50 ROI 10% 5.5% 6% 5% ==Uses== * Rates of return are useful for making investment decisions.

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Consider the initial value problem $$ y^{\prime \prime}+2 a y^{\prime}+\left(a^2+1\right) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 . $$ For $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). We are trying to find a formula for y(t) that satisfies these two equations. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. The solution is X(t) = e^{At}X(0) with e^{At} = e^{at}\begin{bmatrix} 1 & at \\\ 0 & 1 \end{bmatrix} . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Third, we consider the case where |t| = \frac{1}{2}. Second, we consider the case where |t|>\frac{1}{2}. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. Suppose we have an ordinary differential equation in the complex domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In this case, \begin{cases} 4 \det A - (\operatorname{tr} A)^2 > 0 \\\ \operatorname{tr} A < 0 \end{cases}. The origin is a source, with integral curves of form y = cx^{b/a} ** Similarly for a, b < 0. ** If a > 0 > b or a < 0 < b, then \det A < 0, and the origin is a saddle point. with integral curves of form y = cx^{-|b/a|}. * \begin{bmatrix} a & 1 \\\ 0 & a \end{bmatrix} where a eq 0. If the derivative at is exactly 1 or −1, then more information is needed in order to decide stability.

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Consider the initial value problem $$ y^{\prime \prime}+\gamma y^{\prime}+y=\delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0, $$ where $\gamma$ is the damping coefficient (or resistance). Find the time $t_1$ at which the solution attains its maximum value.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Springer-Verlag, Berlin, 2008. x+506 pp. * Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. A strong maximum principle for parabolic equations. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. The essence of the maximum principle is the simple observation that if each eigenvalue is positive (which amounts to a certain formulation of "ellipticity" of the differential equation) then the above equation imposes a certain balancing of the directional second derivatives of the solution. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. We are trying to find a formula for y(t) that satisfies these two equations. After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the perturbation; sending the perturbation to zero leads to a variational inequality from which the maximum principle follows. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. The characteristic equation of this dynamic equation is \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, whose solutions are the characteristic values \lambda_1,\dots , \lambda_k; these are used in the solution equation :x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Here the constants c_1, \dots , c_k are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition x_t Is known. ===Continuous time=== A differential equation system of the first order with n variables stacked in a vector X is :\frac{dX}{dt}=AX. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. If the final state x(T) is not fixed (i.e., its differential variation is not zero), it must also be that These four conditions in (1)-(4) are the necessary conditions for an optimal control. On the maximum principle of optimal control in economic processes, 1969 (Trondheim, NTH, Sosialøkonomisk institutt https://www.worldcat.org/title/on-the-maximum-principle-of-optimal-control-in- economic-processes/oclc/23714026) ===Textbooks=== * * Evans, Lawrence C. Partial differential equations. Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. The sign of c is relevant, as also seen in the one-dimensional case; for instance the solutions to are exponentials, and the character of the maxima of such functions is quite different from that of sinusoidal functions. == See also == * Maximum modulus principle * Hopf maximum principle ==Notes== ==References== ===Research articles=== * Calabi, E. This differential equation cannot be solved exactly.

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Consider the initial value problem $$ y^{\prime}+\frac{2}{3} y=1-\frac{1}{2} t, \quad y(0)=y_0 . $$ Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in MATLAB. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Another solution is given by : y_s(x) = 0 . The solution y_s is tangent to every curve y_c(x) at the point (c,0). We are trying to find a formula for y(t) that satisfies these two equations. Colors vary from blue to yellow with time. thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in Mathematica. As such, these points satisfy x = 0. ==Using equations== If the curve in question is given as y= f(x), the y-coordinate of the y-intercept is found by calculating f(0). The condition of intersection is : ys(x) = yc(x). If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Functions which are undefined at x = 0 have no y-intercept. The y-intercept of ƒ(x) is indicated by the red dot at (x=0, y=1). As such, these points satisfy y=0. We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. We calculate the derivatives: : y_c'(-2 c) = c \,\\! : y_s'(-2 c) = -\tfrac{1}{2} x |_{x = -2 c} = c. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y.

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A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \mathrm{mg}$ of thorium-234 decays to $82.04 \mathrm{mg}$ in 1 week, determine the decay rate $r$.

The decay constant is \frac{\ln(2)}{t_{1/2}} where "t_{1/2}" is the half-life of the radioactive material of interest. ==Example== The decay correct might be used this way: a group of 20 animals is injected with a compound of interest on a Monday at 10:00 a.m. (A simple way to check if you are using the decay correct formula right is to put in the value of the half-life in place of "t". The exponential time-constant for the process is \tau = R \, C , so the half-life is R \, C \, \ln(2) . For example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days. == Solution of the differential equation == The equation that describes exponential decay is :\frac{dN}{dt} = -\lambda N or, by rearranging (applying the technique called separation of variables), :\frac{dN}{N} = -\lambda dt. The solution to this equation (see derivation below) is: :N(t) = N_0 e^{-\lambda t}, where is the quantity at time , is the initial quantity, that is, the quantity at time . == Measuring rates of decay== === Mean lifetime === If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. If the radiation measured has dropped by half between the 4 hour sample and the 24 hour sample we might think that the concentration of compound in that organ has dropped by half; but applying the decay correct we see that the concentration is 0.5*2.82 so it has actually increased by 40% in that period. ==References== Category:Radioactivity A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. After a day, the decay heat falls to 0.4%, and after a week, it will be only 0.2%. After a day, the decay heat falls to 0.4%, and after a week it will be only 0.2%. This value is in the denominator of the decay correcting fraction, so it is the same as multiplying the numerator by its inverse ( {1 \over 0.3546} ), which is 2.82. Some minerals that contain thorium include apatite, sphene, zircon, allanite, monazite, pyrochlore, thorite, and xenotime. ==Decay== Thorium-232 has a half-life of 14 billion years and mainly decays by alpha decay to radium-228 with a decay energy of 4.0816 MeV. The intermediates in the thorium-232 decay chain are all relatively short- lived; the longest-lived intermediate decay products are radium-228 and thorium-228, with half lives of 5.75 years and 1.91 years, respectively. Substitute 12.7 (hours, the half- life of copper-64) for t_{1/2} , giving {0.693 \over 12.7} = 0.0546. Decay energy is usually quoted in terms of the energy units MeV (million electronvolts) or keV (thousand electronvolts): : Q \text{ [MeV]} = -931.5 \Delta M \text{ [Da]},~~(\text{where }\Delta M = \Sigma M_\text{products} - \Sigma M_\text{reactants}). Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac{dN}{dt} = -\lambda N. The units of the decay constant are s−1. === Derivation of the mean lifetime === Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime, \tau, (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly. The following table lists the intermediate decay products in the thorium-232 decay chain: nuclide decay mode half-life (a=year) energy released, MeV product of decay 232Th α 1.4 a 4.081 228Ra 228Ra β− 5.75 a 0.046 228Ac 228Ac β− 6.15 h 2.134 228Th 228Th α 1.9116 a 5.520 224Ra 224Ra α 3.6319 d 5.789 220Rn 220Rn α 55.6 s 6.405 216Po 216Po α 0.145 s 6.906 212Pb 212Pb β− 10.64 h 0.569 212Bi 212Bi β− 64.06% α 35.94% 60.55 min 2.252 6.207 212Po 208Tl 212Po α 299 ns 8.954 208Pb 208Tl β− 3.053 min 4.999 208Pb 208Pb stable . . . ===Rare decay modes=== Although thorium-232 mainly decays by alpha decay, it also undergoes spontaneous fission 1.1% of the time. The half-life can be written in terms of the decay constant, or the mean lifetime, as: :t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2). The total decay rate of the quantity N is given by the sum of the decay routes; thus, in the case of two processes: :-\frac{dN(t)}{dt} = N\lambda _1 + N\lambda _2 = (\lambda _1 + \lambda _2)N. Therefore, the mean lifetime \tau is equal to the half-life divided by the natural log of 2, or: : \tau = \frac{t_{1/2}}{\ln (2)} \approx 1.44 \cdot t_{1/2}. The decay energy is the energy change of a nucleus having undergone a radioactive decay. Quantitatively, at the moment of reactor shutdown, decay heat from these radioactive sources is still 6.5% of the previous core power if the reactor has had a long and steady power history.

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Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\circ} \mathrm{F}$ when freshly poured, and $1 \mathrm{~min}$ later has cooled to $190^{\circ} \mathrm{F}$ in a room at $70^{\circ} \mathrm{F}$, determine when the coffee reaches a temperature of $150^{\circ} \mathrm{F}$.

Rather, using today's terms, Newton noted after some mathematical manipulation that the rate of temperature change of a body is proportional to the difference in temperatures between the body and its surroundings. In that case, Newton's law only approximates the result when the temperature difference is relatively small. In the study of heat transfer, Newton's law of cooling is a physical law which states that > The rate of heat loss of a body is directly proportional to the difference > in the temperatures between the body and its environment. When the heat transfer coefficient is independent, or relatively independent, of the temperature difference between object and environment, Newton's law is followed. The equation to describe this change in (relatively uniform) temperature inside the object, is the simple exponential one described in Newton's law of cooling expressed in terms of temperature difference (see below). In the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature differences. This second system of measurement led Newton to derive his law of convective heat transfer, also known as Newton's law of cooling. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. When stated in terms of temperature differences, Newton's law (with several further simplifying assumptions, such as a low Biot number and a temperature-independent heat capacity) results in a simple differential equation expressing temperature-difference as a function of time. Radiative cooling is better described by the Stefan–Boltzmann law in which the heat transfer rate varies as the difference in the 4th powers of the absolute temperatures of the object and of its environment. == Mathematical formulation of Newton's law == The statement of Newton's law used in the heat transfer literature puts into mathematics the idea that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. Analytic methods for handling these problems, which may exist for simple geometric shapes and uniform material thermal conductivity, are described in the article on the heat equation. == Application of Newton's law of transient cooling == Simple solutions for transient cooling of an object may be obtained when the internal thermal resistance within the object is small in comparison to the resistance to heat transfer away from the object's surface (by external conduction or convection), which is the condition for which the Biot number is less than about 0.1. Typically, this type of analysis leads to simple exponential heating or cooling behavior ("Newtonian" cooling or heating) since the internal energy of the body is directly proportional to its temperature, which in turn determines the rate of heat transfer into or out of it. By comparison to Newton's original data, they concluded that his measurements (from 1692 to 1693) had been "quite accurate". == Relationship to mechanism of cooling== Convection cooling is sometimes said to be governed by "Newton's law of cooling." This final simplest version of the law, given by Newton himself, was partly due to confusion in Newton's time between the concepts of heat and temperature, which would not be fully disentangled until much later.History of Newton's cooling law In 2020, Maruyama and Moriya repeated Newton's experiments with modern apparatus, and they applied modern data reduction techniques. This characteristic decay of the temperature-difference is also associated with Newton's law of cooling. == Historical background == Isaac Newton published his work on cooling anonymously in 1701 as "Scala graduum Caloris. thumb|A small cup of ice coffee from 85°C Bakery Café. 85 °C Bakery Cafe, also brand-named 85 Cafe, 85 °C Daily Cafe, or 85 Degrees C (), is a Taiwanese international chain of retailers selling coffee, tea, and cakes, as well as desserts, smoothies, fruit juices, souvenirs, and bakery products. Newton's law is most closely obeyed in purely conduction-type cooling. Another situation that does not obey Newton's law is radiative heat transfer. In heat conduction, Newton's Law is generally followed as a consequence of Fourier's law. The name "85°C" refers to Wu's belief that is the optimal temperature to serve coffee. A temperature coefficient describes the relative change of a physical property that is associated with a given change in temperature. A correction to Newton's law concerning convection for larger temperature differentials by including an exponent, was made in 1817 by Dulong and Petit.

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D

Solve the initial value problem $4 y^{\prime \prime}-y=0, y(0)=2, y^{\prime}(0)=\beta$. Then find $\beta$ so that the solution approaches zero as $t \rightarrow \infty$.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. The Newmark-beta method is a method of numerical integration used to solve certain differential equations. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Using the extended mean value theorem, the Newmark-\beta method states that the first time derivative (velocity in the equation of motion) can be solved as, :\dot{u}_{n+1}=\dot{u}_n+ \Delta t~\ddot{u}_\gamma \, where :\ddot{u}_\gamma = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1 therefore :\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Beta regression is a form of regression which is used when the response variable, y, takes values within (0, 1) and can be assumed to follow a beta distribution. Beta (Translation: Son) is a 1992 Indian Hindi drama film, directed by Indra Kumar and written by Naushir Khatau and Kamlesh Pandey. Suppose we have an ordinary differential equation in the complex domain. Average constant acceleration (Middle point rule) (\gamma=0.5 and \beta=0.25 ) is unconditionally stable. ==References== Category:Numerical differential equations This solution has a branchpoint at x=c, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant c). In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. Any given solution y(x) of this equation may well have singularities at various points (i.e. points at which it is not a regular holomorphic function, such as branch points, essential singularities or poles). An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. For example the equation : \frac{dy}{dx} = \frac{1}{2y} has solution y=\sqrt{x-c} for any constant c.

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Consider the initial value problem (see Example 5) $$ y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=2, \quad y^{\prime}(0)=\beta $$ where $\beta>0$. Determine the smallest value of $\beta$ for which $y_m \geq 4$.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. For the closest approximation, the optimum values for \alpha and \beta are \alpha_0 = \frac{2 \cos \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.960433870103... and \beta_0 = \frac{2 \sin \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.397824734759..., giving a maximum error of 3.96%. \alpha\,\\! \beta\,\\! thumb|The locus of points that give the same value in the algorithm, for different values of alpha and beta The alpha max plus beta min algorithm is a high-speed approximation of the square root of the sum of two squares. In the shortest common supersequence problem, two sequences X and Y are given, and the task is to find a shortest possible common supersequence of these sequences. The Newmark-beta method is a method of numerical integration used to solve certain differential equations. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Beta regression is a form of regression which is used when the response variable, y, takes values within (0, 1) and can be assumed to follow a beta distribution. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. Using the extended mean value theorem, the Newmark-\beta method states that the first time derivative (velocity in the equation of motion) can be solved as, :\dot{u}_{n+1}=\dot{u}_n+ \Delta t~\ddot{u}_\gamma \, where :\ddot{u}_\gamma = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1 therefore :\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}. Thus, :u_{n+1}=u_n + \Delta t~\dot{u}_n+\begin{matrix} \frac 1 2 \end{matrix} \Delta t^2~\ddot{u}_\beta where again :\ddot{u}_\beta = (1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}~~~~0\leq 2\beta\leq 1 The discretized structural equation becomes \begin{aligned} &\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}\\\ &u;_{n+1}=u_n + \Delta t~\dot{u}_n + \frac{\Delta t^2}{2}\left((1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}\right)\\\ &M;\ddot{u}_{n+1} + C\dot{u}_{n+1} + f^{\textrm{int}}(u_{n+1}) = f_{n+1}^{\textrm{ext}} \, \end{aligned} Explicit central difference scheme is obtained by setting \gamma=0.5 and \beta=0 Average constant acceleration (Middle point rule) is obtained by setting \gamma=0.5 and \beta=0.25 == Stability Analysis == A time-integration scheme is said to be stable if there exists an integration time-step \Delta t_0 > 0 so that for any \Delta t \in (0, \Delta t_0], a finite variation of the state vector q_n at time t_n induces only a non-increasing variation of the state- vector q_{n+1} calculated at a subsequent time t_{n+1}. Largest error (%) Mean error (%) 1/1 1/2 11.80 8.68 1/1 1/4 11.61 3.20 1/1 3/8 6.80 4.25 7/8 7/16 12.50 4.91 15/16 15/32 6.25 3.08 \alpha_0 \beta_0 3.96 2.41 800px|centre ==Improvements== When \alpha < 1, |z| becomes smaller than \mathbf{Max} (which is geometrically impossible) near the axes where \mathbf{Min} is near 0. We are trying to find a formula for y(t) that satisfies these two equations. Average constant acceleration (Middle point rule) (\gamma=0.5 and \beta=0.25 ) is unconditionally stable. ==References== Category:Numerical differential equations A lower \alpha and higher \beta can therefore increase precision further. In general, the closer the observed y values are to the (a, b) extremes, the more significant the choice of link function.

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A home buyer can afford to spend no more than $\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. Determine the total interest paid during the term of the mortgage.

Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. If the length of the periods are equal (monthly payments) then the summations can be simplified using the formula for a geometric series. For a theoretical continuous payment savings annuity we can only calculate an annual rate of payment: :M_a=\frac{500000 \times 12\%}{e^{0.12\cdot 10}-1}=25860.77 At this point there is a temptation to simply divide by 12 to obtain a monthly payment. "A capital budgeting model of the supply and demand of loanable funds", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. For example, for interest rate of 6% (0.06/12), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives: = PMT(0.06/12, 25 * 12, -150000, 0, 0) = $966.45 ====Approximate formula for monthly payment==== A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (I<8\% and terms T=10–30 years), the monthly note rate is small compared to 1: r << 1 so that the \ln(1+r)\approx r which yields a simplification so that c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}} which suggests defining auxiliary variables Y\equiv n r = IT c_0\equiv \frac{P}{n} . thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. So (for example) the time constant when the interest rate is 10% is 10 years and the period of a home loan should be determined - within the bounds of affordability - as a minimum multiple of this if the objective is to minimise interest paid on the loan. ==Mortgage difference and differential equation== The conventional difference equation for a mortgage loan is relatively straightforward to derive - balance due in each successive period is the previous balance plus per period interest less the per period fixed payment. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator.

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Find the fundamental period of the given function: $$f(x)=\left\{\begin{array}{ll}(-1)^n, & 2 n-1 \leq x<2 n, \\ 1, & 2 n \leq x<2 n+1 ;\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right.$$

Period 2 is the first period in the periodic table from which periodic trends can be drawn. The constant function , where is independent of , is periodic with any period, but lacks a fundamental period. Since we are calculating a sine series, a_n=0\ \quad \forall n Now, b_n= \frac{2}{\pi} \int_0^\pi \cos(x)\sin(nx)\,\mathrm{d}x = \frac{2n((-1)^n+1)}{\pi(n^2-1)}\quad \forall n\ge 2 When n is odd, b_n=0 When n is even, b_n={4n \over \pi(n^2-1)} thus b_{2k}={8k \over \pi(4k^2-1)} With the special case b_1=0, hence the required Fourier sine series is \cos(x) = {{8 \over \pi} \sum_{n=1}^{\infty} {n \over(4n^2-1)}\sin(2nx)} Category:Fourier series In a quantum mechanical description of atomic structure, this period corresponds to the filling of the second () shell, more specifically its 2s and 2p subshells. The fundamental series is a set of spectral lines in a set caused by transition between d and f orbitals in atoms. Example Calculate the half range Fourier sine series for the function f(x)=\cos(x) where 0. This is a list of some well-known periodic functions. A period 2 element is one of the chemical elements in the second row (or period) of the periodic table of the chemical elements. Period 2 only has two metals (lithium and beryllium) of eight elements, less than for any subsequent period both by number and by proportion. The second period contains the elements lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, and neon. Year 186 (CLXXXVI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. Period 2 elements (carbon, nitrogen, oxygen, fluorine and neon) obey the octet rule in that they need eight electrons to complete their valence shell (lithium and beryllium obey duet rule, boron is electron deficient.), where at most eight electrons can be accommodated: two in the 2s orbital and six in the 2p subshell. ==Periodic trends== thumb|Calculated atomic radii of period 2 elements in picometers. In mathematics, a half range Fourier series is a Fourier series defined on an interval [0,L] instead of the more common [-L,L], with the implication that the analyzed function f(x), x\in[0,L] should be extended to [-L,0] as either an even (f(-x)=f(x)) or odd function (f(-x)=-f(x)). Period 1, which only contains two elements (hydrogen and helium), is too small to draw any conclusive trends from it, especially because the two elements behave nothing like other s-block elements. The fundamental series was described as badly-named. thumb|The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... Since the function is defined for all , one can still take the limit as x approaches 1, and this is the definition of the Abel sum: \lim_{x\rightarrow 1^{-}}\sum_{n=1}^\infty n(-x)^{n-1} = \lim_{x\rightarrow 1^{-}}\frac{1}{(1+x)^2} = \frac14. ===Euler and Borel=== right|thumb|Euler summation to − . All period 2 elements completely obey the Madelung rule; in period 2, lithium and beryllium fill the 2s subshell, and boron, carbon, nitrogen, oxygen, fluorine, and neon fill the 2p subshell. They became known as the fundamental series. In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. In 1909 W. M. Hicks produced approximate formulas for the various series and noticed that this series had a simpler formula than the others and thus called it the "fundamental series" and used the letter F. In the case where an = bn = (−1)n, the terms of the Cauchy product are given by the finite diagonal sums \begin{array}{rcl} c_n & = &\displaystyle \sum_{k=0}^n a_k b_{n-k}=\sum_{k=0}^n (-1)^k (-1)^{n-k} \\\\[1em] & = &\displaystyle \sum_{k=0}^n (-1)^n = (-1)^n(n+1). \end{array} The product series is then \sum_{n=0}^\infty(-1)^n(n+1) = 1-2+3-4+\cdots.

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A homebuyer wishes to finance the purchase with a \$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \$900?

thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? For a 24-month loan, the denominator is 300. The FHA went one step further, and set restrictions on the terms and interest rates of qualifying mortgages, typically requiring fully amortizing mortgages to carry terms to maturity in excess of 15 years, with interest rates exceeding 5% annually in only isolated cases. Some mortgage lenders are known to allow as high as 55%. == Monthly payment formula == The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. "A capital budgeting model of the supply and demand of loanable funds", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. For home owners with mortgages the important parameter to keep in mind is the time constant of the equation which is simply the reciprocal of the annual interest rate r. For example, $100,000 mortgaged (without fees, since they add into the calculation in a different way) over 15 years costs a total of $193,429.80 (interest is 93.430% of principal), but over 30 years, costs a total of $315,925.20 (interest is 215.925% of principal). The minimum possible payment rate is that which just covers the loan interest – a borrower would in theory pay this amount forever because there is never any decrease in loan capital. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. S&Ls;, on the other hand, tended to offer 11 to 12 year fully amortizing mortgages, and would generally write mortgages with loan-to-value ratios well in excess of 50%. That is, the APR for a 30-year loan cannot be compared to the APR for a 20-year loan. A simple explanation would be as follows: suppose that the total finance charge for a 12-month loan was $78.00. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. If the fee is not considered, this loan has an effective APR of approximately 80% (1.0512 = 1.7959, which is approximately an 80% increase). Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments.

131

9.73

25.6773

135.36

0.0761

B

A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?

Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. :For a 30-year loan with monthly payments, n = 30 \text{ years} \times 12 \text{ months/year} = 360\text{ months} Note that the interest rate is commonly referred to as an annual percentage rate (e.g. 8% APR), but in the above formula, since the payments are monthly, the rate i must be in terms of a monthly percent. thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. For example, $100,000 mortgaged (without fees, since they add into the calculation in a different way) over 15 years costs a total of $193,429.80 (interest is 93.430% of principal), but over 30 years, costs a total of $315,925.20 (interest is 215.925% of principal). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. The expansion P\approx P_0 \left(1 + X + \frac{X^2}{3}\right) is valid to better than 1% provided X\le 1 . ====Example of mortgage payment==== For a $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find: T=30 I=0.045 which gives X=\frac{1}{2}IT=.675 so that P\approx P_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96 The exact payment amount is P=\$608.02 so the approximation is an overestimate of about a sixth of a percent. ===Investing: monthly deposits=== Given a principal (initial) deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Some mortgage lenders are known to allow as high as 55%. == Monthly payment formula == The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. "A capital budgeting model of the supply and demand of loanable funds", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. A money factor of .0030 is equivalent to a monthly interest rate of 0.6% and an APR of 7.2%. For home owners with mortgages the important parameter to keep in mind is the time constant of the equation which is simply the reciprocal of the annual interest rate r. For a 24-month loan, the denominator is 300. During the second month the borrower has use of two $1000 (2/3) amounts and so the payment should be $1000 plus two $10 interest fees. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). thumb|279px|Parts of total cost and effective APR for a 12-month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. At the end of the month, the borrower pays back one $1000 and the $30 interest. If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The second value evaluated within the context of the above formula will provide the required interest rate. In that case the formula becomes: :: S -A = R (1 + \mathrm{APR}/100)^{-t_N} + \sum_{k=1}^N A_k (1 + \mathrm{APR}/100)^{-t_k} :where: :: S is the borrowed amount or principal amount.

−2

8.87

15.757

27

804.62

E

Let a metallic rod $20 \mathrm{~cm}$ long be heated to a uniform temperature of $100^{\circ} \mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\circ} \mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \mathrm{~s}$ if the bar is made of silver.

The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. While thermal bars can form in both fall and spring, most studies of the thermal bar have investigated aspects of the feature in the spring, when the lake is warming up and the summer thermocline is beginning to form. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. Additional studies have been carried out in Lake Ladoga,S. S. Zilitinkevich, K. D. Kreiman, and A. Yu. Terzhevik, “The Thermal Bar,” Journal of Fluid Mechanics 236, no. 1 (1992): 27-42. In metallurgy, cold forming or cold working is any metalworking process in which metal is shaped below its recrystallization temperature, usually at the ambient temperature. This could be used to model heat conduction in a rod. The distribution approaches equilibrium over time. thumb|The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). Thus, if is the temperature, tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point. Unlike the elastic and electromagnetic waves, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too. ==Specific examples== ===Heat flow in a uniform rod=== For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy . Since the thermal conductivity of ice is so small (1.6 - 2.4 W mK−1) compared with most every other ceramic (ex. Al2O3= 40 W mK−1), the growing ice will have a significant insulative effect on the localized thermal conditions within the slurry. This experiment is possible for ice at −10 °C or cooler, and while essentially valid, the details of the process by which the wire passes through the ice are complex. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. thumb|Classic experiment involving regelation of an ice block as a tensioned wire passes through it. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. Therefore, it is sometimes more economical to cold work a less costly and weaker metal than to hot work a more expensive metal that can be heat treated, especially if precision or a fine surface finish is required as well. In large lakes this condition may persist for weeks, during which a temperature front known as a thermal bar forms between the stratified and unstratified areas of the lake. The thermal resistance of the ceramic is significantly smaller than that of the ice however, so the apparent resistance can be expressed as the lower Rceramic. The name comes from the highly polished appearance of the rods; there is no silver in the alloy. The melting point of ice falls by 0.0072 °C for each additional atm of pressure applied. It has been shown that a linearly decreasing temperature on one side of a freeze-cast will result in near- constant solidification velocity, yielding ice crystals with an almost constant thickness along the SSZ of an entire sample. This approach enables a prediction of the ice-front velocity from the thermal parameters of the suspension.

2598960

7

35.91

-9.54

310

C

Find $\gamma$ so that the solution of the initial value problem $x^2 y^{\prime \prime}-2 y=0, y(1)=1, y^{\prime}(1)=\gamma$ is bounded as $x \rightarrow 0$.

right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in MATLAB. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Colors vary from blue to yellow with time. thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in Mathematica. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In the end, this result shows the interval of definition of the solution does not depend on the Lipschitz constant of the field, but only on the interval of definition of the field and its maximum absolute value. == Other existence theorems == The Picard–Lindelöf theorem shows that the solution exists and that it is unique. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Notice the drastic change in the solutions with respect to the solution obtained with MATLAB. thumb|A chaotic attractor found with parameter values \alpha=1.1 and \gamma=0.87 and initial conditions x_0=-1, y_0=-0, and z_0=0.5, using the default ODE solver in Mathematica. For non-positive integers, the gamma function is not defined. A simple proof of existence of the solution is obtained by successive approximations. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem.

2.8

2

'-6.8'

840

30

B

A tank contains 100 gal of water and $50 \mathrm{oz}$ of salt. Water containing a salt concentration of $\frac{1}{4}\left(1+\frac{1}{2} \sin t\right) \mathrm{oz} / \mathrm{gal}$ flows into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture in the tank flows out at the same rate. The long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?

The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. If used correctly, the Leeson equation gives a useful prediction of oscillator performance in this range. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference. ==Formula for phase of an oscillation or a periodic signal== The phase of an oscillation or signal refers to a sinusoidal function such as the following: :\begin{align} x(t) &= A\cdot \cos( 2 \pi f t + \varphi ) \\\ y(t) &= A\cdot \sin( 2 \pi f t + \varphi ) = A\cdot \cos\left( 2 \pi f t + \varphi - \tfrac{\pi}{2}\right) \end{align} where \textstyle A, \textstyle f, and \textstyle \varphi are constant parameters called the amplitude, frequency, and phase of the sinusoid. However it is possible that the concentrations of some reaction intermediates oscillate, and also that the rate of formation of products oscillates.Espenson, J.H. Chemical Kinetics and Reaction Mechanisms (2nd ed., McGraw-Hill 2002) p.190 Theoretical models of oscillating reactions have been studied by chemists, physicists, and mathematicians. For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude z at a given excitation frequency. For waves on a string, or in a medium such as water, the amplitude is a displacement. Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum. Nonlinear Oscillations is a quarterly peer-reviewed mathematical journal that was established in 1998. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). thumb|Duffing oscillator plot, containing phase plot, trajectory, strange attractor, Poincare section, and double well potential plot. At values of t when the difference is zero, the two signals are said to be in phase, otherwise they are out of phase with each other. thumb|right|A stirred BZ reaction mixture showing changes in color over time A chemical oscillator is a complex mixture of reacting chemical compounds in which the concentration of one or more components exhibits periodic changes. The square of the amplitude is proportional to the intensity of the wave. For arguments t when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. Nonlinear Oscillations is a translation of the Ukrainian journal Neliniyni Kolyvannya (). The common misunderstanding, that is the oscillator output level, may result from derivations that are not completely general. The parameters are (0.5, 0.0625, 0.1, 2.5, 2.0) thumb|The strange attractor of the Duffing oscillator, through 4 periods (8pi time). In the above equation, if is set to zero the equation represents a linear analysis of a feedback oscillator in the general case (and flicker noise is not included), it is for this that Leeson is most recognised, showing a -20 dB/ decade of offset frequency slope. An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos. Derivation of the frequency response Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form: :x = a\, \cos(\omega t) + b\, \sin(\omega t) = z\, \cos(\omega t - \phi), with z^2=a^2+b^2 and \tan\phi = \frac{b}{a}. As a result, : \begin{align} & -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 = \gamma \qquad \text{and} \\\ & -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b = 0. \end{align} Squaring both equations and adding leads to the amplitude frequency response: :\left[\left(\omega^2-\alpha-\frac{3}{4}\beta z^2\right)^{2}+\left(\delta\omega\right)^2\right]\,z^2=\gamma^{2}, as stated above. The amplitude of a non-periodic signal is its magnitude compared with a reference value.

1590

0.24995

'-1.46'

10.4

9

B

A mass weighing $8 \mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\gamma$. Determine the value of $\gamma$ for which the system is critically damped; be sure to give the units for $\gamma$

The graph shows the effect of a tuned mass damper on a simple spring–mass–damper system, excited by vibrations with an amplitude of one unit of force applied to the main mass, m1. Now considering m2 = , the blue line shows the motion of the damping mass and the red line shows the motion of the primary mass. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the "effective total mass" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon "external" factors such as the acceleration of gravity along it. The natural frequency of the tuned mass damper is basically defined by its spring constant and the damping ratio determined by the dashpot. As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). The effective mass of the spring can be determined by finding its kinetic energy. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. As the frequency increases m2 moves out of phase with m1 until at around 9.5 Hz it is 180° out of phase with m1, maximizing the damping effect by maximizing the amplitude of x2 − x1, this maximizes the energy dissipated into c2 and simultaneously pulls on the primary mass in the same direction as the motor mounts. ==Mass dampers in automobiles== ===Motorsport=== The tuned mass damper was introduced as part of the suspension system by Renault on its 2005 F1 car (the Renault R25), at the 2005 Brazilian Grand Prix. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations The split between the two peaks can be changed by altering the mass of the damper (m2). This energy represents the amount of mechanical energy being converted to heat in a volume of material resulting in damping. thumb|Damping capacity ==References== Category:Materials science alt=mass connected to the ground with a spring and damper in parallel|thumb|Classic model used for deriving the equations of a mass spring damper model The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Critically damped means the machine has the right amount of damping torque and is ready to be used for experiments. ==References== 1\. The tuned parameter of the tuned mass damper enables the auxiliary mass to oscillate with a phase shift with respect to the motion of the structure. They are frequently used in power transmission, automobiles and buildings. ==Principle== thumb|A schematic of a simple spring–mass–damper system used to demonstrate the tuned mass damper system Tuned mass dampers stabilize against violent motion caused by harmonic vibration. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *"The Effective Mass of an Oscillating Spring" Am. J. Phys., 38, 98 (1970) *"Effective Mass of an Oscillating Spring" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass Dampers were fitted in response. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Damping torque is provided by indicating instrument. The heights of the two peaks can be adjusted by changing the stiffness of the spring in the tuned mass damper. Finally, if it is critically damped, it has an equal amount of deflection and controlling torque, thus allowing the pointer to quickly find the correct value, without the system oscillating past that value.

8

1000

'-1.78'

773

0.366

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Your swimming pool containing 60,000 gal of water has been contaminated by $5 \mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \mathrm{gal} / \mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \mathrm{~g} / \mathrm{gal}$.

500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. This depends on: the microorganism, the disinfectant being used, the concentration of the disinfectant, the contact time, and the temperature and pH of the water. ==Kinetics== The disinfection kinetics are conventionally calculated via the Chick-Watson model, named for the work of Harriette Chick and H. E. Watson. A portion of such a table is reproduced below. ==Example CT Table== CT Values for the Inactivation of Giardia Cysts by Free Chlorine at 5 °C and pH ≈ 7.0: Chlorine Concentration (mg/L) 1 log inactivation (mg·L−1·min) 2 log inactivation (mg·L−1·min) 3 log inactivation (mg·L−1·min) 0.6 48 95 143 1.2 51 101 152 1.8 54 108 162 2.4 57 115 172 Full tables are much larger than this example and should be obtained from the regulatory agency for a particular jurisdiction. ==See also== * Chlorination * Disinfectant ==References== ==External links== * Category:Water treatment Category:Chlorine A CT value is the product of the concentration of a disinfectant (e.g. free chlorine) and the contact time with the water being disinfected. John Martin-Dye (21 May 1940 – 31 December 2022John Martin-Dye) was a retired British swimmer. ==Swimming career== He won a silver medal in the 4 × 100 m freestyle relay at the 1962 European Aquatics Championships. CT Values are an important part of calculating disinfectant dosage for the chlorination of drinking water. The calculated CT value is the product of the disinfectant residual (in mg/L) and the detention time (in minutes), through the section at peak hourly flow. Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. For example if Lake Michigan was emptied, it would take 99 years for its tributaries to completely refill the lake. ==List of residence times of lake water== The residence time listed is taken from the infobox in the associated article unless otherwise specified. In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. These tables express the required CT values to achieve a desired removal of microorganisms of interest in drinking water (e.g. Giardia lamblia cysts) for a given disinfectant under constant temperature and pH conditions. Where: * (\frac{N}{N_0}) \\! is the survival ratio for the microorganisms being killed * \Lambda_{CW} \\! is the Chick-Watson coefficient of specific lethality * C \\! is the concentration of the disinfectant (typically in mg/L) * n \\! is the coefficient of dilution, frequently assumed to be 1 * t \\! is the contact time (typically in minutes or seconds) The survival ratio is commonly expressed as an inactivation ratio (in %) or as the number of reductions in the order of magnitude of the microorganism concentration. Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. In theory it would be possible to integrate a system of hydrodynamic equations with variable boundary conditions over a very long period sufficient for inflowing water particles to exit the lake. Lake retention time (also called the residence time of lake water, or the water age or flushing time) is a calculated quantity expressing the mean time that water (or some dissolved substance) spends in a particular lake. The concentration of this admixture should be small and the gradient of this concentration should be also small.

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For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity. If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.

The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. The most appropriate are the Reynolds number, given by \mathrm{Re} = \frac{u\sqrt{A}}{ u} and the drag coefficient, given by c_{\rm d} = \frac{F_{\rm d}}{\frac12 \rho A u^2}. The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. An object moving downward faster than the terminal velocity (for example because it was thrown downwards, it fell from a thinner part of the atmosphere, or it changed shape) will slow down until it reaches the terminal velocity. As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). The terminal speed of an object changes due to the properties of the fluid, the mass of the object and its projected cross-sectional surface area. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. Air is 1.293 kg/m3 at 0°C and 1 atmosphere *u is the flow velocity relative to the object, *A is the reference area, and *c_{\rm d} is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass m has to be reduced by the displaced fluid mass \rho V, with V the volume of the object. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. Other than fixed properties of the system, this form of the equation shows that the impact force depends only on the fall factor. For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000.Drag Force For smooth bodies, like a cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 107 (ten million).See Batchelor (1967), p. 341. == Discussion == The equation is easier understood for the idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area. The mass m0 used in the fall is 80 kg. Based on air resistance, for example, the terminal speed of a skydiver in a belly-to-earth (i.e., face down) free fall position is about . The expression for the drag force given by equation () is called Stokes' law.

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Consider the initial value problem $$ 3 u^{\prime \prime}-u^{\prime}+2 u=0, \quad u(0)=2, \quad u^{\prime}(0)=0 $$ For $t>0$ find the first time at which $|u(t)|=10$.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. These problems can come from a more typical initial value problem :u'(t) = f(u(t)), \qquad u(t_0)=u_0, after linearizing locally about a fixed or local state u^*: : L = \frac{\partial f}{\partial u}(u^*); \qquad N = f(u) - L u. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. For example, for the equation (a<0), the stationary solution is , which is obtained for the initial condition . We use their notation, and assume that the unknown function is u, and that we have a known solution u_n at time t_n. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Originally developed for solving stiff differential equations, the methods have been used to solve partial differential equations including hyperbolic as well as parabolic problems, such as the heat equation. ==Introduction== We consider initial value problems of the form, :u'(t) = L u(t) + N(u(t) ), \qquad u(t_0)=u_0, \qquad \qquad (1) where L is composed of linear terms, and N is composed of the non-linear terms. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0.

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Consider the initial value problem $$ 9 y^{\prime \prime}+12 y^{\prime}+4 y=0, \quad y(0)=a>0, \quad y^{\prime}(0)=-1 $$ Find the critical value of $a$ that separates solutions that become negative from those that are always positive.

By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. More generally one might expect :f(\tau)=A \tau^k \left(1+b\tau ^{k_1} + \cdots\right) ==The most important critical exponents== Let us assume that the system has two different phases characterized by an order parameter , which vanishes at and above . A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Now we shall check when these solutions are singular solutions. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Critical exponents describe the behavior of physical quantities near continuous phase transitions. Another solution is given by : y_s(x) = 0 . Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Consider the disordered phase (), ordered phase () and critical temperature () phases separately. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Furthermore, for a given x ot=0, this is the unique solution going through (x,y(x)). ==Failure of uniqueness== Consider the differential equation : y'(x)^2 = 4y(x) . If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Following the standard convention, the critical exponents related to the ordered phase are primed. The set on which a solution is singular may be as small as a single point or as large as the full real line. These can be found by taking two constant c_1 < c_2 and defining a solution y(x) to be (x-c_1)^2 when x < c_1, to be 0 when c_1\leq x\leq c_2, and to be (x-c_2)^2 when x > c_2. Uniqueness fails for these solutions on the interval c_1\leq x\leq c_2, and the solutions are singular, in the sense that the second derivative fails to exist, at x=c_1 and x=c_2. ==Further example of failure of uniqueness== The previous example might give the erroneous impression that failure of uniqueness is directly related to y(x)=0.

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In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) $\vec{a}, 2.0 \mathrm{~km}$ due east (directly toward the east); (b) $\vec{b}, 2.0 \mathrm{~km} 30^{\circ}$ north of east (at an angle of $30^{\circ}$ toward the north from due east); (c) $\vec{c}, 1.0 \mathrm{~km}$ due west. Alternatively, you may substitute either $-\vec{b}$ for $\vec{b}$ or $-\vec{c}$ for $\vec{c}$. What is the greatest distance you can be from base camp at the end of the third displacement? (We are not concerned about the direction.)

The distance from the camp post to the market is (1/4-1/15), which will be repeated 3 times by the camel, and the maximum amount of bananas tranported to the market is 2-(1/4-1/15)*3=1.45 units; *# For m=1/2>(1/3+1/15), another midway camp post is necessary at a distance of 1/3 units from the first camp post, where a total of 1 unit of bananas will accrue. The distance from the second camp post to the market is [1/2-(1/4+1/18)], which will be repeated twice by the camel, and the maximum amount of bananas tranported to the market is 1-[1/2-(1/4+1/18)]*2=0.61 units. The distance from the second camp post to the market is [1/2-(1/3+1/15)], which will be traveled only once by the camel, and the maximum amount of bananas tranported to the market is 1-[1/2-(1/3+1/15)]*1=0.9 units. The other base is at m units of distance away. The other base is at m units of distance away. The distance from the camp post to the market is (1/4-1/18), which will be repeated 4 times by the camel, and the maximum amount of bananas tranported to the market is 2-(1/4-1/18)*4=1.22 units; *# For m=1/2>(1/4+1/18), another midway camp post is necessary at a distance of 1/4 units from the first camp post, where a total of 1 unit of bananas (plus 1/4 units for the camel's final return) will accrue. Here \\{n\\}=n-\lfloor n \rfloor is the fractional part of . === Other variants of the problem === In the camel and bananas problem, assuming the merchant has a total of n=7/3 units of bananas at the starting base and the market is at m units of distance away: * If m> \mathrm{cross}(n)=1/15+1/3+1, no solution exists for this problem; * If m\leq \mathrm{cross}(n)-\mathrm{cross}(\lfloor n\rfloor)=1/15, no midway camp post is necessary for the transport, and the distance m will be repeated for 2 \lceil n \rceil-1=5 times by the camel, so the maximum amount of bananas tranported to the market is n-m\times (2 \lceil n \rceil-1)=7/3-5m; * If \mathrm{cross}(n)-\mathrm{cross}(\lfloor n\rfloor), the optimal solution to transport that maximum amount of bananas requires some midway camp posts: *# For m=1/4<(1/3+1/15), only one midway camp post is necessary at a distance of 1/15 units away from the starting base, where a total of 2 units of bananas will accrue. *At the vertex (0,0), the degree is one: the only possible direction for both climbers to go is onto the mountain. If the camel is required to eventually return to the starting base, then the \mathrm{explore}(n) function applies (still assuming n=7/3): *If m> \mathrm{explore}(n)=1/18+1/4+1/2, no solution exists for this problem; * If m\leq \mathrm{explore}(n)-\mathrm{explore}(\lfloor n\rfloor)=1/18, no midway camp post is necessary for the transport, and the distance m will be repeated for 2 \lceil n \rceil=6 times by the camel, so the maximum amount of bananas tranported to the market is n-m\times (2 \lceil n \rceil)=7/3-6m; * If \mathrm{explore}(n)-\mathrm{explore}(\lfloor n\rfloor), the optimal solution to transport that maximum amount of bananas requires some midway camp posts: *# For m=1/4<(1/4+1/18), only one midway camp post is necessary at a distance of 1/18 units away from the starting base, where a total of 2 units of bananas (plus 1/18 units for the camel's final return) will accrue. A route summit is the highest point on a transportation route crossing higher ground. In summary, the maximum distance reachable by the jeep (with a fuel capacity for 1 unit of distance at any time) in n trips (with n-1 midway fuel dumps and consuming a total of n units of fuel) is * \mathrm{explore}(n)= \frac{1}{2}H_{n}=\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots + \frac{1}{2n}, for exploring the desert where the jeep must return to the base at the end of every trip; * \mathrm{cross}(n)=H_{2n-1}-\frac{1}{2}H_{n-1}=1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} , for crossing the desert where the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel. The maximum distance achievable for the "cross the desert" problem with units of fuel is :\mathrm{cross}(n) = \int_0^n \frac{\mathrm{d}f}{2 \lceil n-f \rceil - 1} with the first fuel dump located at \\{n\\}/(2 \lceil n \rceil-1) units of distance away from the starting base, the second one at 1/(2 \lceil n \rceil-3) units of distance away from the first fuel dump, the third one at 1/(2 \lceil n \rceil-5) units of distance away from the second fuel dump, and so on. thumb|250px|Plot of amount of fuel f vs distance from origin d for exploring (1–3) and crossing (I–III) versions of the jeep problem for three units of fuel - coloured arrows denote depots, diagonal segments denote travel and vertical segments denote fuel transfer The jeep problem, desert crossing problem or exploration problem"Exploration problems. In the general case, the maximum distance achievable for the "explore the desert" problem with units of fuel is :\mathrm{explore}(n) = \int_0^n \frac{\mathrm{d}f}{2 \lceil n-f \rceil} with the first fuel dump located at \\{n\\}/(2 \lceil n \rceil) units of distance away from the starting base, the second one at 1/(2 \lceil n \rceil-2) units of distance away from the first fuel dump, the third one at 1/(2 \lceil n \rceil-4) units of distance away from the second fuel dump, and so on. In either case the objective is to maximize the distance traveled by the jeep on its final trip. *At a vertex where both climbers are at peaks or both climbers are at valleys, the degree is four: both climbers may choose independently of each other which direction to go. If the last and final traveler also needs to return to the starting base, then he would only travel 1/(n+1) unit alone so that he has n/(n+1) units of supply to return, so the longest distance n travelers can reach is :\mathrm{travel_{return}}(n)=\frac{n}{n+1}=1-\frac{1}{n+1} Equating this to m, one may solve for the minimum number of travelers needed to travel m units of distance. A strategy that maximizes the distance traveled on the final trip for the "exploring the desert" variant is as follows: *The jeep makes n trips. A variant of this problem gives the total number of cars available, and asks for the maximum distance that can be reached. ==Solution== thumb|250px|Solution to "exploring the desert" variant for n = 3, showing fuel contents of jeep and fuel dumps at start of each trip and at turnround point on each trip. Mathematical Recreations and Essays, Thirteenth Edition, Dover, p32. . is a mathematics problem in which a jeep must maximize the distance it can travel into a desert with a given quantity of fuel. The problem asks for the minimum number of accompanying travelers needed to reach the other base. thumb|Approach (α) and departure angle (β) of a vehicle Approach angle is the maximum angle of a ramp onto which a vehicle can climb from a horizontal plane without interference.

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"Top gun" pilots have long worried about taking a turn too tightly. As a pilot's body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to loss of brain function. There are several warning signs. When the centripetal acceleration is $2 g$ or $3 g$, the pilot feels heavy. At about $4 g$, the pilot's vision switches to black and white and narrows to "tunnel vision." If that acceleration is sustained or increased, vision ceases and, soon after, the pilot is unconscious - a condition known as $g$-LOC for " $g$-induced loss of consciousness." What is the magnitude of the acceleration, in $g$ units, of a pilot whose aircraft enters a horizontal circular turn with a velocity of $\vec{v}_i=(400 \hat{\mathrm{i}}+500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ and $24.0 \mathrm{~s}$ later leaves the turn with a velocity of $\vec{v}_f=(-400 \hat{\mathrm{i}}-500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ ?

Incidents of acceleration-induced loss of consciousness have caused fatal accidents in aircraft capable of sustaining high-g for considerable periods. Skilled pilots can use this loss of vision as their indicator that they are at maximum turn performance without losing consciousness. The danger of g-LOC to aircraft pilots is magnified because on relaxation of g-force there is a period of disorientation before full sensation is re-gained. If G-LOC occurs at low altitude, this momentary lapse can prove fatal and even highly experienced pilots can pull straight to a G-LOC condition without first perceiving the visual onset warnings that would normally be used as the sign to back off from pulling any more gs. A further increase in g-forces will cause g-LOC where consciousness is lost. g-force induced loss of consciousness (abbreviated as G-LOC, pronounced "JEE- lock") is a term generally used in aerospace physiology to describe a loss of consciousness occurring from excessive and sustained g-forces draining blood away from the brain causing cerebral hypoxia. If g-forces increase further, complete loss of vision will occur, while consciousness remains. High-g training for pilots of high performance aircraft or spacecraft often includes ground training for G-LOC in special centrifuges, with some profiles exposing pilots to 9 gs for a sustained period. ==Effects of g-forces== Under increasing positive g-force, blood in the body will tend to move from the head toward the feet. In general, when the g-force pushes the body forwards (colloquially known as 'eyeballs in'NASA Physiological Acceleration Systems ) a much higher tolerance is shown than when g-force is pushing the body backwards ('eyeballs out') since blood vessels in the retina appear more sensitive to that direction.NASA Technical note D-337, Centrifuge Study of Pilot Tolerance to Acceleration and the Effects of Acceleration on Pilot Performance, by Brent Y. Creer, Captain Harald A. Smedal, USN (MC), and Rodney C. Wingrove ==G-suits== A g-suit is worn by aviators and astronauts who are subject to high levels of acceleration and, hence, increasing positive g. thumb|upright=1.2|If the cockpit lost pressure while the aircraft was above the Armstrong limit, even a positive pressure oxygen mask could not sustain pilot consciousness. This is not true in 0 g when you strafe up; that is an eyeballs-down maneuver, which is the same force as a blackout where blood rushes to the feet, and this force is parallel to the spine. A pilot aiming for a 500-foot per minute descent, for example, may find themselves descending more rapidly than intended. This is doubly dangerous because, on recovery as g is reduced, a period of several seconds of disorientation occurs, during which the aircraft can dive into the ground. thumb|A pilot in a Cessna 152 performing a steep turn as seen from the cockpit. The reverse effect is experienced in advanced aerobatic maneuvers under negative g-forces, where excess blood moves towards the brain and eyes ("redout"). Black-out and g-LOC have caused a number of fatal aircraft accidents.Amos, Smith, "Report: Blue Angels pilot became disoriented" , Military Times, January 16, 2008. == Operation == If blood is allowed to pool in the lower areas of the body, the brain will be deprived of blood. By using a modern g-suit in combination with anti-g strain techniques, a trained pilot is now expected to endure accelerations of up to nine g without blacking out. A trained, fit individual wearing a g suit and practising the straining maneuver can, with some difficulty, sustain up to 9 g without loss of consciousness. A trained, fit individual wearing a g suit and practicing the straining manoeuvre can, with some difficulty, sustain up to 12-14g without loss of consciousness. Upon regaining cerebral blood flow, the G-LOC victim usually experiences myoclonic convulsions (often called the ‘funky chicken’) and often full amnesia of the event is experienced. The most dangerous pilot-induced oscillations can occur during landing. * G-LOC – where consciousness is lost.

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The world’s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length L of the string in the ball?

If the string should be raised off the ground, all the way along the equator, how much longer would the string be? Even more surprising is that the size of the sphere or circle around which the string is spanned is irrelevant, and may be anything from the size of an atom to the Milky Way -- the result depends only on the amount it is raised. The scale length of a string instrument is the maximum vibrating length of the strings that produce sound, and determines the range of tones that string can produce at a given tension. The question that is then posed is whether the gap between string and Earth will allow the passage of a car, a cat or a thin knife blade. ==Solution== thumb|Visual analogue using a square As the string must be raised all along the entire circumference, one might expect several kilometres of additional string. 300px|thumb|right|A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1). 275px|thumb|Stick figure of 1.75 meters standing next to a violin string of .33 meters and a long string instrument string of 10 meters. On instruments in which strings are not "stopped" (typically by frets or the player's fingers) or divided in length (such as in the piano), it is the actual length of string between the nut and the bridge. More formally, let c be the Earth's circumference, r its radius, Δc the added string length and Δr the added radius. has a circumference of 2R , : \begin{align} c + \varDelta c & = 2 \pi (r + \varDelta r) \\\ 2 \pi r + \varDelta c & = 2 \pi r + 2 \pi \varDelta r \\\ \varDelta c & = 2 \pi \varDelta r \\\ \therefore\; \varDelta r & = \frac{\varDelta c}{2 \pi} \end{align} regardless of the value of c . In a version of this puzzle, string is tightly wrapped around the equator of a perfectly spherical Earth. For example if a string 0.33 meters long, of given mass and tension, produces A440, a string with identical mass and tension but eight times as long, 2.64 meters, produces 55 hertz. thumb|300px|Visualisation showing that the length added to the circumference (blue) is dependent only on the additional radius (red) and not the original circumference (grey) String girdling Earth is a mathematical puzzle with a counterintuitive solution. Elevenstring may refer to: * The eleven-string alto guitar * A fictional musical instrument in The Hydrogen Sonata by Iain M. Banks It is also called string length. Alternatively, of string is spliced into the original string, and the extended string rearranged so that it is at a uniform height above the equator. The long-string instrument is a musical instrument in which the string is of such a length that the fundamental transverse wave is below what a person can hear as a tone (±20 Hz). In the second phrasing, considering that is almost negligible compared with the circumference, the first response may be that the new position of the string will be no different from the original surface-hugging position. This long-string instrument's range is centered on the octave of middle C and extends above and below this by an octave. The string spelled by the edges from the root to such a node is a longest repeated substring. If the tension on a string is ten lbs., it must be increased to 40 lbs. for a pitch an octave higher. Springer. p.10. . ==Equations== The natural frequency is: *a) Inversely proportional to the length of the string (the law of Pythagoras), *b) Proportional to the square root of the stretching force, and *c) Inversely proportional to the square root of the mass per length. : f_0 \propto \tfrac{1}{L}. (equation 26) : f_0 \propto \sqrt{F}. (equation 27) : f_0 \propto \frac{1}{\sqrt{\mu}}. (equation 28) Thus, for example, all other properties of the string being equal, to make the note one octave higher (2/1) one would need either to decrease its length by half (1/2), to increase the tension to the square (4), or to decrease its mass per length by the inverse square (1/4). The Long- String Instrument. The strings of the bass octave extend the instrument's full 90 feet.

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You drive a beat-up pickup truck along a straight road for $8.4 \mathrm{~km}$ at $70 \mathrm{~km} / \mathrm{h}$, at which point the truck runs out of gasoline and stops. Over the next $30 \mathrm{~min}$, you walk another $2.0 \mathrm{~km}$ farther along the road to a gasoline station. What is your overall displacement from the beginning of your drive to your arrival at the station?

A displacement may be also described as a relative position (resulting from the motion), that is, as the final position of a point relative to its initial position . Displacement is usually measured in units of tonnes or long tons. ==Definitions== There are terms for the displacement of a vessel under specified conditions: ===Loaded displacement=== *Loaded displacement is the weight of the ship including cargo, passengers, fuel, water, stores, dunnage and such other items necessary for use on a voyage. thumb|Displacement versus distance travelled along a path In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. Engine displacement is the measure of the cylinder volume swept by all of the pistons of a piston engine, excluding the combustion chambers. right|thumb|upright=1.3|Load lines, by showing how low a ship is sitting in the water, make it possible to determine displacement. In fluid mechanics, displacement occurs when an object is largely immersed in a fluid, pushing it out of the way and taking its place. The displacement or displacement tonnage of a ship is its weight. The process of determining a vessel's displacement begins with measuring its draft.George, 2005. p.5. Ship displacement varies by a vessel's degree of load, from its empty weight as designed (known as "lightweight tonnage") to its maximum load. A displacement may be identified with the translation that maps the initial position to the final position. For this reason displacement is one of the measures often used in advertising, as well as regulating, motor vehicles. The corresponding displacement vector can be defined as the difference between the final and initial positions: s = x_\textrm{f} - x_\textrm{i} = \Delta{x} In considering motions of objects over time, the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The weight of the displaced fluid can be found mathematically. Therefore, the weight of the displaced fluid can be expressed as . The ship's hydrostatic tables show the corresponding volume displaced.George, 2005. p. 465. The fluid displaced has a weight , where is acceleration due to gravity. Ship displacement should not be confused with measurements of volume or capacity typically used for commercial vessels and measured by tonnage: net tonnage and gross tonnage. ==Calculation== thumb|Shipboard stability computer programs can be used to calculate a vessel's displacement. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering and physics. For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity, which is a vector, and differs thus from the average speed, which is a scalar quantity. == Rigid body == In dealing with the motion of a rigid body, the term displacement may also include the rotations of the body. Therefore, they are generally taxed as 1.5 times their stated physical displacement (1.3 litres becomes effectively 2.0, 2.0 becomes effectively 3.0), although actual power outputs can be higher than suggested by this conversion factor. == Automotive model names == Historically, many car model names have included their engine displacement. The formula is: : \text{Displacement} = \text{stroke length} \times \pi \left(\frac{\text{bore}}{2}\right)^2 \times \text{number of cylinders} Using this formula for non-typical types of engine, such as the Wankel design and the oval-piston type used in Honda NR motorcycles, can sometimes yield misleading results when attempting to compare engines.

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A heavy object can sink into the ground during an earthquake if the shaking causes the ground to undergo liquefaction, in which the soil grains experience little friction as they slide over one another. The ground is then effectively quicksand. The possibility of liquefaction in sandy ground can be predicted in terms of the void ratio $e$ for a sample of the ground: $$ e=\frac{V_{\text {voids }}}{V_{\text {grains }}} . $$ Here, $V_{\text {grains }}$ is the total volume of the sand grains in the sample and $V_{\text {voids }}$ is the total volume between the grains (in the voids). If $e$ exceeds a critical value of 0.80 , liquefaction can occur during an earthquake. What is the corresponding sand density $\rho_{\text {sand }}$ ? Solid silicon dioxide (the primary component of sand) has a density of $\rho_{\mathrm{SiO}_2}=2.600 \times 10^3 \mathrm{~kg} / \mathrm{m}^3$.

Eng., 139(3), 407–419. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000743 ] #The earthquake load, measured as cyclic stress ratio CSR=\frac{\tau_{av}}{\sigma'_{v}}=0,65\frac{a_{max}}{g}\frac{\sigma_{v}}{\sigma'_{v}}r_d Evaluation of soil liquefaction from surface analysis #the capacity of the soil to resist liquefaction, expressed in terms of the cyclic resistance ratio (CRR) ==Earthquake liquefaction== Pressures generated during large earthquakes can force underground water and liquefied sand to the surface. Soil liquefaction may occur in partially saturated soil when it is shaken by an earthquake or similar forces. Aluminium, for example, has a density of about 2.7 grams per cubic centimeter, but a piece of aluminium will float on top of quicksand until motion causes the sand to liquefy. The saturated sediment may appear quite solid until a change in pressure or a shock initiates the liquefaction, causing the sand to form a suspension with each grain surrounded by a thin film of water. "Liquefaction of sands and its evaluation.", Proceedings of the 1st International Conference on Earthquake Geotechnical Engineering, Tokyo The resistance of the cohesionless soil to liquefaction will depend on the density of the soil, confining stresses, soil structure (fabric, age and cementation), the magnitude and duration of the cyclic loading, and the extent to which shear stress reversal occurs. ==Liquefaction potential: simplified empirical analysis== Three parameters are needed to assess liquefaction potential using the simplified empirical method: #A measure of soil resistance to liquefaction: Standard Penetration Resistance (SPT), [Cetin, K.O., Seed, R.B., Armen Der Kiureghian, M., Tokimatsu, K., Harder, L.F. Jr., Kayen, R.E., Moss, R.E.S. (2004) SPT-Based Probabilistic and Deterministic Assessment of Seismic Soil Liquefaction Potential, Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 130, No. 12, December 2004, pp. 1314-1340. http://ascelibrary.org/doi/abs/10.1061/%28ASCE%291090-0241%282004%29130%3A12%281314%29 ] [I.M. Idriss, Ross W. Boulanger, 2nd Ishihara Lecture: SPT- and CPT-based relationships for the residual shear strength of liquefied soils, Soil Dynamics and Earthquake Engineering, Volume 68, 2015, Pages 57 68, ISSN 0267-7261, https://doi.org/10.1016/j.soildyn.2014.09.010.] Thus, shear strains are significantly less than a true state of soil liquefaction. ==Occurrence== Liquefaction is more likely to occur in loose to moderately saturated granular soils with poor drainage, such as silty sands or sands and gravels containing impermeable sediments. Soil liquefaction occurs when a cohesionless saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress such as shaking during an earthquake or other sudden change in stress condition, in which material that is ordinarily a solid behaves like a liquid. One positive aspect of soil liquefaction is the tendency for the effects of earthquake shaking to be significantly damped (reduced) for the remainder of the earthquake. Liquefaction is more likely to occur in sandy or non-plastic silty soils, but may in rare cases occur in gravels and clays (see quick clay). In the case of earthquakes, the shaking force can increase the pressure of shallow groundwater, liquefying sand and silt deposits. If the soil strain-hardens, e.g. moderately dense to dense sand, flow liquefaction will generally not occur. Soil liquefaction induced by earthquake shaking is a major contributor to urban seismic risk. ==Effects== The effects of soil liquefaction on the built environment can be extremely damaging. Objects in liquefied sand sink to the level at which the weight of the object is equal to the weight of the displaced soil/water mix and the submerged object floats due to its buoyancy. Deposits most susceptible to liquefaction are young (Holocene-age, deposited within the last 10,000 years) sands and silts of similar grain size (well- sorted), in beds at least metres thick, and saturated with water. When water in the sand cannot escape, it creates a liquefied soil that loses strength and cannot support weight. When the water trapped in the batch of sand cannot escape, it creates liquefied soil that can no longer resist force. The saturated sediment may appear quite solid until a sudden change in pressure or shock initiates liquefaction. Recent Advances in Soil Liquefaction Engineering: A Unified and Consistent Framework, 26th Annual ASCE Los Angeles Geotechnical Spring Seminar, Long Beach, California, April 30, 2003, Earthquake Engineering Research Center ==External links== * Soil Liquefaction * Liquefaction – Pacific Northwest Seismic Network * Liquefaction in Chiba, Japan on YouTube recorded during the 2011 Tohoku earthquake Category:Earthquake engineering Liquifaction, soil Category:Sedimentology Category:Seismology Category:Soil mechanics Category:Natural disasters By weight, its mineral composition is about 40–40–20% concentration of sand–silt–clay, respectively. The phenomenon of dilatancy can be observed in a drained simple shear test on a sample of dense sand. Examples of soil liquefaction include quicksand, quick clay, turbidity currents and earthquake- induced liquefaction. To move within the quicksand, a person or object must apply sufficient pressure on the compacted sand to re-introduce enough water to liquefy it.

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What is the angle $\phi$ between $\vec{a}=3.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}$ and $\vec{b}=$ $-2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{k}}$ ? (Caution: Although many of the following steps can be bypassed with a vector-capable calculator, you will learn more about scalar products if, at least here, you use these steps.)

===Vector rejection=== By definition, \mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 Hence, \mathbf{a}_2 = \mathbf{a} - \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}}. ==Properties== ===Scalar projection=== The scalar projection on is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees. Hence, a_2 = \left\|\mathbf{a}\right\| \sin \theta = \frac {\mathbf{a} \cdot \mathbf{b}^\perp} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a}_y \mathbf{b}_x - \mathbf{a}_x \mathbf{b}_y} {\left\|\mathbf{b}\right\| }. Which finally gives: \mathbf{a}_1 = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\|^2}{\mathbf{b}} = \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}} ~ . In mathematics, the scalar projection of a vector \mathbf{a} on (or onto) a vector \mathbf{b}, also known as the scalar resolute of \mathbf{a} in the direction of \mathbf{b}, is given by: :s = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b}, where the operator \cdot denotes a dot product, \hat{\mathbf{b}} is the unit vector in the direction of \mathbf{b}, \left\|\mathbf{a}\right\| is the length of \mathbf{a}, and \theta is the angle between \mathbf{a} and \mathbf{b}. Namely, it is defined as \mathbf{a}_1 = a_1 \mathbf{\hat b} = (\left\|\mathbf{a}\right\| \cos \theta) \mathbf{\hat b} where a_1 is the corresponding scalar projection, as defined above, and \mathbf{\hat b} is the unit vector with the same direction as : \mathbf{\hat b} = \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|} ===Vector rejection=== By definition, the vector rejection of on is: \mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 Hence, \mathbf{a}_2 = \mathbf{a} - \left(\left\|\mathbf{a}\right\| \cos \theta\right) \mathbf{\hat b} ==Definitions in terms of a and b== When is not known, the cosine of can be computed in terms of and , by the following property of the dot product \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta ===Scalar projection=== By the above- mentioned property of the dot product, the definition of the scalar projection becomes: a_1 = \left\|\mathbf{a}\right\| \cos \theta = \frac {\mathbf{a} \cdot \mathbf{b}} { \left\|\mathbf{b}\right\|}. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. thumb|300px|Geometric interpretation of the angle between two vectors defined using an inner product alt=Scalar product spaces, inner product spaces, Hermitian product spaces.|thumb|300px|Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Multiplying the scalar projection of \mathbf{a} on \mathbf{b} by \mathbf{\hat b} converts it into the above-mentioned orthogonal projection, also called vector projection of \mathbf{a} on \mathbf{b}. ==Definition based on angle θ== If the angle \theta between \mathbf{a} and \mathbf{b} is known, the scalar projection of \mathbf{a} on \mathbf{b} can be computed using :s = \left\|\mathbf{a}\right\| \cos \theta . (s = \left\|\mathbf{a}_1\right\| in the figure) ==Definition in terms of a and b== When \theta is not known, the cosine of \theta can be computed in terms of \mathbf{a} and \mathbf{b}, by the following property of the dot product \mathbf{a} \cdot \mathbf{b}: : \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta By this property, the definition of the scalar projection s becomes: : s = \left\|\mathbf{a}_1\right\| = \left\|\mathbf{a}\right\| \cos \theta = \left\|\mathbf{a}\right\| \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| }\, ==Properties== The scalar projection has a negative sign if 90^\circ < \theta \le 180^\circ. The scalar projection is a scalar, equal to the length of the orthogonal projection of \mathbf{a} on \mathbf{b}, with a negative sign if the projection has an opposite direction with respect to \mathbf{b}. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. More exactly, if the vector projection is denoted \mathbf{a}_1 and its length \left\|\mathbf{a}_1\right\|: : s = \left\|\mathbf{a}_1\right\| if 0^\circ \le \theta \le 90^\circ, : s = -\left\|\mathbf{a}_1\right\| if 90^\circ < \theta \le 180^\circ. ==See also== * Scalar product * Cross product * Vector projection ==Sources== *Dot products - www.mit.org *Scalar projection - Flexbooks.ck12.org *Scalar Projection & Vector Projection - medium.com *Lesson Explainer: Scalar Projection | Nagwa Category:Operations on vectors In turn, the scalar projection is defined as a_1 = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b} where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and θ is the angle between and . It coincides with the length of the vector projection if the angle is smaller than 90°. Any vector field can be written in terms of the unit vectors as: \mathbf A = A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z} = A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi} The spherical unit vectors are related to the Cartesian unit vectors by: \begin{bmatrix}\boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\\ -\sin\phi & \cos\phi & 0 \end{bmatrix} \begin{bmatrix} \mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. It is a vector parallel to , defined as \mathbf{a}_1 = a_1\mathbf{\hat b} where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. In two dimensions, this becomes a_1 = \frac {\mathbf{a}_x \mathbf{b}_x + \mathbf{a}_y \mathbf{b}_y} {\left\|\mathbf{b}\right\|}. ===Vector projection=== Similarly, the definition of the vector projection of onto becomes: \mathbf{a}_1 = a_1 \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|}, which is equivalent to either \mathbf{a}_1 = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b}, or \mathbf{a}_1 = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\|^2}{\mathbf{b}} = \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}} ~ . ===Scalar rejection=== In two dimensions, the scalar rejection is equivalent to the projection of onto \mathbf{b}^\perp = \begin{pmatrix}-\mathbf{b}_y & \mathbf{b}_x\end{pmatrix}, which is \mathbf{b} = \begin{pmatrix}\mathbf{b}_x & \mathbf{b}_y\end{pmatrix} rotated 90° to the left. The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . This means that \mathbf{A} = \mathbf{P} = \rho \mathbf{\hat \rho} + z \mathbf{\hat z}. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. The vector projection of on and the corresponding rejection are sometimes denoted by and , respectively. ==Definitions based on angle θ== ===Scalar projection=== The scalar projection of on is a scalar equal to a_1 = \left\|\mathbf{a}\right\| \cos \theta , where θ is the angle between and .

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During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement $\vec{d}=(-3.0 \mathrm{~m}) \hat{\mathrm{i}}$ while a steady wind pushes against the crate with a force $\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}+(-6.0 \mathrm{~N}) \hat{\mathrm{j}}$. The situation and coordinate axes are shown in Figure. If the crate has a kinetic energy of $10 \mathrm{~J}$ at the beginning of displacement $\vec{d}$, what is its kinetic energy at the end of $\vec{d}$ ?

Kinetic energy is the movement energy of an object. The velocity v of the car can be determined from the length of the skid using the work–energy principle, kWs = \frac{W}{2g} v^2,\quad\text{or}\quad v = \sqrt{2ksg}. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. Where e_k is the specific kinetic energy and v is velocity. The work of the net force is calculated as the product of its magnitude and the particle displacement. The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass. Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). It is known as the work–energy principle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = m \int_{t_1}^{t_2} \mathbf{a} \cdot \mathbf{v}dt = \frac{m}{2} \int_{t_1}^{t_2} \frac{d v^2}{dt}\,dt = \frac{m}{2} \int_{v^2_1}^{v^2_2} d v^2 = \frac{mv_2^2}{2} - \frac{mv_1^2}{2} = \Delta E_\text{k} The identity \mathbf{a} \cdot \mathbf{v} = \frac{1}{2} \frac{d v^2}{dt} requires some algebra. John Wiley & Sons :E_\text{k} = \frac{1}{2} mv^2 Dividing by V, the unit of volume: :\begin{align} \frac{E_\text{k}}{V} &= \frac{1}{2} \frac{m}{V}v^2 \\\ q &= \frac{1}{2} \rho v^2 \end{align} where q is the dynamic pressure, and ρ is the density of the incompressible fluid. ===Frame of reference=== The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. Specific kinetic energy is kinetic energy of an object per unit of mass. For example: the energy of TNT is 4.6 MJ/kg, and the energy of a kinetic kill vehicle with a closing speed of is 50 MJ/kg. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. The work done is given by the dot product of the two vectors. Assuming the object was at rest at time 0, we integrate from time 0 to time t because the work done by the force to bring the object from rest to velocity v is equal to the work necessary to do the reverse: :E_\text{k} = \int_0^t \mathbf{F} \cdot d \mathbf{x} = \int_0^t \mathbf{v} \cdot d (m \mathbf{v}) = \int_0^t d \left(\frac{m v^2}{2}\right) = \frac{m v^2}{2}.

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When the force on an object depends on the position of the object, we cannot find the work done by it on the object by simply multiplying the force by the displacement. The reason is that there is no one value for the force-it changes. So, we must find the work in tiny little displacements and then add up all the work results. We effectively say, "Yes, the force varies over any given tiny little displacement, but the variation is so small we can approximate the force as being constant during the displacement." Sure, it is not precise, but if we make the displacements infinitesimal, then our error becomes infinitesimal and the result becomes precise. But, to add an infinite number of work contributions by hand would take us forever, longer than a semester. So, we add them up via an integration, which allows us to do all this in minutes (much less than a semester). Force $\vec{F}=\left(3 x^2 \mathrm{~N}\right) \hat{\mathrm{i}}+(4 \mathrm{~N}) \hat{\mathrm{j}}$, with $x$ in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates $(2 \mathrm{~m}, 3 \mathrm{~m})$ to $(3 \mathrm{~m}, 0 \mathrm{~m})$ ? Does the speed of the particle increase, decrease, or remain the same?

The work of the net force is calculated as the product of its magnitude and the particle displacement. The small amount of work by the forces over the small displacements can be determined by approximating the displacement by so \delta W = \mathbf{F}_1\cdot\mathbf{V}_1\delta t+\mathbf{F}_2\cdot\mathbf{V}_2\delta t + \ldots + \mathbf{F}_n\cdot\mathbf{V}_n\delta t or \delta W = \sum_{i=1}^n \mathbf{F}_i\cdot (\boldsymbol{\omega}\times(\mathbf{X}_i-\mathbf{d}) + \dot{\mathbf{d}})\delta t. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The actual force increased from 0 to F during the deformation; the work done can be computed by integration in distance. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. The small amount of work that occurs over an instant of time is calculated as \delta W = \mathbf{F} \cdot d\mathbf{s} = \mathbf{F} \cdot \mathbf{v}dt where the is the power over the instant . Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector), and the velocity vector of the point of application. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The time derivative of the integral for work yields the instantaneous power, \frac{dW}{dt} = P(t) = \mathbf{F}\cdot \mathbf{v} . ===Path independence=== If the work for an applied force is independent of the path, then the work done by the force, by the gradient theorem, defines a potential function which is evaluated at the start and end of the trajectory of the point of application. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. Therefore, work need only be computed for the gravitational forces acting on the bodies. Notice that the work done by gravity depends only on the vertical movement of the object. The work done by a conservative force is equal to the negative of change in potential energy during that process.

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The charges of an electron and a positron are $-e$ and $+e$. The mass of each is $9.11 \times 10^{-31} \mathrm{~kg}$. What is the ratio of the electrical force to the gravitational force between an electron and a positron?

The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. While the electron has a negative electric charge, the positron has a positive electric charge, and is produced naturally in certain types of radioactive decay. It turns out that gravitational forces are negligible on the subatomic level, due to the extremely small masses of subatomic particles. === Electron === The electron charge-to-mass quotient, -e/m_{e}, is a quantity that may be measured in experimental physics. Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. In this approximation, the energy levels are different because of a different effective mass, μ, in the energy equation (see electron energy levels for a derivation): E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2}, where: * is the charge magnitude of the electron (same as the positron), * is the Planck constant, * is the electric constant (otherwise known as the permittivity of free space), * is the reduced mass: \mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2}, where and are, respectively, the mass of the electron and the positron (which are the same by definition as antiparticles). Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. So finally, the energy levels of positronium are given by E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}. Approximately: * ~60% of positrons will directly annihilate with an electron without forming positronium. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Positron paths in a cloud- chamber trace the same helical path as an electron but rotate in the opposite direction with respect to the magnetic field direction due to their having the same magnitude of charge-to-mass ratio but with opposite charge and, therefore, opposite signed charge-to-mass ratios. As with any two charged objects, electrons and positrons may also interact with each other without annihilating, in general by elastic scattering. ==Low-energy case== There are only a very limited set of possibilities for the final state. Rearranging, it is possible to solve for the charge-to-mass ratio of an electron as \frac{e}{m} = \frac{4 \pi c}{B(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})}\frac{\delta D}{D\Delta D} \, . ==See also== *Gyromagnetic ratio *Thomson (unit) == References== == Bibliography == * * * CC. However, because of the reduced mass, the frequencies of the spectral lines are less than half of those for the corresponding hydrogen lines. ==States== The mass of positronium is 1.022 MeV, which is twice the electron mass minus the binding energy of a few eV. Corrections that involved higher orders were then calculated in a non-relativistic quantum electrodynamics. == Exotic compounds == Molecular bonding was predicted for positronium. Positronium can also be considered by a particular form of the two-body Dirac equation; Two particles with a Coulomb interaction can be exactly separated in the (relativistic) center-of-momentum frame and the resulting ground-state energy has been obtained very accurately using finite element methods of Janine Shertzer. The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to- charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. Another difficulty was the difference in masses of the electron and the proton. This gives : = , which is much larger than the length scale associated with the electron's charge. By doing this, he showed that the electron was in fact a particle with a mass and a charge, and that its mass-to-charge ratio was much smaller than that of the hydrogen ion H+. thumb|350px|Naturally occurring electron-positron annihilation as a result of beta plus decay Electron–positron annihilation occurs when an electron () and a positron (, the electron's antiparticle) collide. Positrons, because of the direction that their paths curled, were at first mistaken for electrons travelling in the opposite direction. In 1897, the mass-to-charge ratio of the electron was first measured by J. J. Thomson.J. J. Thomson (1856–1940) Philosophical Magazine, 44, 293 (1897).

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In Fig. 21-26, particle 1 of charge $+q$ and particle 2 of charge $+4.00 q$ are held at separation $L=9.00 \mathrm{~cm}$ on an $x$ axis. If particle 3 of charge $q_3$ is to be located such that the three particles remain in place when released, what must be the $x$ coordinate of particle 3?

The position coordinates xj and xk are replaced > by their relative position rjk = xj − xk and by the vector to their center > of mass Rjk = (mj qj \+ mkqk)/(mj \+ mk). thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. A trion is a localized excitation which consists of three charged particles. Electrostatic separation is a process that uses electrostatic charges to separate crushed particles of material. An electrostatic separator is a device for separating particles by mass in a low energy charged beam. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. thumb|540x540px|Oppositely charged particles interact as they are moved through a column. This way, when a mixture of particles falls past a repelling object, the particles with the correct charge fall away from the other particles when they are repelled by the similarly charged object. The vector \boldsymbol{r}_N is the center of mass of all the bodies and \boldsymbol{r}_1 is the relative coordinate between the particles 1 and 2: The result one is left with is thus a system of N-1 translationally invariant coordinates \boldsymbol{r}_1, \dots, \boldsymbol{r}_{N-1} and a center of mass coordinate \boldsymbol{r}_N, from iteratively reducing two-body systems within the many- body system. In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. Electrostatic separation is a preferred sorting method when dealing with separating conductors from electrostatic separation non- conductors. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. Generally, electrostatic charges are used to attract or repel differently charged material.Science Direct.com When electrostatic separation uses the force of attraction to sort particles, conducting particles stick to an oppositely charged object, such as a metal drum, thereby separating them from the particle mixture. The order of children indicates the relative coordinate points > from xk to xj. For example, the surface charge of adsorbent is described by the ion that lies on the surface of the particle (adsorbent) structure like image. In a similar way to that in which electrostatic separation sorts particles with different electrostatic charges magnetic beneficiation sorts particles that respond to a magnetic field. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. A negative trion consists of two electrons and one hole and a positive trion consists of two holes and one electron. Surface Charging and Points of Zero Charge. The charged particle beams that can be manipulated in particle accelerators can be subdivided into electron beams, ion beams and proton beams. ==Common types== * Electron beam, or cathode ray, such as in a scanning electron microscope or in accelerators such as the Large Electron–Positron Collider or synchrotron light sources. An industrial process used to separate large amounts of material particles, electrostatic separating is most often used in the process of sorting mineral ore. Experiments showing electrostatic sorting in action can help make the process more clear.

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Two charged particles are fixed to an $x$ axis: Particle 1 of charge $q_1=2.1 \times 10^{-8} \mathrm{C}$ is at position $x=20 \mathrm{~cm}$ and particle 2 of charge $q_2=-4.00 q_1$ is at position $x=70 \mathrm{~cm}$. At what coordinate on the axis (other than at infinity) is the net electric field produced by the two particles equal to zero?

Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The particle located experiences an interaction with the electric field. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . Surface Charging and Points of Zero Charge. Electric field work is the work performed by an electric field on a charged particle in its vicinity. In the example both charges are positive; this equation is applicable to any charge configuration (as the product of the charges will be either positive or negative according to their (dis)similarity). 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. To move q+ closer to Q+ (starting from r_0 = \infty , where the potential energy=0, for convenience), we would have to apply an external force against the Coulomb field and positive work would be performed. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. : W = Q \int_{a}^{b} \mathbf{E} \cdot \, d \mathbf{r} = Q \int_{a}^{b} \frac{\mathbf{F_E}}{Q} \cdot \, d \mathbf{r}= \int_{a}^{b} \mathbf{F_E} \cdot \, d \mathbf{r} where :Q is the electric charge of the particle :E is the electric field, which at a location is the force at that location divided by a unit ('test') charge :FE is the Coulomb (electric) force :r is the displacement :\cdot is the dot product operator ==Mathematical description== Given a charged object in empty space, Q+. In physics, a charged particle is a particle with an electric charge. thumb|Circuit diagram of a charge qubit circuit. As the field lines are pulled together tightly by gluons, they do not "bow" outwards as much as an electric field between electric charges. The work per unit of charge, when moving a negligible test charge between two points, is defined as the voltage between those points. The work per unit of charge is defined by moving a negligible test charge between two points, and is expressed as the difference in electric potential at those points. The modern version of these equations is called Maxwell's equations. ====Electrostatics==== A charged test particle with charge q experiences a force F based solely on its charge. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. So, integrating and using Coulomb's Law for the force: :U(r) = \Delta U = -\int_{r_0}^{r} \mathbf{F}_{ext} \cdot \, d \mathbf{r}= -\int_{r_0}^{r} \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{\mathbf{r^2}} \cdot \, d \mathbf{r}= \frac{q_1q_2}{4\pi\varepsilon_0}\left(\frac{1}{r_0}- \frac{1}{r}\right) = -\frac{q_1q_2}{4\pi\varepsilon_0} \frac{1}{r} Now, use the relationship : W = -\Delta U \\! A related concept in electrochemistry is the electrode potential at the point of zero charge. The current generated by the second particle is \mathbf J = q_2 \mathbf v_2 \delta{\left( \mathbf r - \mathbf r_2 \right)}, which has a Fourier transform \mathbf J\left( \mathbf k \right) \equiv \int d^3r \exp\left( -i\mathbf k \cdot \mathbf r \right) \mathbf J\left( \mathbf r \right) = q_2 \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right). thumb|Electrical double layer (EDL) around a negatively charged particle in suspension in water. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte.

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The volume charge density of a solid nonconducting sphere of radius $R=5.60 \mathrm{~cm}$ varies with radial distance $r$ as given by $\rho=$ $\left(14.1 \mathrm{pC} / \mathrm{m}^3\right) r / R$. What is the sphere's total charge?

Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into "free" and "bound" charges. The volume can also be expressed in terms of A_n, the area of the unit -sphere. == Formulas == The first volumes are as follows: Dimension Volume of a ball of radius Radius of a ball of volume 0 1 (all 0-balls have volume 1) 1 2R \frac{V}{2}=0.5\times V 2 \pi R^2 \approx 3.142\times R^2 \frac{V^{1/2}}{\sqrt{\pi}}\approx 0.564\times V^{\frac{1}{2}} 3 \frac{4\pi}{3} R^3 \approx 4.189\times R^3 \left(\frac{3V}{4\pi}\right)^{1/3}\approx 0.620\times V^{1/3} 4 \frac{\pi^2}{2} R^4 \approx 4.935\times R^4 \frac{(2V)^{1/4}}{\sqrt{\pi}}\approx 0.671\times V^{1/4} 5 \frac{8\pi^2}{15} R^5\approx 5.264\times R^5 \left(\frac{15V}{8\pi^2}\right)^{1/5}\approx 0.717\times V^{1/5} 6 \frac{\pi^3}{6} R^6 \approx 5.168 \times R^6 \frac{(6V)^{1/6}}{\sqrt{\pi}}\approx 0.761\times V^{1/6} 7 \frac{16\pi^3}{105} R^7 \approx 4.725\times R^7 \left(\frac{105V}{16\pi^3}\right)^{1/7}\approx 0.801\times V^{1/7} 8 \frac{\pi^4}{24} R^8 \approx 4.059\times R^8 \frac{(24V)^{1/8}}{\sqrt{\pi}}\approx 0.839\times V^{1/8} 9 \frac{32\pi^4}{945} R^9 \approx 3.299\times R^9 \left(\frac{945V}{32\pi^4}\right)^{1/9}\approx 0.876\times V^{1/9} 10 \frac{\pi^5}{120} R^{10} \approx 2.550\times R^{10} \frac{(120V)^{1/10}}{\sqrt{\pi}}\approx 0.911\times V^{1/10} 11 \frac{64\pi^5}{10395} R^{11} \approx 1.884\times R^{11} \left(\frac{10395V}{64\pi^5}\right)^{1/11}\approx 0.944\times V^{1/11} 12 \frac{\pi^6}{720} R^{12} \approx 1.335\times R^{12} \frac{(720V)^{1/12}}{\sqrt{\pi}}\approx 0.976\times V^{1/12} 13 \frac{128\pi^6}{135135} R^{13} \approx 0.911\times R^{13} \left(\frac{135135V}{128\pi^6}\right)^{1/13}\approx 1.007\times V^{1/13} 14 \frac{\pi^7}{5040} R^{14} \approx 0.599\times R^{14} \frac{(5040V)^{1/14}}{\sqrt{\pi}}\approx 1.037\times V^{1/14} 15 \frac{256\pi^7}{2027025} R^{15} \approx 0.381\times R^{15} \left(\frac{2027025V}{256\pi^7}\right)^{1/15}\approx 1.066\times V^{1/15} n Vn(R) Rn(V) === Two-dimension recurrence relation === As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation: : V_n(R) = \begin{cases} 1 &\text{if } n=0,\\\\[0.5ex] 2R &\text{if } n=1,\\\\[0.5ex] \dfrac{2\pi}{n}R^2 \times V_{n-2}(R) &\text{otherwise}. \end{cases} This allows computation of in approximately steps. === Closed form === The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/5.19#E4, Release 1.0.6 of 2013-05-06. The volume of a -ball of radius is R^nV_n, where V_n is the volume of the unit -ball, the -ball of radius . The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then: :A_{n-1}(r) = r^{n-1} A_{n-1}(1). In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. However, if then the correction factor is : the surface area of an sphere of radius in is times the derivative of the volume of an ball. The radius can also be calculated as :r_{\rm s}= \left(\frac{3M}{4\pi Z \rho N_{\rm A}}\right)^\frac{1}{3}\,, where M is molar mass, Z is amount of free electrons per atom, \rho is mass density, and N_{\rm A} is the Avogadro constant. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. The volume of an odd-dimensional ball is :V_{2k+1}(R) = \frac{2(2\pi)^k}{(2k + 1)!!}R^{2k+1}. The volume of an ball of radius is :V^p_n(R) = \frac{\Bigl(2\,\Gamma\bigl(\tfrac1p + 1\bigr)\Bigr)^n}{\Gamma\bigl(\tfrac np + 1\bigr)}R^n. To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation : :V_n(R) = \int_0^R \frac{2\pi^{n/2}}{\Gamma\bigl(\tfrac n2\bigr)} \,r^{n-1}\,dr = \frac{2\pi^{n/2}}{n\,\Gamma\bigl(\tfrac n2\bigr)}R^n = \frac{\pi^{n/2}}{\Gamma\bigl(\tfrac n2 + 1\bigr)}R^n. === Geometric proof === The relations V_{n+1}(R) = \frac{R}{n+1}A_n(R) and A_{n+1}(R) = (2\pi R)V_n(R) and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. "How Small Is a Unit Ball?", Mathematics Magazine, Volume 62, Issue 2, 1989, pp. 101–107, https://doi.org/10.1080/0025570X.1989.11977419. === Relation with surface area === Let denote the hypervolume of the -sphere of radius . The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by: :A_{n-1}(R) = \frac{d}{dR} V_{n}(R) = \frac{n}{R}V_{n}(R). The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. The Pauthenier equation states K. Adamiak, "Rate of charging of spherical particles by monopolar ions in electric fields", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. As always, the integral of the charge density over a region of space is the charge contained in that region. Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume: :\begin{align} R_n(V) &= \frac{\Gamma\bigl(\tfrac n2 + 1\bigr)^{1/n}}{\sqrt{\pi}}V^{1/n} \\\ &= \left(\frac{n!! This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... Also, A_{n-1}(R) = \frac{dV_n(R)}{dR} because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε.

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Two charged concentric spher- ical shells have radii $10.0 \mathrm{~cm}$ and $15.0 \mathrm{~cm}$. The charge on the inner shell is $4.00 \times 10^{-8} \mathrm{C}$, and that on the outer shell is $2.00 \times 10^{-8} \mathrm{C}$. Find the electric field at $r=12.0 \mathrm{~cm}$.

The "spherium" model consists of two electrons trapped on the surface of a sphere of radius R. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. As before, the Rabi problem is solved by assuming that the electric field E is oscillatory with constant magnitude E0: E = E_0 (e^{i\omega t} + \text{c.c.}). thumb|right|FitzHugh-Nagumo model in phase space, with a = 0.7, b = 0.8, \tau = 12.5, R = 0.1, I_{ext} = 0.5. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It follows that the exact correlation energy for R = \sqrt{3}/2 is E_{\rm corr} = 1-2/\sqrt{3} \approx -0.1547 which is much larger than the limiting correlation energies of the helium-like ions (-0.0467) or Hooke's atoms (-0.0497). The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Values from other sources may differ significantly (see text) The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. To apply this to the Rabi problem, one assumes that the electric field E is oscillatory in time and constant in space: : E = E_0[e^{i\omega t} + e^{-i\omega t}], and xa is decomposed into a part ua that is in-phase with the driving E field (corresponding to dispersion) and a part va that is out of phase (corresponding to absorption): : x_a = x_0 (u_a \cos \omega t + v_a \sin \omega t). Gallery figures: FitzHugh-Nagumo model, with a = 0.7, \tau = 12.5, R = 0.1, and varying b, I_{ext}. This confirms the view that electron correlation on the surface of a sphere is qualitatively different from that in three-dimensional physical space. ==Spherium on a 3-sphere== Loos and Gill considered the case of two electrons confined to a 3-sphere repelling Coulombically. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. * For N = 12, electrons reside at the vertices of a regular icosahedron. , Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are all congruent equilateral triangles. Dividing by the Avogadro constant gives V = = 0.8685 Å, corresponding to r = 0.59 Å. === Polarizability === The polarizability α of a gas is related to its electric susceptibility χ by the relation \alpha = {\varepsilon_0 k_{\rm B}T\over p}\chi_{\rm e} and the electric susceptibility may be calculated from tabulated values of the relative permittivity ε using the relation χ = ε − 1\. The single electron may reside at any point on the surface of the unit sphere. Configurations reproduced in * * * This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us. Category:Electrostatics Category:Electron Category:Circle packing Category:Unsolved problems in mathematics

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Assume that a honeybee is a sphere of diameter 1.000 $\mathrm{cm}$ with a charge of $+45.0 \mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \mu \mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \mathrm{pC}$ on the near side of the grain and a charge of $+1.00 \mathrm{pC}$ on the far side. What is the magnitude of the net electrostatic force on the grain due to the bee?

thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. If r is the distance (in meters) between two charges, then the force (in newtons) between two point charges q and Q (in coulombs) is: :F = \frac{1}{4\pi \varepsilon_0}\frac{qQ}{r^2}= k_\text{e}\frac{qQ}{r^2}\, , where ε0 is the vacuum permittivity, or permittivity of free space: :\varepsilon_0 \approx \mathrm{8.854\ 187\ 817 \times 10^{-12} ~C^2{\cdot}N^{-1}{\cdot}m^{-2}}. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This force is the average of the discontinuous electric field at the surface charge. Adult worker honey bees consume 3.4–4.3 mg of pollen per day to meet a dry matter requirement of 66–74% protein. A drone is a male honey bee. A honey bee (also spelled honeybee) is a eusocial flying insect within the genus Apis of the bee clade, all native to mainland Afro-Eurasia. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. Defense can vary based on the habitat of the bee. According to Nikolaides, the electrostatic force engenders a long range capillary attraction. Cambridge, Massachusetts and London, England: Harvard University Press.. ==Pollination== thumb|Hind leg of a honey bee with pollen pellet stuck on the pollen basket or corbicula. Adult worker honey bees require 4 mg of utilizable sugars per day and larvae require about 59.4 mg of carbohydrates for proper development. Electrostatic phenomena arise from the forces that electric charges exert on each other. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces are relatively large. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The familiar phenomenon of a static "shock" is caused by the neutralization of charge built up in the body from contact with insulated surfaces. ==Coulomb's law== Coulomb's law states that: 'The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.' In this case, the surface charge density decreases upon approach. Drones depend on worker bees to feed them.

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In the radioactive decay of Eq. 21-13, $\mathrm{a}^{238} \mathrm{U}$ nucleus transforms to ${ }^{234} \mathrm{Th}$ and an ejected ${ }^4 \mathrm{He}$. (These are nuclei, not atoms, and thus electrons are not involved.) When the separation between ${ }^{234} \mathrm{Th}$ and ${ }^4 \mathrm{He}$ is $9.0 \times 10^{-15} \mathrm{~m}$, what are the magnitudes of the electrostatic force between them?

See also: H. Geiger and J.M. Nuttall (1912) "The ranges of α particles from uranium," Philosophical Magazine, Series 6, vol. 23, no. 135, pages 439-445. in its modern form the Geiger–Nuttall law is :\log_{10}T_{1/2}=\frac{A(Z)}{\sqrt{E}}+B(Z) where T_{1/2} is the half-life, E the total kinetic energy (of the alpha particle and the daughter nucleus), and A and B are coefficients that depend on the isotope's atomic number Z. The nuclear force is powerfully attractive between nucleons at distances of about 0.8 femtometre (fm, or 0.8×10−15 metre), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. The 4s electrons in iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge. At distances less than 0.7 fm, the nuclear force becomes repulsive. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). In this case, the effective nuclear charge can be calculated by Coulomb's law. In atomic physics, the effective nuclear charge is the actual amount of positive (nuclear) charge experienced by an electron in a multi-electron atom. thumb|right|300px|Woods–Saxon potential for , relative to V0 with a and R=4.6 fm The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus. In nuclear physics, the Geiger–Nuttall law or Geiger–Nuttall rule relates the decay constant of a radioactive isotope with the energy of the alpha particles emitted. In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the πNN coupling constant, improved phase-shift analysis, high-precision NN data, high-precision NN potentials, NN scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD. ==The nuclear force as a residual of the strong force== The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. * Nucleons near the surface of the nucleus (i.e. having within a distance of order a) experience a large force towards the center. There are only strong attractions when the total isospin of the set of interacting particles is 0, which is confirmed by experiment.Griffiths, David, Introduction to Elementary Particles Our understanding of the nuclear force is obtained by scattering experiments and the binding energy of light nuclei. At short distances (less than 1.7 fm or so), the attractive nuclear force is stronger than the repulsive Coulomb force between protons; it thus overcomes the repulsion of protons within the nucleus. In contrast, the effective nuclear charge is the attractive positive charge of nuclear protons acting on valence electrons, which is always less than the total number of protons present in a nucleus due to the shielding effect. ==See also== * Atomic orbitals * Core charge * d-block contraction (or scandide contraction) * Electronegativity * Lanthanide contraction * Shielding effect * Slater-type orbitals * Valence electrons * Weak charge ==References== ==Resources== *Brown, Theodore; intekhab khan, H.E.; & Bursten, Bruce (2002). Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. thumb|A conjectural example of an interaction between two neutrons and a proton, the triton or hydrogen-3, which is beta unstable. However, electrons further away are screened from the nucleus by other electrons in between, and feel less electrostatic interaction as a result. Nuclear interaction length is the mean distance travelled by a hadronic particle before undergoing an inelastic nuclear interaction. ==See also== *Nuclear collision length *Radiation length ==External links== *Particle Data Group site Category:Experimental particle physics These nuclear forces are very weak compared to direct gluon forces ("color forces" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance.

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The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$ at the point $(3.0,3.0) \mathrm{cm}$ and $100 \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}$ at the point $(2.0,0) \mathrm{cm}$. What is the $x$ coordinate of the particle?

Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. In Cartesian coordinate system \vec{r}=x\,\hat{e}_x+y\,\hat{e}_y+z\,\hat{e}_z. Several other definitions are in use, and so care must be taken in comparing different sources.Wolfram Mathworld, spherical coordinates == Cylindrical coordinate system == === Vector fields === Vectors are defined in cylindrical coordinates by (ρ, φ, z), where * ρ is the length of the vector projected onto the xy- plane, * φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), * z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by: \begin{bmatrix} \rho \\\ \phi \\\ z \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2} \\\ \operatorname{arctan}(y / x) \\\ z \end{bmatrix},\ \ \ 0 \le \phi < 2\pi, thumb or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} \rho\cos\phi \\\ \rho\sin\phi \\\ z \end{bmatrix}. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. This is the intersection between the reference plane and the axis. The dot is the point with radial distance , angular coordinate , and height . * The azimuth is the angle between the reference direction on the chosen plane and the line from the origin to the projection of on the plane. The coordinates of the vector r with respect to the basis vectors ei are xi. The relative direction between two points is their relative position normalized as a unit vector: :\Delta \mathbf{\hat{r}}=\Delta \mathbf{r} / \|\Delta \mathbf{r}\| where the denominator is the distance between the two points, \| \Delta \mathbf{r} \|. ==Definition== ===Three dimensions=== In three dimensions, any set of three- dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used. Any vector field can be written in terms of the unit vectors as: \mathbf A = A_x \mathbf{\hat x} + A_y \mathbf{\hat y} + A_z \mathbf{\hat z} = A_\rho \mathbf{\hat \rho} + A_\phi \boldsymbol{\hat \phi} + A_z \mathbf{\hat z} The cylindrical unit vectors are related to the Cartesian unit vectors by: \begin{bmatrix}\mathbf{\hat \rho} \\\ \boldsymbol{\hat\phi} \\\ \mathbf{\hat z}\end{bmatrix} = \begin{bmatrix} \cos\phi & \sin\phi & 0 \\\ -\sin\phi & \cos\phi & 0 \\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The line element is \mathrm{d}\boldsymbol{r} = \mathrm{d}\rho\,\boldsymbol{\hat{\rho}} + \rho\,\mathrm{d}\varphi\,\boldsymbol{\hat{\varphi}} + \mathrm{d}z\,\boldsymbol{\hat{z}}. Then the -coordinate is the same in both systems, and the correspondence between cylindrical and Cartesian are the same as for polar coordinates, namely \begin{align} x &= \rho \cos \varphi \\\ y &= \rho \sin \varphi \\\ z &= z \end{align} in one direction, and \begin{align} \rho &= \sqrt{x^2+y^2} \\\ \varphi &= \begin{cases} \text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\\ \arcsin\left(\frac{y}{\rho}\right) & \text{if } x \geq 0 \\\ -\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y \ge 0\\\ -\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y < 0 \end{cases} \end{align} in the other. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position, or axial position. A cylindrical coordinate system is a three- dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane containing the purple section). Its acceleration is \boldsymbol{a} = \frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \left( \ddot{\rho} - \rho\,\dot\varphi^2 \right)\boldsymbol{\hat \rho} + \left( 2\dot{\rho}\,\dot\varphi + \rho\,\ddot\varphi \right) \hat{\boldsymbol\varphi } + \ddot{z}\,\hat{\boldsymbol{z}} ==See also== *List of canonical coordinate transformations *Vector fields in cylindrical and spherical coordinates *Del in cylindrical and spherical coordinates ==References== ==Further reading== * * * * * * ==External links== * *MathWorld description of cylindrical coordinates *Cylindrical Coordinates Animations illustrating cylindrical coordinates by Frank Wattenberg Category:Three- dimensional coordinate systems Category:Orthogonal coordinate systems de:Polarkoordinaten#Zylinderkoordinaten ro:Coordonate polare#Coordonate cilindrice fi:Koordinaatisto#Sylinterikoordinaatisto The inverse projection is then given by: :\begin{align}\varphi &= \frac{\pi}{2} - \rho \\\ \lambda &=\frac{E \rho} {\sin \varphi_1 \sin \rho} \end{align} where :\rho = \sqrt{ (x\sin \varphi_1)^2 + \left(\varphi_1 - y + \cot \varphi_1\right)^2 }, \qquad E= \tan^{-1}\left(\frac{x\sin \varphi_1}{\varphi_1 - y + \cot \varphi_1}\right). thumb|Radius vector \vec{r} represents the position of a point \mathrm{P}(x,y,z) with respect to origin O. thumb|240px|A cylindrical coordinate system with origin , polar axis , and longitudinal axis .

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A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude $E=K r^4$, directed radially outward from the center of the sphere. Here $r$ is the radial distance from that center, and $K$ is a constant. What is the volume density $\rho$ of the charge distribution?

Let the first charge distribution \rho_1(\mathbf{r}') be centered on the origin and lie entirely within the second charge distribution \rho_2(\mathbf{r}'). In the interior case, where r' > r, the result is: \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^\ell I_{\ell m} r^\ell \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell m}(\theta, \phi) , with the interior multipole moments defined as I_{\ell m} \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \frac{\rho(\mathbf{r}')}{\left( r' \right)^{\ell+1}} \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell m}^{*}(\theta', \phi'). ==Interaction energies of spherical multipoles== A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals to charged colloids. As always, the integral of the charge density over a region of space is the charge contained in that region. We also use spherical coordinates throughout, e.g., the vector \mathbf r' has coordinates ( r', \theta', \phi') where r' is the radius, \theta' is the colatitude and \phi' is the azimuthal angle. ==Spherical multipole moments of a point charge== thumb|right|Figure 1: Definitions for the spherical multipole expansion The electric potential due to a point charge located at \mathbf{r'} is given by \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^2 + r^{\prime 2} - 2 r' r \cos \gamma}}. where R \ \stackrel{\mathrm{def}}{=}\ \left|\mathbf{r} - \mathbf{r'} \right| is the distance between the charge position and the observation point and \gamma is the angle between the vectors \mathbf{r} and \mathbf{r'}. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. For three dimensions, this normalization is the number density of the system ( \rho ) multiplied by the volume of the spherical shell, which symbolically can be expressed as \rho \, 4\pi r^2 dr. Using the mathematical identity P_\ell(\cos \theta) \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell 0}(\theta, \phi) the exterior multipole expansion becomes \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} \left( \frac{Q_\ell}{r^{\ell+1}} \right) P_\ell(\cos \theta) where the axially symmetric multipole moments are defined Q_\ell \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \rho(\mathbf{r}') \left( r' \right)^\ell P_\ell(\cos \theta') In the limit that the charge is confined to the z-axis, we recover the exterior axial multipole moments. For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density \rho(\mathbf r'). Similarly the interior multipole expansion becomes \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} I_\ell r^\ell P_\ell(\cos \theta) where the axially symmetric interior multipole moments are defined I_\ell \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \frac{\rho(\mathbf{r}')}{\left( r' \right)^{\ell+1}} P_\ell(\cos \theta') In the limit that the charge is confined to the z-axis, we recover the interior axial multipole moments. ==See also== * Solid harmonics * Laplace expansion * Multipole expansion * Legendre polynomials * Axial multipole moments * Cylindrical multipole moments Category:Electromagnetism Category:Potential theory Category:Moment (physics) In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. Taking particle 0 as fixed at the origin of the coordinates, \textstyle \rho g(\mathbf{r}) d^3r = \mathrm{d} n (\mathbf{r}) is the average number of particles (among the remaining N-1) to be found in the volume \textstyle d^3r around the position \textstyle \mathbf{r}. Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein-Zernike equation, using approximative closure relations like the Percus-Yevick approximation or the Hypernetted Chain Theory. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein-Zernike equation or structure-optimized potential refinement. ==Definition== Consider a system of N particles in a volume V (for an average number density \rho =N/V) and at a temperature T (let us also define \textstyle \beta = \frac{1}{kT}). This is expressed by a continuity equation which links the rate of change of charge density \rho(\boldsymbol{x}) and the current density \boldsymbol{J}(\boldsymbol{x}). Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. The delta function has the sifting property for any function f: \int_R d^3 \mathbf{r} f(\mathbf{r})\delta(\mathbf{r} - \mathbf{r}_0) = f(\mathbf{r}_0) so the delta function ensures that when the charge density is integrated over R, the total charge in R is q: Q =\int_R d^3 \mathbf{r} \, \rho_q =\int_R d^3 \mathbf{r} \, q \delta(\mathbf{r} - \mathbf{r}_0) = q \int_R d^3 \mathbf{r} \, \delta(\mathbf{r} - \mathbf{r}_0) = q This can be extended to N discrete point-like charge carriers.

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Two particles, each with a charge of magnitude $12 \mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the third vertex if both charges are positive?

thumb|upright=1.25|triangle ABC exsymmedians (red): e_a, e_b, e_c symmedians (green): s_a, s_b, s_c exsymmedian points (red): E_a, E_b, E_c The exsymmedians are three lines associated with a triangle. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). thumb|This triangle diagram is forbidden by Furry's theorem in quantum electrodynamics. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|upright=1.0|Distance from the origin O to the line E calculated with the Hesse normal form. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. Specifically the following formulas apply: :\begin{align} k_a&=a\cdot \frac{2\triangle}{c^2+b^2-a^2} \\\\[6pt] k_b&=b\cdot \frac{2\triangle}{c^2+a^2-b^2} \\\\[6pt] k_c&=c\cdot \frac{2\triangle}{a^2+b^2-c^2} \end{align} Here \triangle denotes the area of the triangle ABC and k_a, k_b, k_c the perpendicular line segments connecting the triangle sides a, b, c with the exsymmedian points E_a, E_b, E_c . == References == * Roger A. Johnson: Advanced Euclidean Geometry. For a triangle ABC with e_a, e_b, e_c being the exsymmedians and s_a, s_b, s_c being the symmedians through the vertices A, B, C two exsymmedians and one symmedian intersect in a common point, that is: :\begin{align} E_a&=e_b \cap e_c \cap s_a \\\ E_b&=e_a \cap e_c \cap s_b \\\ E_c&=e_a \cap e_b \cap s_c \end{align} The length of the perpendicular line segment connecting a triangle side with its associated exsymmedian point is proportional to that triangle side. The triangle formed by the three exsymmedians is the tangential triangle and its vertices, that is the three intersections of the exsymmedians are called exsymmedian points. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. More precisely for a given triangle the exsymmedians are the tangent lines on the triangle's circumcircle through the three vertices of the triangle. The trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse, so one of its angles is obtuse and hence has a negative cosine. ==Properties== Any two polar circles of two triangles in an orthocentric system are orthogonal. As the field lines are pulled together tightly by gluons, they do not "bow" outwards as much as an electric field between electric charges. It has an eccentricity equal to \sqrt{2}. ==In polygons== ===Triangles=== The legs of a right triangle are perpendicular to each other. To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. We can similarly describe the electric field E so that . * All four angles are equal. ==In computing distances== == Graph of functions == In the two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1. The altitudes of a triangle are perpendicular to their respective bases. Harcourt's theorem concerns the relationship of line segments through a vertex and perpendicular to any line tangent to the triangle's incircle. ===Quadrilaterals=== In a square or other rectangle, all pairs of adjacent sides are perpendicular. The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r2 – 4p2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).College Mathematics Journal 29(4), September 1998, p. 331, problem 635. The magnitude |\vec r_s| of {\vec r_s} is the shortest distance from the origin to the plane. ==References== == External links == * Category:Analytic geometry

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How much work is required to turn an electric dipole $180^{\circ}$ in a uniform electric field of magnitude $E=46.0 \mathrm{~N} / \mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \times$ $10^{-25} \mathrm{C} \cdot \mathrm{m}$ and the initial angle is $64^{\circ} ?$

This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque \boldsymbol{\tau} are given by U = - \mathbf{p} \cdot \mathbf{E},\qquad\ \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}, The scalar dot "" product and the negative sign shows the potential energy minimises when the dipole is parallel with field and is maximum when antiparallel while zero when perpendicular. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . thumb|upright=1.2|In the discrete dipole approximation, a larger object is approximated in terms of discrete radiating electric dipoles. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some angle with it. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. However in a non- uniform electric field a dipole may indeed receive a net force since the force on one end of the dipole no longer balances that on the other end. Note that a dipole in such a uniform field may twist and oscillate but receives no overall net force with no linear acceleration of the dipole. Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. However, when making measurements at a distance much larger than the charge separation, the dipole gives a good approximation of the actual electric field. The torque tends to align the dipole with the field. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin). (See electron electric dipole moment). In general the total dipole moment have contributions coming from translations and rotations of the molecules in the sample, \mathcal{M}_\text{Tot} = \mathcal{M}_\text{Trans} + \mathcal{M}_\text{Rot}. It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix. Further improvements, or a positive result, would place further limits on which theory takes precedence. == Formal definition == As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that : \mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} depends on the point {\mathbf r}_0 about which the moment of the charge distribution \rho({\mathbf r}) is taken.

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We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by $0.00010 \%$. With what force would two copper coins, placed $1.0 \mathrm{~m}$ apart, repel each other? Assume that each coin contains $3 \times 10^{22}$ copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.) What do you conclude?

This is not the case with copper. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. thumb|upright=1.35|Coin of Tennes. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. And although we would all be walking around with a few $1 coins, they would be replacing several quarters".Barro, Robert J. and Stevenson, Betsey: Do You Want That In Paper, or Metal? , The Wall Street Journal, Nov. 6, 1997 In advocating abolition of the penny, the Coin Coalition cites three penny-related costs that are passed on to consumers:Should the penny go?, Annelena Lobb, CNN Money, Apr. 11, 2002 *Wrapping charges (stores pay about 60 cents for each roll of 50 pennies) *Lost store productivity from penny users slowing the checkout line *Lost wages paid to clerks counting pennies in the register on each shift James C. Benfield, a partner with Bracy Williams and Company (Washington, D.C.), led the Coalition from 1987 until his death in 2002. The Coin Coalition is an organization supporting the elimination of pennies and dollar bills from U.S. currency. thumb|upright=1.5|Coin of Amyntas. Copper has the highest electrical conductivity rating of all non-precious metals: the electrical resistivity of copper = 16.78 nΩ•m at 20 °C. What is unique about copper is its long mean free path (approximately 100 atomic spacings at room temperature). A dollar coin is a coin valued at one dollar in a given currency. Roughly half of all copper mined is used to manufacture electrical wire and cable conductors. == Properties of copper == ===Electrical conductivity=== Electrical conductivity is a measure of how well a material transports an electric charge. Aluminium has 61% of the conductivity of copper. Manufacturers converted machines to accept the dollar coin at great expense, but the unwillingness of the U.S. government to phase out the dollar bill prevented the coin from becoming popular.$1 Coin Makes No "Cents" If $1 Bills Remain , Randy Chilton, Replay Magazine, March 2001 Although copper miners and other interest groups backed the Coin Coalition on this issue, they were unable to match the influence of Save the Greenback, a rival organization supporting continued dollar-bill production. Although 6-nines copper (99.9999% pure) has been produced in small quantities, it is extremely expensive and probably unnecessary for most commercial applications such as magnet, telecommunications, and building wire. Silver, a precious metal, is the only metal with a higher electrical conductivity than copper. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. If the two metals are joined, a galvanic reaction can occur. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Copper has a higher ductility than alternate metal conductors with the exception of gold and silver.Rich, Jack C., 1988, The Materials and Methods of Sculpture. This copper is at least 99.90% pure and has an electrical conductivity of at least 101% IACS.

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What must be the distance between point charge $q_1=$ $26.0 \mu \mathrm{C}$ and point charge $q_2=-47.0 \mu \mathrm{C}$ for the electrostatic force between them to have a magnitude of $5.70 \mathrm{~N}$ ?

The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). Then the distance between p and q is given by: d(p,q) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2}. Thus if p and q are two points on the real line, then the distance between them is given by: d(p,q) = |p-q|. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. If the polar coordinates of p are (r,\theta) and the polar coordinates of q are (s,\psi), then their distance is given by the law of cosines: d(p,q)=\sqrt{r^2 + s^2 - 2rs\cos(\theta-\psi)}. thumb|upright=1.35|Using the Pythagorean theorem to compute two-dimensional Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. In general, for points given by Cartesian coordinates in n-dimensional Euclidean space, the distance is d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2}. When p and q are expressed as complex numbers in the complex plane, the same formula for one-dimensional points expressed as real numbers can be used, although here the absolute value sign indicates the complex norm: d(p,q)=|p-q|. === Higher dimensions === thumb|upright=1.2|Deriving the n-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem In three dimensions, for points given by their Cartesian coordinates, the distance is d(p,q)=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. As an equation, the squared distance can be expressed as a sum of squares: d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2. Once the potential profile is known, the force per unit area between the plates expressed as the disjoining pressure Π can be obtained as follows. For unequally charged objects and eventually at shorted distances, these forces may also be attractive.

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Three charged particles form a triangle: particle 1 with charge $Q_1=80.0 \mathrm{nC}$ is at $x y$ coordinates $(0,3.00 \mathrm{~mm})$, particle 2 with charge $Q_2$ is at $(0,-3.00 \mathrm{~mm})$, and particle 3 with charge $q=18.0$ $\mathrm{nC}$ is at $(4.00 \mathrm{~mm}, 0)$. In unit-vector notation, what is the electrostatic force on particle 3 due to the other two particles if $Q_2$ is equal to $80.0 \mathrm{nC}$?

The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. #Lines of application of the actual forces {\scriptstyle \vec{F}_{1}} and \scriptstyle \vec{F}_{2} in the leftmost illustration intersect. Now let H denote the orthocenter of the triangle, then connection vector \overrightarrow{OH} is equal to the sum of the three vectors:Roger A. Johnson: Advanced Euclidean Geometry. After vector addition "at the location of \scriptstyle\vec{F}_{2}", the net force is translated to the appropriate line of application, whereof it becomes the resultant force \scriptstyle \vec{F}_{R}. After vector addition is performed "at the location of \scriptstyle \vec{F}_{1}", the net force obtained is translated so that its line of application passes through the common intersection point. #Illustration in the middle of the diagram shows two parallel actual forces. With respect to that point all torques are zero, so the torque of the resultant force \scriptstyle \vec{F}_{R} is equal to the sum of the torques of the actual forces. A trion is a localized excitation which consists of three charged particles. Pressure is force magnitude applied over an area. thumb|500px|Graphical placing of the resultant force In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The term resultant force should be understood to refer to both the forces and torques acting on a rigid body, which is why some use the term resultant force–torque. ==Illustration== The diagram illustrates simple graphical methods for finding the line of application of the resultant force of simple planar systems. If this condition is satisfied then there is a point of application for the resultant which results in a pure force. thumb|upright=1.5|sum of three equal lengthed vectors Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. Then the arguments from the previous case are applied to the forces and their components to demonstrate the torque relationships. This point is defined by the property : \mathbf{R} \times \mathbf{F} = \sum_{i=1}^n \mathbf{R}_i \times \mathbf{F}_i, where F is resultant force and Fi form the system of forces. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The sum of these forces and torques yields the resultant force- torque. ==Associated torque== If a point R is selected as the point of application of the resultant force F of a system of n forces Fi then the associated torque T is determined from the formulas : \mathbf{F} = \sum_{i=1}^n \mathbf{F}_i, and : \mathbf{T} = \sum_{i=1}^n (\mathbf{R}_i-\mathbf{R})\times \mathbf{F}_i. With the electrostatic force being proportional to r^{-2}, individual particle-particle interactions are long- range in nature, presenting a challenging computational problem in the simulation of particulate systems. It is useful to note that the point of application R of the resultant force may be anywhere along the line of action of F without changing the value of the associated torque. Notice that the case of two equal but opposite forces F and -F acting at points A and B respectively, yields the resultant W=(F-F, A×F - B× F) = (0, (A-B)×F). The point of application of the resultant force determines its associated torque. Notice that this equation for R has a solution only if the sum of the individual torques on the right side yield a vector that is perpendicular to F.

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A particle of charge $-q_1$ is at the origin of an $x$ axis. (a) At what location on the axis should a particle of charge $-4 q_1$ be placed so that the net electric field is zero at $x=2.0 \mathrm{~mm}$ on the axis?

The figure at right shows the electric field lines of two equal charges of opposite sign. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. There, the field vanishes and the lines coming axially from the charges end. Surface Charging and Points of Zero Charge. thumb|Field lines depicting the electric field created by a positive charge (left), negative charge (center), and uncharged object (right). The electric field lines in this case are straight lines that emanate from the charge uniformly in all directions in three-dimensional space. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|Circuit diagram of a charge qubit circuit. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. However, if the electric field lines for this setup were just drawn on a two-dimensional plane, their two-dimensional density would be proportional to 1/r, an incorrect result for this situation.A. Wolf, S. J. Van Hook, E. R. Weeks, Electric field line diagrams don't work Am. J. Phys., Vol. 64, No. 6. (1996), pp. 714–724 DOI 10.1119/1.18237 == Construction == thumb|upright=1.3|Construction of a field line Given a vector field \mathbf{F}(\mathbf{x}) and a starting point \mathbf{x}_\text{0} a field line can be constructed iteratively by finding the field vector at that point \mathbf{F}(\mathbf{x}_\text{0}). The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. Related values associated with the soil characteristics exist along with the pzc value, including zero point of charge (zpc), point of zero net charge (pznc), etc. == Term definition of point of zero charge == The point of zero charge is the pH for which the net surface charge of adsorbent is equal to zero. A related concept in electrochemistry is the electrode potential at the point of zero charge. The field line can be extended in the opposite direction from \mathbf{x}_\text{0} by taking each step in the opposite direction by using a negative step -ds. == Examples == thumb|420px|Different ways to depict the field of a magnet. thumb|right|250px|The slope field of \frac{dy}{dx}=x^{2}-x-2, with the blue, red, and turquoise lines being \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x+4, \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x, and \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x-4, respectively. For example, Gauss's law states that an electric field has sources at positive charges, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. Points where the field is zero or infinite have no field line through them, since direction cannot be defined there, but can be the endpoints of field lines. For example, electric field lines begin on positive charges and end on negative charges. The modern version of these equations is called Maxwell's equations. ====Electrostatics==== A charged test particle with charge q experiences a force F based solely on its charge. As the field lines are pulled together tightly by gluons, they do not "bow" outwards as much as an electric field between electric charges. The point of zero charge (pzc) is generally described as the pH at which the net charge of total particle surface (i.e. absorbent's surface) is equal to zero, which concept has been introduced in the studies dealt with colloidal flocculation to explain pH affecting the phenomenon.

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An electron is released from rest in a uniform electric field of magnitude $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$. Calculate the acceleration of the electron. (Ignore gravitation.)

Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. The acceleration voltage is an important quantity for the design of microwave cavities for particle accelerators. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . thumb|upright=1.3|right|Launch of Electron in start of the "Birds of the Feather" mission. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. Electron is a two-stage small-lift launch vehicle built and operated by Rocket Lab. Relativistic electron beams are streams of electrons moving at relativistic speeds. The term runaway electrons (RE) is used to denote electrons that undergo free fall acceleration into the realm of relativistic particles. Using Searle's formula (1897) for the electromagnetic energy increase of charged bodies with velocity, he calculated the increase of the electron's electromagnetic mass as a function of velocity: :\phi(\beta)=\frac{3}{4\beta^{2}}\left[\frac{1}{\beta}\lg\frac{1-\beta}{1+\beta}+\frac{2}{1-\beta^{2}}\right],\;\beta=\frac{v}{c}, Kaufmann noticed that the observed increase cannot be explained by this formula, so he separated the measured total mass into a mechanical (true) mass and an electromagnetic (apparent) mass, the mechanical mass being considerably greater than the electromagnetic one. Kaufmann's measurements of 1901 (corrected in 1902) showed that the charge-to-mass ratio diminishes and thus the electron's momentum (or mass) increases with velocity. Since the ratio doesn't vary for resting electrons, the data points should be on a single horizontal line (see Fig. 6). It has been suggested that relativistic electron beams could be used to heat and accelerate the reaction mass in electrical rocket engines that Dr. Robert W. Bussard called quiet electric-discharge engines (QEDs). ==References== ==External links== *PEARL Lab @ UHawaii *Applying REBs for the development of high-powered microwaves (HPM) Category:Electron beam Category:Quantum mechanics Category:Special relativity First flight of Electron with a fully autonomous flight termination system on the rocket. === 2020 === First launch for the National Reconnaissance Office in January 2020. In 1906 and 1907, Planck published his own conclusion on the behavior of the inertial mass of electrons with high speeds. In his theory, the longitudinal mass m_L= {{\gamma}^3}m and the transverse mass m_T= {\gamma}m, where \gamma is the Lorentz factor and m is the rest mass of the electron. (a) This front view of the apparatus illustrates the uniform acceleration imposed by the charged condenser plates on the beta particles. Suborbital Flight-1 was the first suborbital launch of the rocket. == Launch statistics == === Launch outcomes === === Launch sites === === Booster tests and recoveries === === Rocket configurations === == Orbital launches == === 2017–2018 === Electron experienced its first successful launch in January 2018, and launched their first mission for NASA in December 2018. In this notation, the effective acceleration voltage |V_\parallel| is often expressed as V_0 T. == Transverse voltage == In symbolic analogy to the longitudinal voltage, one can define effective voltages in two orthogonal directions x,y that are transversal to the particle trajectory V_{x,y} = \frac{1}{q} \vec e_{x,y} \cdot \int \vec F_L(s) \exp\left(i k_\beta s\right)\,\mathrm d s which describe the integrated forces that deflect the particle from its design path. Accelerationen (Accelerations), op. 234, is a waltz composed by Johann Strauss II in 1860 for the Engineering Students' Ball at the Sofienbad-Saal in Vienna.

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Identify $\mathrm{X}$ in the following nuclear reactions: (a) ${ }^1 \mathrm{H}+$ ${ }^9 \mathrm{Be} \rightarrow \mathrm{X}+\mathrm{n} ;$ (b) ${ }^{12} \mathrm{C}+{ }^1 \mathrm{H} \rightarrow \mathrm{X} ;$ (c) ${ }^{15} \mathrm{~N}+{ }^1 \mathrm{H} \rightarrow{ }^4 \mathrm{He}+\mathrm{X}$. Appendix F will help.

Jonathan Feng et al. attribute this 6.8-σ anomaly to a 17 MeV protophobic X-boson dubbed the X17 particle. The X17 particle is a hypothetical subatomic particle proposed by Attila Krasznahorkay and his colleagues to explain certain anomalous measurement results. Krasznahorkay (2019) posted a preprint announcing that he and his team at ATOMKI had successfully observed the same anomalies in the decay of stable helium atoms as had been observed in beryllium-8, strengthening the case for the existence of the X17 particle. Hexanitrostilbene (HNS), also called JD-X, is an organic compound with the formula [(O2N)3C6H2CH]2. The molecular formula C8H11N (molar mass: 121.18 g/mol) may refer to: * Bicyclo(2.2.1)heptane-2-carbonitrile * Collidines (trimethylpyridines) ** 2,3,4-Trimethylpyridine ** 2,3,5-Trimethylpyridine ** 2,3,6-Trimethylpyridine ** 2,4,5-Trimethylpyridine ** 2,4,6-Trimethylpyridine ** 3,4,5-Trimethylpyridine * Dimethylaniline * Phenethylamine * 1-Phenylethylamine * Xylidines ** 2,3-Xylidine ** 2,4-Xylidine ** 2,5-Xylidine ** 2,6-Xylidine ** 3,4-Xylidine ** 3,5-Xylidine Beryllium-8 (8Be, Be-8) is a radionuclide with 4 neutrons and 4 protons. HNC may refer to: *Hydrogen isocyanide, a molecule with the formula HNC that is important to the field of astrochemistry *Heptanitrocubane, an experimental high explosive *Higher National Certificate, a higher education qualification in the United Kingdom *High Negotiations Committee, a Syrian political- military opposition bloc headquartered in Riyadh *Classical-map Hyper-Netted- Chain equation, a method in many-body theoretical physics for interacting uniform electron liquids in two and three dimensions *Hypernetted-chain equation, a closure relation to solve the Ornstein-Zernike equation commonly applied in statistical mechanics and fluid theory *Hopkins-Nanjing Center, a joint educational venture between Nanjing University and Johns Hopkins University located in Nanjing, China *Habits & Contradictions, album by Schoolboy Q *Huddersfield Narrow Canal, Northern England Feng et al. (2016) proposed that a "protophobic" X boson, with a mass of , suppressed couplings to protons relative to neutrons and electrons at femtometer range, could explain the data. The X17 particle could be the force carrier for a postulated fifth force, possibly connected with dark matter, and has been described as a protophobic (i.e., ignoring protons) vector boson with a mass near . The NA64 experiment at CERN looks for the proposed X17 particle by striking the electron beams from the Super Proton Synchrotron on fixed target nuclei. ==History== In 2015, Krasznahorkay and his colleagues at ATOMKI, the Hungarian Institute for Nuclear Research, posited the existence of a new, light boson with a mass of about (i.e., 34 times heavier than the electron). However, stable 8Be would enable alternative reaction pathways in helium burning (such as 8Be + 4He and 8Be + 8Be; constituting a "beryllium burning" phase) and possibly affect the abundance of the resultant 12C, 16O, and heavier nuclei, though 1H and 4He would remain the most abundant nuclides. CXOU J061705.3+222127 is a neutron star. While further experiments are needed to corroborate these observations, the influence of a fifth boson has been proposed as "the most straightforward possibility". == Role in stellar nucleosynthesis == In stellar nucleosynthesis, two helium-4 nuclei may collide and fuse into a single beryllium-8 nucleus. Furthermore, while other alpha nuclides have similar short-lived resonances, 8Be is exceptionally already in the ground state. The nucleus of helium-4 is particularly stable, having a doubly magic configuration and larger binding energy per nucleon than 8Be. Owing to the instability of 8Be, the triple-alpha process is the only reaction in which 12C and heavier elements may be produced in observed quantities. If the beryllium-8 collides with a helium-4 nucleus before decaying, they can fuse into a carbon-12 nucleus. Beryllium-8 has an extremely short half-life (8.19 seconds), and decays back into two helium-4 nuclei. They reported that this populated a nucleus with A = 8 that near- instantaneously decays into two alpha particles. In an effort to find a dark photon, the team fired protons at thin targets of lithium-7, which created unstable beryllium-8 nuclei that then decayed and produced pairs of electrons and positrons. The X‑17 particle is not consistent with the Standard Model, so its existence would need to be explained by another theory. ==See also== * Axion * List of particles * 750 GeV diphoton excess ==References== Category:Bosons Category:Dark matter Category:Hypothetical elementary particles Category:Force carriers This activity was observed again several months later, and was inferred to originate from 8Be. == Properties == left|thumb|300px|Triple-alpha process Beryllium-8 is unbound with respect to alpha emission by 92 keV; it is a resonance having a width of 6 eV.

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The nucleus of a plutonium-239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the magnitude of the electric field produced by the protons?

This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... The proton radius is approximately one femtometre = . It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton Plutonium is a radioactive chemical element with the symbol Pu and atomic number 94. When one of these neutrons strikes the nucleus of another 238U atom, it is absorbed by the atom, which becomes 239U. Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). Later studies found an empirical relation between the charge radius and the mass number, A, for heavier nuclei (A > 20): :R ≈ r0A where the empirical constant r0 of 1.2–1.5 fm can be interpreted as the Compton wavelength of the proton. The proton radius puzzle is an unanswered problem in physics relating to the size of the proton. Plutonium-239 (239Pu or Pu-239) is an isotope of plutonium. The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. The known isotopes of plutonium range in mass number from 228 to 247. The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. Occasionally, when an atom of 238U is exposed to neutron radiation, its nucleus will capture a neutron, changing it to 239U. [It] was stocked with 24 pounds of plutonium that produced about 240 watts of electricity when it left Earth in 2006, according to Ryan Bechtel, an engineer from the Department of Energy who works on space nuclear power. For the proton, the two radii are the same. ==History== The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909,. under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester, UK. The plutonium grade is usually listed as percentage of 240Pu. The result is again ~5% smaller than the previously-accepted proton radius. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. The fission of one atom of 239Pu generates 207.1 MeV = 3.318 × 10−11 J, i.e. 19.98 TJ/mol = 83.61 TJ/kg, or about 23 gigawatt hours/kg. radiation source (thermal fission of 239Pu) average energy released [MeV] Kinetic energy of fission fragments 175.8 Kinetic energy of prompt neutrons 5.9 Energy carried by prompt γ-rays 7.8 Total instantaneous energy 189.5 Energy of β− particles 5.3 Energy of antineutrinos 7.1 Energy of delayed γ-rays 5.2 Total from decaying fission products 17.6 Energy released by radiative capture of prompt neutrons 11.5 Total heat released in a thermal-spectrum reactor (anti-neutrinos do not contribute) 211.5 == Production == Plutonium is made from uranium-238. 239Pu is normally created in nuclear reactors by transmutation of individual atoms of one of the isotopes of uranium present in the fuel rods. A 5 kg mass of 239Pu contains about atoms. Nucleons have a radius of about 0.8 fm. Weapons- grade plutonium is defined as containing no more than 7% 240Pu; this is achieved by only exposing 238U to neutron sources for short periods of time to minimize the 240Pu produced.

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A nonconducting spherical shell, with an inner radius of $4.0 \mathrm{~cm}$ and an outer radius of $6.0 \mathrm{~cm}$, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density $\rho$ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell $\rho=b / r$, where $r$ is the distance in meters from the center of the shell and $b=3.0 \mu \mathrm{C} / \mathrm{m}^2$. What is the net charge in the shell?

It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry The key to the effectiveness of the hollow charge is its diameter. A typical modern shaped charge, with a metal liner on the charge cavity, can penetrate armor steel to a depth of seven or more times the diameter of the charge (charge diameters, CD), though depths of 10 CD and above have been achieved. A shell is a three-dimensional solid structural element whose thickness is very small compared to its other dimensions, and characterized in structural terms by mid-plane stress which is both coplanar and normal to the surface. Octol-loaded charges with a rounded cone apex generally had higher surface temperatures with an average of 810 K, and the temperature of a tin- lead liner with Comp-B fill averaged 842 K. A 66-pound shaped charge accelerated the gas in a 3-cm glass-walled tube 2 meters in length. Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. A shaped charge is an explosive charge shaped to focus the effect of the explosive's energy. The Pauthenier equation states K. Adamiak, "Rate of charging of spherical particles by monopolar ions in electric fields", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. The History of Shaped Charges", Technical Report BRL-TR-3158, U.S. Army Laboratory Command, Ballistic Research Laboratory (Aberdeen Proving Ground, Maryland), p. Available on-line at: Defense Technical Information Center During World War II, shaped-charge munitions were developed by Germany (Panzerschreck, Panzerfaust, Panzerwurfmine, Mistel), Britain (PIAT, Beehive cratering charge), the Soviet Union (RPG-43, RPG-6), the U.S. (bazooka),Donald R. Kennedy, "History of the Shaped Charge Effect: The First 100 Years ", D.R. Kennedy and Associates, Inc., Mountain View, California, 1983 and Italy (Effetto Pronto Speciale shells for various artillery pieces).https://comandosupremo.com/forums/index.php?attachments/ep_eps-jpg.59/ The development of shaped charges revolutionized anti-tank warfare. In probability and statistics, a spherical contact distribution function, first contact distribution function,D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. *William P. Walters (September 1990) "The Shaped Charge Concept. A description of Munroe's first shaped-charge experiment appears on p. 453. > Among the experiments made ... was one upon a safe twenty-nine inches cube, > with walls four inches and three quarters thick, made up of plates of iron > and steel ... Shaped charges are used most extensively in the petroleum and natural gas industries, in particular in the completion of oil and gas wells, in which they are detonated to perforate the metal casing of the well at intervals to admit the influx of oil and gas. The location of the charge relative to its target is critical for optimum penetration for two reasons. Contrary to a misconception (possibly resulting from the acronym for high- explosive anti-tank, HEAT) the shaped charge does not depend in any way on heating or melting for its effectiveness; that is, the jet from a shaped charge does not melt its way through armor, as its effect is purely kinetic in nature – however the process does create significant heat and often has a significant secondary incendiary effect after penetration. ==Munroe effect== The Munroe or Neumann effect is the focusing of blast energy by a hollow or void cut on a surface of an explosive. Although Munroe's experiment with the shaped charge was widely publicized in 1900 in Popular Science Monthly, the importance of the tin can "liner" of the hollow charge remained unrecognized for another 44 years.Kennedy (1990), p. The limit of this approximation is the shell integral. In general, shaped charges can penetrate a steel plate as thick as 150% to 700%Jane's Ammunition Handbook 1994, pp. 140–141, addresses the reported ≈700 mm penetration of the Swedish 106 3A-HEAT-T and Austrian RAT 700 HEAT projectiles for the 106 mm M40A1 recoilless rifle. of their diameter, depending on the charge quality.

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An electron is shot directly Figure 23-50 Problem 40. toward the center of a large metal plate that has surface charge density $-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^2$. If the initial kinetic energy of the electron is $1.60 \times 10^{-17} \mathrm{~J}$ and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?

thumb|upright=1.3|right|Launch of Electron in start of the "Birds of the Feather" mission. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. Electron is a two-stage small-lift launch vehicle built and operated by Rocket Lab. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. First mid-air helicopter capture attempt of an Electron first stage following launch. First flight of Electron with a fully autonomous flight termination system on the rocket. === 2020 === First launch for the National Reconnaissance Office in January 2020. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. In physics, a projectile launched with specific initial conditions will have a range. thumb|Scheme of two types of electron capture. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Second stage engine shut down early causing the mission to be lost, but Electron's first stage safely completed a successful splashdown under parachute. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. First, as previously noted, the electron must absorb an amount of energy equivalent to the energy difference between the electron's current energy level and an unoccupied, higher energy level in order to be promoted to that energy level. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The second solution is the useful one for determining the range of the projectile. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance: * constrained global optimization (Altschuler et al. 1994), * steepest descent (Claxton and Benson 1966, Erber and Hockney 1991), * random walk (Weinrach et al. 1990), * genetic algorithm (Morris et al. 1996) While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest. ===Continuous spherical shell charge=== thumb|The extreme upper energy limit of the Thomson Problem is given by N^2/2 for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Flight 26 was the first Electron flight to attempt a full catch recovery using a mid-air helicopter catch. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge. Surface Charging and Points of Zero Charge. The energy released is equal to the difference in energy levels between the electron energy states. Suborbital Flight-1 was the first suborbital launch of the rocket. == Launch statistics == === Launch outcomes === === Launch sites === === Booster tests and recoveries === === Rocket configurations === == Orbital launches == === 2017–2018 === Electron experienced its first successful launch in January 2018, and launched their first mission for NASA in December 2018.

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A square metal plate of edge length $8.0 \mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \times 10^{-6} \mathrm{C}$. Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \mathrm{~mm}$ from the center) by assuming that the charge is spread uniformly over the two faces of the plate.

We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. An obvious choice for this fraction is ½ (−3 dB), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. As an example, consider a charged spherical shell of negligible thickness, with a uniformly distributed charge and radius . We can similarly describe the electric field E so that . Uniform field theory is a formula for determining the effective electrical resistance of a parallel wire system. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. Because the field close to the sheet can be approximated as constant, the pillbox is oriented in a way so that the field lines penetrate the disks at the ends of the field at a perpendicular angle and the side of the cylinder are parallel to the field lines. ==See also== * Area * Surface area * Vector calculus * Integration * Divergence theorem * Faraday cage * Field theory * Field line == References == * * ==Further reading== * Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ==External links== * Fields - a chapter from an online textbook Category:Surfaces Category:Electrostatics Category:Carl Friedrich Gauss Thereby is the electrical charge enclosed by the Gaussian surface. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example "field near infinite line charge" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. With the same example, using a larger Gaussian surface outside the shell where , Gauss's law will produce a non-zero electric field.

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E

A neutron consists of one "up" quark of charge $+2 e / 3$ and two "down" quarks each having charge $-e / 3$. If we assume that the down quarks are $2.6 \times 10^{-15} \mathrm{~m}$ apart inside the neutron, what is the magnitude of the electrostatic force between them?

Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. This same force is much weaker between neutrons and protons, because it is mostly neutralized within them, in the same way that electromagnetic forces between neutral atoms (van der Waals forces) are much weaker than the electromagnetic forces that hold electrons in association with the nucleus, forming the atoms. A Pauli repulsion also occurs between quarks of the same flavour from different nucleons (a proton and a neutron). ===Field strength=== At distances larger than 0.7 fm the force becomes attractive between spin-aligned nucleons, becoming maximal at a center–center distance of about 0.9 fm. Beyond this distance the force drops exponentially, until beyond about 2.0 fm separation, the force is negligible. The strong attraction between nucleons was the side-effect of a more fundamental force that bound the quarks together into protons and neutrons. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). In the context of atomic nuclei, the same strong interaction force (that binds quarks within a nucleon) also binds protons and neutrons together to form a nucleus. The strong interaction is the attractive force that binds the elementary particles called quarks together to form the nucleons (protons and neutrons) themselves. On the smaller scale (less than about 0.8 fm, the radius of a nucleon), it is the force (carried by gluons) that holds quarks together to form protons, neutrons, and other hadron particles. The strong force acts between quarks. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Unlike all other forces (electromagnetic, weak, and gravitational), the strong force does not diminish in strength with increasing distance between pairs of quarks. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. On a larger scale (of about 1 to 3 fm), it is the force (carried by mesons) that binds protons and neutrons (nucleons) together to form the nucleus of an atom. For identical nucleons (such as two neutrons or two protons) this repulsion arises from the Pauli exclusion force. It was known that the nucleus was composed of protons and neutrons and that protons possessed positive electric charge, while neutrons were electrically neutral. The residual strong force is thus a minor residuum of the strong force that binds quarks together into protons and neutrons. The larger the neutron cross section, the more likely a neutron will react with the nucleus. thumb|A conjectural example of an interaction between two neutrons and a proton, the triton or hydrogen-3, which is beta unstable. The nuclear force is nearly independent of whether the nucleons are neutrons or protons. After the verification of the quark model, strong interaction has come to mean QCD. ==Nucleon–nucleon potentials== Two- nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the NN force. In nuclear physics, the concept of a neutron cross section is used to express the likelihood of interaction between an incident neutron and a target nucleus. Most of the mass of a common proton or neutron is the result of the strong interaction energy; the individual quarks provide only about 1% of the mass of a proton.

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E

In an early model of the hydrogen atom (the Bohr model), the electron orbits the proton in uniformly circular motion. The radius of the circle is restricted (quantized) to certain values given by where $a_0=52.92 \mathrm{pm}$. What is the speed of the electron if it orbits in the smallest allowed orbit?

In the simplest atom, hydrogen, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. This means that the innermost electrons orbit at approximately 1/2 the Bohr radius. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. In atomic physics, the Bohr model or Rutherford–Bohr model of the atom, presented by Niels Bohr and Ernest Rutherford in 1913, consists of a small, dense nucleus surrounded by orbiting electrons. :The smallest possible value of r in the hydrogen atom () is called the Bohr radius and is equal to: :: r_1 = \frac{\hbar^2}{k_\mathrm{e} e^2 m_\mathrm{e}} \approx 5.29 \times 10^{-11}~\mathrm{m}. Bohr considered circular orbits. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. Once an electron is in this lowest orbit, it can get no closer to the nucleus. This outer electron should be at nearly one Bohr radius from the nucleus. Bohr described angular momentum of the electron orbit as 1/2h while de Broglie's wavelength of described h divided by the electron momentum. This equation determines the electron's speed at any radius: :: v = \sqrt{\frac{Zk_\mathrm{e} e^2}{m_\mathrm{e} r}}. For a hydrogen atom, the classical orbits have a period T determined by Kepler's third law to scale as r3/2. In modern quantum mechanics, the electron in hydrogen is a spherical cloud of probability that grows denser near the nucleus. The de Broglie wavelength of an electron is : \lambda = \frac{h}{mv}, which implies that : \frac{nh}{mv} = 2 \pi r, or : \frac{nh}{2 \pi} = mvr, where mvr is the angular momentum of the orbiting electron. It does not work for (neutral) helium. == Refinements == thumb|Elliptical orbits with the same energy and quantized angular momentum Several enhancements to the Bohr model were proposed, most notably the Sommerfeld or Bohr–Sommerfeld models, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. The lowest value of n is 1; this gives the smallest possible orbital radius of 0.0529 nm known as the Bohr radius. An electron in the lowest energy level of hydrogen () therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree is considered a "coincidence".

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At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk?

thumb|upright=1.35|An ellipse, its minimum bounding box, and its director circle. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. An obvious choice for this fraction is ½ (−3 dB), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). This measure is strictly a ratio of diameters and should not be confused with the covered fraction of the apparent area (disk) of the eclipsed body. A line segment through a circle's center bisecting a chord is perpendicular to the chord. thumb|right|170px Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. thumb|An annular solar eclipse has a magnitude of less than 1.0 The magnitude of eclipse is the fraction of the angular diameter of a celestial body being eclipsed. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. In an annular solar eclipse, the magnitude of the eclipse is the ratio between the apparent angular diameters of the Moon and that of the Sun during the maximum eclipse, yielding a ratio less than 1.0. It has the same center as the ellipse, with radius \sqrt{a^2+b^2}, where a and b are the semi- major axis and semi-minor axis of the ellipse. This can be calculated in a single integral similar to the following: :\pi\int_a^b\left(R_\mathrm{O}(x)^2 - R_\mathrm{I}(x)^2\right)\,dx where is the function that is farthest from the axis of rotation and is the function that is closest to the axis of rotation. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. It has an eccentricity equal to \sqrt{2}. ==In polygons== ===Triangles=== The legs of a right triangle are perpendicular to each other. In many cases, it makes more sense to take the distance between points where the intensity falls to 1/e2 = 0.135 times the maximum value. We can similarly describe the electric field E so that . * Line c is perpendicular to line a. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. Note that when integrating along an axis other than the , the graph of the function that is farthest from the axis of rotation may not be that obvious. The magnitude of a partial or annular solar eclipse is always between 0.0 and 1.0, while the magnitude of a total solar eclipse is always greater than or equal to 1.0.

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B

Of the charge $Q$ on a tiny sphere, a fraction $\alpha$ is to be transferred to a second, nearby sphere. The spheres can be treated as particles. What value of $\alpha$ maximizes the magnitude $F$ of the electrostatic force between the two spheres?

The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This equation can be extended to more highly charged particles by reinterpreting the charge Q as an effective charge. The Pauthenier equation states K. Adamiak, "Rate of charging of spherical particles by monopolar ions in electric fields", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). Minimum bounding circle may refer to: * Bounding sphere * Smallest circle problem For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). The interaction free energy involving two spherical particles within the DH approximation follows the Yukawa or screened Coulomb potential : U = \frac{Q^2}{4 \pi \epsilon \epsilon_0} \left( \frac{e^{\kappa a}}{1+\kappa a} \right)^2 \frac{e^{-\kappa r}}{r} where r is the center-to- center distance, Q is the particle charge, and a the particle radius. The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to- charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. "Min-Energy Configurations of Electrons On A Sphere". In this case, the surface charge density decreases upon approach. At larger distances, oppositely charged surfaces repel and equally charged ones attract. ==Charge regulating surfaces== While the superposition approximation is actually exact at larger distances, it is no longer accurate at smaller separations.

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In a spherical metal shell of radius $R$, an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of $6.90 \times 10^{-13} \mathrm{C} / \mathrm{m}^2$. What is the magnitude of the electron's acceleration when it reaches radial distances $r=0.500 R$?

The radius r is then defined to be the classical electron radius, r_\text{e}, and one arrives at the expression given above. This asymmetric distribution of charge within the particle gives rise to a small negative squared charge radius for the particle as a whole. The electrostatic potential at a distance r from a charge q is :V(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Note that this derivation does not say that r_\text{e} is the actual radius of an electron. It is customary when charge radius takes an imaginary numbered value to report the negative valued square of the charge radius, rather than the charge radius itself, for a particle. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The classical electron radius is given as :r_\text{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\text{e}} c^2} = 2.817 940 3227(19) \times 10^{-15} \text{ m} = 2.817 940 3227(19) \text{ fm} , where e is the elementary charge, m_{\text{e}} is the electron mass, c is the speed of light, and \varepsilon_0 is the permittivity of free space.David J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, 1995, p. 155. If the sphere is assumed to have constant charge density, \rho, then :q = \rho \frac{4}{3} \pi r^3 and dq = \rho 4 \pi r^2 dr. Integrating for r from zero to the final radius r yields the expression for the total energy U, necessary to assemble the total charge q into a uniform sphere of radius r: :U = \frac{1}{4\pi\varepsilon_0} \frac{3}{5} \frac{q^2}{r}. This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The single electron may reside at any point on the surface of the unit sphere. The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. Although it is sometimes stated that all the electrons in a shell have the same energy, this is an approximation. This numerical value is several times larger than the radius of the proton. The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. The best known particle with a negative squared charge radius is the neutron. According to modern understanding, the electron is a point particle with a point charge and no spatial extent. The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length. Surface Charging and Points of Zero Charge.

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A particle of charge $+3.00 \times 10^{-6} \mathrm{C}$ is $12.0 \mathrm{~cm}$ distant from a second particle of charge $-1.50 \times 10^{-6} \mathrm{C}$. Calculate the magnitude of the electrostatic force between the particles.

If is the distance between the charges, the magnitude of the force is |\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2}, where is the electric constant. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. Coulomb's law in vector form states that the electrostatic force \mathbf{F}_1 experienced by a charge, q_1 at position \mathbf{r}_1, in the vicinity of another charge, q_2 at position \mathbf{r}_2, in a vacuum is equal to \mathbf{F}_1 = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{r}_1-\mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}_{12}}{|\mathbf{r}_{12}|^2} where \boldsymbol{r}_{12} = \boldsymbol{r}_1 - \boldsymbol{r}_2 is the vectorial distance between the charges, \widehat{\mathbf{r}}_{12}=\frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|} a unit vector pointing from q_2 to and \varepsilon_0 the electric constant. The force is along the straight line joining the two charges. If both charges have the same sign (like charges) then the product q_1q_2 is positive and the direction of the force on q_1 is given by \widehat{\mathbf{r}}_{12}; the charges repel each other. If the product is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. == Vector form == thumb|right|350px|In the image, the vector is the force experienced by , and the vector is the force experienced by . The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. If the charges have opposite signs then the product q_1q_2 is negative and the direction of the force on q_1 is _{12};}} the charges attract each other. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In 1767, he conjectured that the force between charges varied as the inverse square of the distance. The scalar form gives the magnitude of the vector of the electrostatic force between two point charges and , but not its direction. The magnitude of the electric field can be derived from Coulomb's law. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. Value Item 10−21 zepto- (zC) 10−20 (−1/3 e) – Charge of down, strange and bottom quarks 10−19 (2/3 e)—Charge of up, charm and top quarks 10−19 The elementary charge e, i.e. the negative charge on a single electron or the positive charge on a single proton 10−18 atto- (aC) ~ Planck chargePlanck Units 10−17 (92 e) – Positive charge on a uranium nucleus (derived: 92 x ) 10−16 Charge on a dust particle in a plasma 10−15 femto- (fC) Charge on a typical dust particle 10−12 pico- (pC) Charge in typical microwave frequency capacitors 10−9 nano- (nC) Charge in typical radio frequency capacitors 10−6 micro- (μC) Charge in typical audio frequency capacitors 10−6 micro- (μC) ~ Static electricity from rubbing materials together 10−3 milli- (mC) Charge in typical power supply capacitors 10−3 milli- (mC) Charge in CH85-2100-105 high voltage capacitor for microwaves 100 C Two like charges, each of , placed one meter apart, would experience a repulsive force of approximately 100 C Supercapacitor for real-time clock (RTC) (1F x 3.6V) 101 deca- (daC) Charge in a typical thundercloud Hasbrouck, Richard. When the electromagnetic theory is expressed in the International System of Units, force is measured in newtons, charge in coulombs and distance in meters. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. Electrostatic charge on an object can be measured by placing it into the Faraday Cup. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field created by a single source point charge Q at a certain distance from it r in vacuum is given by |\mathbf{E}| = k_\text{e} \frac{|q|}{r^2} A system N of charges q_i stationed at \mathbf{r}_i produces an electric field whose magnitude and direction is, by superposition \mathbf{E}(\mathbf{r}) = {1\over4\pi\varepsilon_0} \sum_{i=1}^N q_i \frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3} == Atomic forces == Coulomb's law holds even within atoms, correctly describing the force between the positively charged atomic nucleus and each of the negatively charged electrons. The force on a small test charge q at position \boldsymbol{r} in vacuum is given by the integral over the distribution of charge \mathbf{F}(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0}\int dq' \frac{\mathbf{r} - \mathbf{r'}}{|\mathbf{r} - \mathbf{r'}|^3}. where it the "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which |\mathbf{r} - \mathbf{r'}| = 0 because that location would directly overlap with the location of a charged particle (e.g. electron or proton) which is not a valid location to analyze the electric field or potential classically. In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by Henry Cavendish of England. This electric force is conventionally called electrostatic force or Coulomb force.

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A charged particle produces an electric field with a magnitude of $2.0 \mathrm{~N} / \mathrm{C}$ at a point that is $50 \mathrm{~cm}$ away from the particle. What is the magnitude of the particle's charge?

Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The voltage between two points is defined as:Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 685–686 {\Delta V} = -\int {\mathbf E \cdot d \boldsymbol \ell} with d \boldsymbol \ell the element of path along the integration of electric field vector E. Electric field work is the work performed by an electric field on a charged particle in its vicinity. List of orders of magnitude for electric charge Factor [Coulomb] SI prefix8th edition of the official brochure of the BIPM (SI units and prefixes). 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. : W = Q \int_{a}^{b} \mathbf{E} \cdot \, d \mathbf{r} = Q \int_{a}^{b} \frac{\mathbf{F_E}}{Q} \cdot \, d \mathbf{r}= \int_{a}^{b} \mathbf{F_E} \cdot \, d \mathbf{r} where :Q is the electric charge of the particle :E is the electric field, which at a location is the force at that location divided by a unit ('test') charge :FE is the Coulomb (electric) force :r is the displacement :\cdot is the dot product operator ==Mathematical description== Given a charged object in empty space, Q+. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. In physics, a charged particle is a particle with an electric charge. The work per unit of charge is defined by moving a negligible test charge between two points, and is expressed as the difference in electric potential at those points. From these two constants, the elementary charge can be deduced: e = \frac{2}{R_\text{K} K_\text{J}}. ===CODATA method=== The relation used by CODATA to determine elementary charge was: e^2 = \frac{2h \alpha}{\mu_0 c} = 2h \alpha \varepsilon_0 c, where h is the Planck constant, α is the fine-structure constant, μ0 is the magnetic constant, ε0 is the electric constant, and c is the speed of light. The work per unit of charge, when moving a negligible test charge between two points, is defined as the voltage between those points. Prior to this change, the elementary charge was a measured quantity whose magnitude was determined experimentally. right|thumb|150px|V, I, and R, the parameters of Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation: V = {E}{\ell} \ \ \text{or} \ \ E = \frac{V}{\ell}. The particle located experiences an interaction with the electric field. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). This article is a progressive and labeled list of the SI charge orders of magnitude, with certain examples appended to some list objects. Later, the name electron was assigned to the particle and the unit of charge e lost its name. In some other natural unit systems the unit of charge is defined as \sqrt{\varepsilon_0\hbar c}, with the result that e = \sqrt{4\pi\alpha}\sqrt{\varepsilon_0 \hbar c} \approx 0.30282212088 \sqrt{\varepsilon_0 \hbar c}, where is the fine-structure constant, is the speed of light, is the electric constant, and is the reduced Planck constant. == Quantization == Charge quantization is the principle that the charge of any object is an integer multiple of the elementary charge. 104 ~ Charge on one mole of electrons (Faraday constant) 105 Automotive battery charge. 50Ah = 106 mega- (MC) Charge needed to produce 1 kg of aluminium from bauxite in an electrolytic cell 107 108 Charge in world's largest battery bank (36 MWh), assuming 220 VAC outputhttp://www.popsci.com/science/article/2012-01/china-builds-worlds- largest-battery-36-megawatt-hour-behemoth - China Builds the World's Largest Battery – 01.04.2012 == References == Charge

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In Millikan's experiment, an oil drop of radius $1.64 \mu \mathrm{m}$ and density $0.851 \mathrm{~g} / \mathrm{cm}^3$ is suspended in chamber C (Fig. 22-16) when a downward electric field of $1.92 \times 10^5 \mathrm{~N} / \mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$.

Using the known electric field, Millikan and Fletcher could determine the charge on the oil droplet. right|thumb|Millikan's setup for the oil drop experiment|300x300px The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the electron). Since electric force, in turn, is the product of the electric charge and the known electric field, the electric charge of the oil drop could be accurately computed. Robert A. Millikan and Harvey Fletcher's oil drop experiment first directly measured the magnitude of the elementary charge in 1909, differing from the modern accepted value by just 0.6%. As of 2015, no evidence for fractional charge particles has been found after measuring over 100 million drops. ==Experimental procedure== === Apparatus=== right|thumb|Simplified scheme of Millikan's oil drop experiment|576x576px right|thumb|Oil drop experiment apparatus|335x335px Millikan's and Fletcher's apparatus incorporated a parallel pair of horizontal metal plates. Now the field is turned back on, and the electric force on the drop is :F_E = q E \, where q is the charge on the oil drop and E is the electric field between the plates. By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. The forces due to gravity and viscosity could be calculated based on the size and velocity of the oil drop, so electric force could be deduced. The limit to the precision of the method is the measurement of F: the best experimental value has a relative uncertainty of 1.6 ppm, about thirty times higher than other modern methods of measuring or calculating the elementary charge. === Oil-drop experiment === A famous method for measuring e is Millikan's oil-drop experiment. With the electrical field calculated, they could measure the droplet's charge, the charge on a single electron being (). So the mass discharged is m = \frac{x M}{v N_{\rm A}} = \frac{Q M}{e N_{\rm A} v} = \frac{Q M}{vF} where * is the Avogadro constant; * is the total charge, equal to the number of electrons () times the elementary charge ; * is the Faraday constant. ==Mathematical form== Faraday's laws can be summarized by :Z = \frac{m}{Q} = \frac{1}{F}\left(\frac{M}{v}\right) = \frac{E}{F} where is the molar mass of the substance (usually given in SI units of grams per mole) and is the valency of the ions . The droplets entered the space between the plates and, because they were charged, could be made to rise and fall by changing the voltage across the plates. ===Method=== thumb|372x372px Initially the oil drops are allowed to fall between the plates with the electric field turned off. Millikan and Fletcher's experiment involved measuring the force on oil droplets in a glass chamber sandwiched between two electrodes, one above and one below. A small drop of oil in an electric field would move at a rate that balanced the forces of gravity, viscosity (of traveling through the air), and electric force. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. By 1937 it was "quite obvious" that Millikan's value could not be maintained any longer, and the established value became or . == References == ==Further reading== * * * ==External links== * Simulation of the oil drop experiment (requires JavaScript) * Thomsen, Marshall, "Good to the Last Drop". The drop is allowed to fall and its terminal velocity v1 in the absence of an electric field is calculated. The experiment entailed observing tiny electrically charged droplets of oil located between two parallel metal surfaces, forming the plates of a capacitor. First, with zero applied electric field, the velocity of a falling droplet was measured. Reasons for a failure to generate a complete observation include annotations regarding the apparatus setup, oil drop production, and atmospheric effects which invalidated, in Millikan's opinion (borne out by the reduced error in this set), a given particular measurement. ==Millikan's experiment as an example of psychological effects in scientific methodology== thumb|A scatter plot of electron charge measurements as suggested by Feynman, using papers published from 1913-1951 In a commencement address given at the California Institute of Technology (Caltech) in 1974 (and reprinted in Surely You're Joking, Mr. Feynman! in 1985 as well as in The Pleasure of Finding Things Out in 1999), physicist Richard Feynman noted: (adapted from the 1974 California Institute of Technology commencement address), Donald Simanek's Pages, Lock Haven University, rev. December 2017. the value of the elementary charge is defined to be exactly . A mist of atomized oil drops was introduced through a small hole in the top plate and was ionized by an x-ray, making them negatively charged. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest.

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The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_1=+3.0 \mu \mathrm{C}, x_1=3.5 \mathrm{~cm}, y_1=0.50 \mathrm{~cm}$, and $q_2=-4.0 \mu \mathrm{C}, x_2=-2.0 \mathrm{~cm}, y_2=1.5 \mathrm{~cm}$. Find the magnitude of the electrostatic force on particle 2 due to particle 1.

The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. Then, by Newton's second law, \ddot{\mathbf{r}} = \ddot{\mathbf{x}}_{1} - \ddot{\mathbf{x}}_{2} = \left( \frac{\mathbf{F}_{21}}{m_{1}} - \frac{\mathbf{F}_{12}}{m_{2}} \right) = \left(\frac{1}{m_{1}} + \frac{1}{m_{2}} \right)\mathbf{F}_{21} The final equation derives from Newton's third law; the force of the second body on the first body (F21) is equal and opposite to the force of the first body on the second (F12). The Darwin interaction term in the Lagrangian is then L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 where again we kept only the lowest order term in . ==Lagrangian equations of motion== The equation of motion for one of the particles is \frac{d}{dt} \frac{\partial}{\partial \mathbf v_1} L\left( \mathbf r_1 , \mathbf v_1 \right) = abla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) \frac{d \mathbf p_1}{dt} = abla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) where is the momentum of the particle. ===Free particle=== The equation of motion for a free particle neglecting interactions between the two particles is \frac{d}{dt} \left[ \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 \right] = 0 \mathbf p_1 = \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 ===Interacting particles=== For interacting particles, the equation of motion becomes \frac{d}{dt} \left[ \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 + \frac{q_1}{c} \mathbf A\left( \mathbf r_1 \right) \right] = \- abla \frac{q_1 q_2}{r} \+ abla \left[ \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 \right] \frac{d \mathbf p_1}{dt} = \frac{q_1 q_2}{r^2} \hat{\mathbf r} \+ \frac{q_1 q_2}{r^2} \frac{1}{2c^2} \left\\{ \mathbf v_1 \left( { \hat\mathbf{r} \cdot \mathbf v_2} \right) \+ \mathbf v_2 \left( { \hat\mathbf{r} \cdot \mathbf v_1}\right) \- \hat\mathbf{r} \left[ \mathbf v_1 \cdot \left( \mathbf 1 +3 \hat\mathbf{r} \hat\mathbf{r}\right)\cdot \mathbf v_2\right] \right\\} \mathbf p_1 = \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1\mathbf v_1 \+ \frac{q_1}{c} \mathbf A\left( \mathbf r_1 \right) \mathbf A \left( \mathbf r_1 \right) = \frac{q_2}{2c} \frac{1}{r} \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 \mathbf r = \mathbf r_1 - \mathbf r_2 ==Hamiltonian for two particles in a vacuum== The Darwin Hamiltonian for two particles in a vacuum is related to the Lagrangian by a Legendre transformation H = \mathbf p_1 \cdot \mathbf v_1 + \mathbf p_2 \cdot \mathbf v_2 - L. The difference vector Δr = rBC − rAB equals ΔvΔt (green line), where Δv = vBC − vAB is the change in velocity resulting from the force at point B. The Hamiltonian becomes H\left( \mathbf r_1 , \mathbf p_1 ,\mathbf r_2 , \mathbf p_2 \right)= \left( 1 - \frac{1}{4} \frac{p_1^2}{m_1^2 c^2} \right) \frac{p_1^2}{2 m_1} \; + \; \left( 1 - \frac{1}{4} \frac{p_2^2}{m_2^2 c^2} \right) \frac{p_2^2}{2 m_2} \; + \; \frac{q_1 q_2}{r} \; - \; \frac{q_1 q_2}{r} \frac{1}{2m_1 m_2 c^2} \mathbf p_1\cdot \left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right] \cdot\mathbf p_2 . ==Hamiltonian equations of motion== The Hamiltonian equations of motion are \mathbf v_1 = \frac{\partial H}{\partial \mathbf p_1} and \frac{d \mathbf p_1}{dt} = - abla_1 H which yield \mathbf v_1 = \left( 1- \frac{1}{2} \frac{p_1^2}{m_1^2 c^2} \right) \frac{\mathbf p_1}{m_1} \- \frac{q_1 q_2}{2m_1m_2 c^2} \frac{1}{r} \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf p_2 and \frac{d \mathbf p_1}{dt} = \frac{q_1 q_2}{r^2}{\hat{\mathbf r}} \; + \; \frac{q_1 q_2}{r^2} \frac{1}{2m_1 m_2 c^2} \left\\{ \mathbf p_1 \left( { {\hat{\mathbf r}}\cdot \mathbf p_2} \right) \+ \mathbf p_2 \left( { {\hat{\mathbf r}}\cdot \mathbf p_1}\right) \- {\hat{\mathbf r}} \left[ \mathbf p_1 \cdot \left( \mathbf 1 +3 {\hat{\mathbf r}}{\hat{\mathbf r}}\right)\cdot \mathbf p_2\right] \right\\} Note that the quantum mechanical Breit equation originally used the Darwin Lagrangian with the Darwin Hamiltonian as its classical starting point though the Breit equation would be better vindicated by the Wheeler–Feynman absorber theory and better yet quantum electrodynamics. ==See also== * Static forces and virtual-particle exchange * Breit equation * Wheeler–Feynman absorber theory ==References== Category:Magnetostatics Category:Equations of physics Thus, the equation of motion for r can be written in the form \mu \ddot{\mathbf{r}} = \mathbf{F} where \mu is the reduced mass \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} = \frac{m_1 m_2}{m_1 + m_2} As a special case, the problem of two bodies interacting by a central force can be reduced to a central-force problem of one body. ==Qualitative properties== ===Planar motion=== thumb|right|alt=The image shows a yellow disc with three vectors. The current generated by the second particle is \mathbf J = q_2 \mathbf v_2 \delta{\left( \mathbf r - \mathbf r_2 \right)}, which has a Fourier transform \mathbf J\left( \mathbf k \right) \equiv \int d^3r \exp\left( -i\mathbf k \cdot \mathbf r \right) \mathbf J\left( \mathbf r \right) = q_2 \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right). Since the potential profile passes through a minimum at the mid-plane, it is easiest to evaluate the disjoining pressure at the midplane. To demonstrate this, let x1 and x2 be the positions of the two particles, and let r = x1 − x2 be their relative position. A way to correct this problem is to shift the force to zero at r_c, thus removing the discontinuity. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The potential at the midplane is thus given by twice the value of this potential at a distance z = h/2. In other words, the azimuthal angles of the two particles are related by the equation φ2(t) = k φ1(t). Once the potential profile is known, the force per unit area between the plates expressed as the disjoining pressure Π can be obtained as follows. The perpendicular force is Ze^2/4\pi\epsilon_0 b^2 at the closest approach and the duration of the encounter is about b/v. If F is a central force, it must be parallel to the vector rB from the center O to the point B (dashed green line); in that case, Δr is also parallel to rB.

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An electron on the axis of an electric dipole is $25 \mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \times 10^{-29} \mathrm{C} \cdot \mathrm{m}$ ? Assume that $25 \mathrm{~nm}$ is much larger than the separation of the charged particles that form the dipole.

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. (See electron electric dipole moment). The SI unit for electric dipole moment is the coulomb-meter (C⋅m). It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with angular momentum , its magnetic dipole moment is given by: \boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\,\mathbf{L}\,, where e is the electron rest mass. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. Further improvements, or a positive result, would place further limits on which theory takes precedence. == Formal definition == As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that : \mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} depends on the point {\mathbf r}_0 about which the moment of the charge distribution \rho({\mathbf r}) is taken. The electric dipole moment vector also points from the negative charge to the positive charge. Experiments have been performed to measure the electric dipole moment of various particles like the electron and the neutron. However, when making measurements at a distance much larger than the charge separation, the dipole gives a good approximation of the actual electric field. The dipole moment is particularly useful in the context of an overall neutral system of charges, for example a pair of opposite charges, or a neutral conductor in a uniform electric field. A nonzero electric dipole moment can only exist if the centers of the negative and positive charge distribution inside the particle do not coincide. See Landé g-factor for details. ==Example: hydrogen atom== For a hydrogen atom, an electron occupying the atomic orbital , the magnetic dipole moment is given by :\mu_\text{L} = -g_\text{L} \frac{\mu_\text{B}}{\hbar}\langle\Psi_{n,\ell,m}|L|\Psi_{n,\ell,m}\rangle = -\mu_\text{B}\sqrt{\ell(\ell + 1)}. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The q^2 = 0 form factor F_1(0) = -e is the electron's charge, \mu = [\,F_1(0)+F_2(0)\,]/[\,2\,m_{\rm e}\,] is its static magnetic dipole moment, and -F_3(0)/[\,2\,m_{\rm e}\,] provides the formal definion of the electron's electric dipole moment. Theoretically, an electric dipole is defined by the first-order term of the multipole expansion; it consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.Many theorists predict elementary particles can have very tiny electric dipole moments, possibly without separated charge. The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin).

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Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?

Consequently, there is always a small current of approximately 2pA per square metre transporting charged particles in the form of atmospheric ions between the ionosphere and the surface. === Fair weather=== This current is carried by ions present in the atmosphere (generated mainly by cosmic rays in the free troposphere and above, and by radioactivity in the lowest 1km or so). The movement of charge between the Earth's surface, the atmosphere, and the ionosphere is known as the global atmospheric electrical circuit. Near the surface of the Earth, the magnitude of the field is on average around 100 V/m. Atmospheric electricity is the study of electrical charges in the Earth's atmosphere (or that of another planet). There is a weak conduction current of atmospheric ions moving in the atmospheric electric field, about 2 picoamperes per square meter, and the air is weakly conductive due to the presence of these atmospheric ions. ===Variations=== Global daily cycles in the atmospheric electric field, with a minimum around 03 UT and peaking roughly 16 hours later, were researched by the Carnegie Institution of Washington in the 20th century. The protonosphere is a layer of the Earth's atmosphere (or any planet with a similar atmosphere) where the dominant components are atomic hydrogen and ionic hydrogen (protons). Atmospheric ions created by cosmic rays and natural radioactivity move in the electric field, so a very small current flows through the atmosphere, even away from thunderstorms. A global atmospheric electrical circuit is the continuous movement of atmospheric charge carriers, such as ions, between an upper conductive layer (often an ionosphere) and surface. The Earth's electrical environment. A ring current is an electric current carried by charged particles trapped in a planet's magnetosphere. thumb|400px|A snapshot of the variation of the Earth's magnetic field from its intrinsic field at 400 km altitude, due to the ionospheric current systems. Because of this, the ionosphere is positively charged relative to the earth. * R Reiter, Relationships Between Atmospheric Electric Phenomena and Simultaneous Meteorological Conditions. 1960 * J. Law, The ionisation of the atmosphere near the ground in fair weather. Global-scale ionospheric circulation establishes a Sq (solar quiet) current system in the E region of the Earth's ionosphere (100-130 km altitude), and a primary eastwards electric field near day-side magnetic equator, where the magnetic field is horizontal and northwards. * R Markson, Modulation of the earth's electric field by cosmic radiation. Over a flat field on a day with clear skies, the atmospheric potential gradient is approximately 120 V/m. This Carnegie curve variation has been described as "the fundamental electrical heartbeat of the planet".Liz Kalaugher, Atmospheric electricity affects cloud height 3 March 2013, physicsworld.com accessed 15 April 2021 Even away from thunderstorms, atmospheric electricity can be highly variable, but, generally, the electric field is enhanced in fogs and dust whereas the atmospheric electrical conductivity is diminished. === Links with biology === The atmospheric potential gradient leads to an ion flow from the positively charged atmosphere to the negatively charged earth surface. The global electrical circuit is also relevant to the study of human health and air pollution, due to the interaction of ions and aerosols. * Bespalov P.A., Chugunov Yu. V. and Davydenko S.S., Planetary electric generator under fair- weather condition with altitude-dependent atmospheric conductivity, Journal of Atmospheric and Terrestrial Physics, v.58, #5,pp. 605–611,1996 * DG Yerg, KR Johnson, Short-period fluctuations in the fair weather electric field. Physics – Doklady, Volume 39, Issue 8, August 1994, pp. 553–555 * Bespalov, P. A.; Chugunov, Yu. V.; Davydenko, S. S. Planetary electric generator under fair-weather conditions with altitude-dependent atmospheric conductivity. Discoveries about the electrification of the atmosphere via sensitive electrical instruments and ideas on how the Earth's negative charge is maintained were developed mainly in the 20th century, with CTR Wilson playing an important part.Encyclopedia of Geomagnetism and Paleomagnetism - Page 359 Current research on atmospheric electricity focuses mainly on lightning, particularly high-energy particles and transient luminous events, and the role of non-thunderstorm electrical processes in weather and climate. ==Description== Atmospheric electricity is always present, and during fine weather away from thunderstorms, the air above the surface of Earth is positively charged, while the Earth's surface charge is negative. Atmospheric electricity is an interdisciplinary topic with a long history, involving concepts from electrostatics, atmospheric physics, meteorology and Earth science.

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