Pendulum (mechanics)
A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple  pendulum allow the equations of motion to be solved analytically for small-angle oscillations. Simple gravity pendulum. A "simple gravity pendulum" is an idealized mathematical model of a real pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in this model there is no frictional energy loss, when given an initial displacement it will swing back and forth at a constant amplitude. The model is based on these assumptions: The differential equation which represents the motion of a simple pendulum is where is the magnitude of the gravitational  field, is the length of the rod or cord, and is the angle from the vertical to the pendulum. First start by defining the torque on the pendulum bob using the force due to gravity. formula_1 where is the length vector of the pendulum and is the force due to gravity. For now just consider the magnitude of the torque on the pendulum. formula_2 where is the mass of the pendulum, is the acceleration due to gravity, is the length of the pendulum, and is the angle between the length vector and the force due to gravity. Next rewrite the  angular momentum. formula_3 Again just consider the magnitude of the angular momentum. formula_4 and its time derivative formula_5 According to , we can get by comparing the magnitudes formula_6 thus: formula_7 which is the same result as obtained through force analysis. where is the vertical distance the pendulum fell. Look at Figure 2, which presents the trigonometry of a simple pendulum. If the pendulum starts its swing from some initial angle , then , the vertical distance from the screw, is given by formula_8 Similarly, for , we have formula_9 Then is the difference of the two formula_10 In terms of gives This equation  is known as the "first integral of motion", it gives the velocity in terms of the location and includes an integration constant related to the initial displacement (). We can differentiate, by applying the chain rule, with respect to time to get the acceleration formula_11 which is the same result as obtained through force analysis. Small-angle approximation. The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However, adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily  obtained. If it is assumed that the angle is much less than 1 radian (often cited as less than 0.1 radians, about 6°), or formula_12 then substituting for into using the small-angle approximation, formula_13 yields the equation for a harmonic oscillator, formula_14 The error due to the approximation is of order (from the Taylor expansion for ). Let the starting angle be . If it is assumed that the pendulum is released with zero angular velocity, the solution becomes The motion is simple harmonic motion where is the amplitude of the oscillation (that is, the maximum angle between the rod  of the pendulum and the vertical). The corresponding approximate period of the motion is then which is known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude ; this is the property of isochronism that Galileo discovered. Rule of thumb for pendulum length. formula_15 gives formula_16 If SI units are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then , and (0.994 is the approximation to 3 decimal places). Therefore, relatively reasonable approximations for the length and period  are: formula_17 where is the number of seconds between "two" beats (one beat for each side of the swing), and is measured in metres. Arbitrary-amplitude period. For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method (), formula_18 and then integrating over one complete cycle, formula_19 or twice the half-cycle formula_20 or four times the quarter-cycle formula_21 which leads to formula_22 Note that this integral diverges as approaches the vertical formula_23 so that a pendulum with just the right energy to  go vertical will never actually get there. (Conversely, a pendulum close to its maximum can take an arbitrarily long time to fall down.) This integral can be rewritten in terms of elliptic integrals as formula_24 where is the incomplete elliptic integral of the first kind defined by formula_25 Or more concisely by the substitution formula_26 expressing in terms of , Here is the complete elliptic integral of the first kind defined by formula_27 For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth ( = ) at initial angle 10 degrees is  formula_28 The linear approximation gives formula_29 The difference between the two values, less than 0.2%, is much less than that caused by the variation of with geographical location. From here there are many ways to proceed to calculate the elliptic integral. Legendre polynomial solution for the elliptic integral. Given and the Legendre polynomial solution for the elliptic integral: formula_30 where denotes the double factorial, an exact solution to the period of a simple pendulum is: formula_31 Figure 4 shows the relative errors using the power series. is the linear approximation, and to include respectively the terms up to the 2nd  to the 10th powers. Power series solution for the elliptic integral. Another formulation of the above solution can be found if the following Maclaurin series: formula_32 is used in the Legendre polynomial solution above. The resulting power series is: formula_33 more fractions available in the On-Line Encyclopedia of Integer Sequences with having the numerators and having the denominators. Arithmetic-geometric mean solution for elliptic integral. Given and the arithmetic–geometric mean solution of the elliptic integral: formula_34 where is the arithmetic-geometric mean of and . This yields an alternative and faster-converging formula for the period: formula_35 The first  iteration of this algorithm gives formula_36 This approximation has the relative error of less than 1% for angles up to 96.11 degrees. Since formula_37 the expression can be written more concisely as formula_38 The second order expansion of formula_39 reduces to formula_40 A second iteration of this algorithm gives formula_41 This second approximation has a relative error of less than 1% for angles up to 163.10 degrees. Approximate formulae for the nonlinear pendulum period. Though the exact period formula_42 can be determined, for any finite amplitude formula_43 rad, by evaluating the corresponding complete elliptic integral formula_44, where formula_45, this is  often avoided in applications because it is not possible to express this integral in a closed form in terms of elementary functions. This has made way for research on simple approximate formulae for the increase of the pendulum period with amplitude (useful in introductory physics labs, classical mechanics, electromagnetism, acoustics, electronics, superconductivity, etc. The approximate formulae found by different authors can be classified as follows: Of course, the increase of formula_42 with amplitude is more apparent when formula_53, as has been observed in many experiments using either a rigid rod or a disc. As accurate timers and sensors are currently  available even in introductory physics labs, the experimental errors found in ‘very large-angle’ experiments are already small enough for a comparison with the exact period and a very good agreement between theory and experiments in which friction is negligible has been found. Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. In 2008, Lima derived a weighted-average formula with this characteristic: formula_54 where formula_55, which presents a maximum error of only 0.6% (at formula_56). Arbitrary-amplitude  angular displacement Fourier series. The Fourier series expansion of formula_57 is given by formula_58 where formula_59 is the elliptic nome, formula_60, and formula_61 the angular frequency. If one defines formula_62 formula_59 can be approximated using the expansion formula_64 (see ). Note that for formula_65 we have formula_66, thus the approximation is applicable even for large amplitudes. Examples. The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the  vertical axis is velocity. With a large enough initial velocity the pendulum does not oscillate back and forth but rotates completely around the pivot. Compound pendulum. A compound pendulum (or physical pendulum) is one where the rod is not massless, and may have extended size; that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the pendulum's period depends on its moment of inertia around the pivot point. The equation of torque gives: formula_67 where: The torque is generated by gravity so: formula_68 where: Hence, under the small-angle approximation , formula_69 where is the moment  of inertia of the body about the pivot point. The expression for is of the same form as the conventional simple pendulum and gives a period of formula_70 And a frequency of formula_71 If the initial angle is taken into consideration (for large amplitudes), then the expression for formula_72 becomes: formula_73 and gives a period of: formula_74 where is the maximum angle of oscillation (with respect to the vertical) and is the complete elliptic integral of the first kind. Physical interpretation of the imaginary period. The Jacobian elliptic function that expresses the position of a pendulum as a function of  time is a doubly periodic function with a real period and an imaginary period. The real period is, of course, the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period: if is the maximum angle of one pendulum and is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other. Coupled pendula. Coupled pendulums can affect each other's motion, either through a direction connection (such as a spring connecting the bobs) or through motions in  a supporting structure (such as a tabletop). The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian Mechanics. The kinetic energy of the system is: formula_75 where formula_76 is the mass of the bobs, formula_77 is the length of the strings, and formula_78, formula_79 are the angular displacements of the two bobs from equilibrium. The potential energy of the system is: formula_80 where formula_81 is the gravitational acceleration, and formula_82 is the spring constant. The displacement formula_83 of the spring from its equilibrium position assumes the small angle approximation . The Lagrangian is then formula_84 which leads to the following set of coupled differential equations: formula_85 Adding and subtracting these two equations in turn, and applying the small angle approximation, gives two harmonic oscillator equations in the variables formula_86 and formula_87: formula_88 with the corresponding solutions formula_89 where formula_90 and formula_91, formula_92, formula_72, formula_94 are constants of integration. Expressing the solutions in terms of formula_78 and formula_79 alone: formula_97 If the bobs are not given an initial push, then the condition formula_98 requires formula_99, which gives (after some rearranging): formula_100