OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Example of Vector Potential

If the solenoidal vector \,${\vec {U}}={\vec {U}}(x,\,y,\,z)$\, is a homogeneous function of degree $\lambda$ ($\neq -2$),\, then it has the vector potential
   ${\begin{matrix}{\vec {A}}={\frac {1}{\lambda \!+\!2}}{\vec {U}}\!\times \!{\vec {r}},\end{matrix}}$
where\, ${\vec {r}}=x{\vec {i}}\!+\!y{\vec {j}}\!+\!z{\vec {k}}$\, is the position vector.
Proof. \, Using the entry nabla acting on products, we first may write $\nabla \times ({\frac {1}{\lambda \!+\!2}}{\vec {U}}\!\times \!{\vec {r}})={\frac {1}{\lambda \!+\!2}}[({\vec {r}}\cdot \nabla ){\vec {U}}-({\vec {U}}\cdot \nabla ){\vec {r}}-(\nabla \cdot {\vec {U}}){\vec {r}}+(\nabla \cdot {\vec {r}}){\vec {U}}].$ In the brackets the first product is, according to Euler's theorem on homogeneous functions, equal to $\lambda {\vec {U}}$.\, The second product can be written as\, </math>U_x\frac{\partial\vec{r}}{\partial x}+ U_y\frac{\partial\vec{r}}{\partial y}+U_z\frac{\partial\vec{r}}{\partial z}$,whichis$U_x\vec{i}+U_y\vec{j}+U_z\vec{k}$,i.e.$\vec{U}$.\,Thethirdproductis,duetothesodenoidalness,equalto\,$0\vec{r} = \vec{0}$.\,Thelastproductequalsto$3\vec{U}$(seethe[[../PositionVector/|firstformula]]forpositionvector).\,Thuswegettheresult<math>\nabla \times ({\frac {1}{\lambda \!+\!2}}{\vec {U}}\!\times \!{\vec {r}})={\frac {1}{\lambda \!+\!2}}[\lambda {\vec {U}}-{\vec {U}}-{\vec {0}}+3{\vec {U}}]={\vec {U}}.$ This means that ${\vec {U}}$ has the vector potential (1).
# PlanetPhysics/Example of Generalized Coordinates for Constrained Motion on a Horizontal Circle

Let a particle of mass $m$, constrained to move on a smooth horizontal circle of radius $a$, be given an initial velocity $V$, and let it be resisted by the air with a force proportional to the square of its velocity.
Here we have one degree of freedom. Let us take as our coordinate $q_{1}$ the angle $\theta$ which the particle has described about the center of its path in the time $t$.
For the kinetic energy $T={\frac {m}{2}}a^{2}{\dot {\theta }}^{2},$
and we have
${\frac {\partial T}{\partial {\dot {\theta }}}}=ma^{2}{\dot {\theta }}.$
\quad Our differential equation is
$ma^{2}{\ddot {\theta }}\delta \theta =-ka^{2}{\dot {\theta }}^{2}a\delta \theta ,$
which reduces to
${\ddot {\theta }}+{\frac {k}{m}}a{\dot {\theta }}^{2}=0,$
or
${\frac {d{\dot {\theta }}}{dt}}+{\frac {k}{m}}a{\dot {\theta }}^{2}=0.$
Separating the variables,
${\frac {d{\dot {\theta }}}{{\dot {\theta }}^{2}}}+{\frac {k}{m}}adt=0.$
Integrating,
$-{\frac {1}{\dot {\theta }}}+{\frac {k}{m}}at=C=-{\frac {a}{V}}.$
${\frac {1}{\dot {\theta }}}={\frac {ma+kVat}{mV}},$
${\frac {d\theta }{dt}}={\frac {mV}{ma+kVat}},$
$\theta ={\frac {m}{ka}}\log \left[m+kVt\right]+C,$
$\theta ={\frac {m}{ka}}\log \left[1+{\frac {kVt}{m}}\right];$
and the problem of the motion is completely solved.
# PlanetPhysics/Example of Mechanical Power

{\mathbf Problem 1.} If a 200 pound man runs up a flight of stairs 20 feet high in seven seconds, at what rate is he working? Express the result in foot-pounds per second, horsepower, and watts.
{\mathbf Answer.}
First find the work done in climbing from one floor to the next, which is equal to the potential energy gained.
$Work=Wh=(200\,pounds)(20\,feet)=4,000\,foot-pounds$
Since the rate of doing this work, that is, the power, is the work divided by the time, we find the quotient obtained from (4,000) foot-pounds) / (7 seconds) or 571 foot-pounds per second. This is equivalent to 571/550 or 1.037 horsepower or in terms of watts, (1.037)(746) = 774 watts. It is perfectly possible for a man to work at the rate of a horsepower for a short time; it is also possible for a horse to work at the rate of many horsepower for a short time; but it takes a first-class horse to work at the rate of one horsepower for an entire day.
{\mathbf Problem 2.} A horse pulls a plow at the rate of two feet per second and exerts a forward force on the plow of 250 pounds. At what rate does the horse work? Express the result in foot-pounds per second, in horsepower, and in watts. How much work does the horse do in five hours?
{\mathbf Answer.}
The rate of doing work is the power, one formula for which is the product of the force and the speed. Since the force is 250 pounds and the speed is 2.00 feet/second, the required power is (250 pounds)(2.00 feet/second) or 500 foot-pounds/second. Since there are 550 foot-pounds/second per horsepower, this power is 500/550 or 0.909 horsepower. It will be noticed that the numerator together with its units is 550 foot-pounds per second per horsepower, or 550 foot-pounds/horsepower-second. When we divide the numerator by the denominator, all the units cancel exept horsepower, which being in the denominator of the denominator can be transferred to the numerator and survives in the result. By multiplying 0.909 horsepower by the conversion factor 746 watts/horsepower, the horsepower cancels giving 678 watts. Since power is the ratio of work to time, it follows that work is the product of power and time. we may therefore multiply any one of our three answers by an amount of time corresponding to five hours and get the work done by the horse. Five hours is 18,000 seconds. When we multiply 500 foot-pounds/second by 18,000 seconds, the seconds cancel and we have 9,000,000 foot-pounds. if we multiply by 0.909 horsepower by 5.00 hours, we have 4.54 horsepower-hours. If we multiply 678 watts by 5.00 hours, we have 3,390 watt-hours. Since 1,000 watts equals one kilowatt, this is equivalent to 3.39 kilowatt-hours.
# PlanetPhysics/Example of Linear Least Squares

The assumption of linear least squares is that there is a linear relationship between our measurements $z$ and the variables to be estimated $x$
   $z=Mx+b$
For this example let us assume that our measurements are given in Table 1 and you can see them plotted in Figure 1.
| x | -3.0 | -2.5 | -2.0 | -1.5 | -1.0 | -0.5 | 0.0 | 0.5 | 1.0 | 1.5 |
| z | -1.0 | -0.25 | 0.0 | 0.25 | 0.4 | 0.7 | 1.0 | 1.1 | 1.4 | 1.8 |
   {\mathbf Table 1:} Example Data
The linear least squares solution to fit the given data is given by the equation
   $x_{fit}=(A^{T}A)^{-1}A^{T}z$
The only not so obvious step before using a tool like Matlab, is to form the $A$ matrix, which is a combination of an identity vector and $x$ as column vectors, such that
$A=[\,\,1\,\,|\,\,x\,\,]$
This is clarified by looking at the example code in Matlab, LinearLeastSquares.m. A plot of fitting the measurement data with a line such that it minimizes the the mean square of the error is shown in Figure 1.
The equation of the line to fit this data is then
$z=0.543x+0.947$
   \includegraphics[scale=.6]{LinearLeastSquares2.eps}
   {\mathbf Figure 1:} Linear Fit of Example Data (Matlab)
   \includegraphics[scale=.8]{least_squares.eps}
   {\mathbf Figure 1:} Linear Fit of Example Data (rlplot)
# PlanetPhysics/Example of Quantum Commutator Algebra

Here we illustrate a simple example of quantum commutator algebra using a one-dimensional quantum system. Let $f(q)$ be a function of $q$. The three commutators of $q$ and of each of the functions $p^{2}f(q)$, $pf(q)p$, and $f(q)p^{2}$ may all be identified (to within the factor $i\hbar$) with the derivative with respect to $p$ of these functions, but they are not the same operators. Indeed, by repeated application of the commutator algebra rule
   $[q_{i},G(p_{1},\dots ,p_{R})]=i\hbar {\frac {\partial G}{\partial p_{i}}}$
we get
$[q,p^{2}f(q)]=2i\hbar pf(q)$ $[q,pfp]=i\hbar (fp+pf)$ $[q,fp^{2}]=2i\hbar fp$
In the same way
$[p,p^{2}f]={\frac {\hbar }{i}}p^{2}f^{\prime }$ $[p,pfp]={\frac {\hbar }{i}}pf^{\prime }p$ $[p,fp^{2}]={\frac {\hbar }{i}}f^{\prime }p^{2}$
# PlanetPhysics/Example of Trace

For the matrix $M,$
$M=\left({\begin{matrix}8&1&6\\3&5&7\\4&9&2\end{matrix}}\right)$
the trace of $M,$ $tr(M)$ is
$tr(M)=8+5+2=15$
# PlanetPhysics/Example of a Matrix Commutator

Calculate the commutator $[A,B]$ of the matrices $A=\left[{\begin{matrix}5&3i&-i\\1&0&2\\i&2i&-1\end{matrix}}\right]\,\,\,B=\left[{\begin{matrix}1&0&1\\-2&2i&3\\3i&1&-1\end{matrix}}\right]$
Ans.
The commutator is calculated by definition as
$[A,B]=AB-BA$
Carrying out the first matrix multiplication gives
$AB=\left[{\begin{matrix}8-6i&-6-i&5+10i\\1+6i&2&-1\\-6i&-5&1+7i\end{matrix}}\right]$
and the second multiplication is
$BA=\left[{\begin{matrix}5+i&5i&-1-i\\-10+5i&0&-3+6i\\1+14i&-9-2i&6\end{matrix}}\right].$
Finally, subtracting the two yields
$[A,B]=AB-BA\left[{\begin{matrix}3-7i&-6-6i&6+11i\\11+i&2&2-6i\\-1-20i&4+2i&-5+7i\end{matrix}}\right].$
Since the commutator is non-zero, we say that $A$ and $B$ do not commute.
# PlanetPhysics/Examples of Constants of the Motion

There exists an observable which always commutes with the Hamiltonian: the Hamiltonian itself. The energy is therefore a constant of the motion of all systems whose Hamiltonian does not depend explicitly upon the time.
As another possible constant of the motion, let us mention parity . We denote under the name of parity the observable $P$ defined by
   $P\psi (q)=\psi (-q)$
It is easily verified that $P$ is Hermitean. Moreover, $P^{2}=1$ and, consequently, the only possible eigenvalues of $P$ are $+1$ and $-1$; even functions are associated with $+1$, and odd functions with $-1$.
When the Hamiltonian is invariant under the substitution of $-q$ for $q$, we obviously have
$[P,H]=0$
Indeed, if
$H\left({\frac {\hbar }{i}}{\frac {d}{dq}},q\right)=H\left(-{\frac {\hbar }{i}}{\frac {d}{dq}},-q\right)$
one has, for any $\psi (q)$,
$PH\psi =H\left(-{\frac {\hbar }{i}}{\frac {d}{dq}},-q\right)\psi (-q)=H\left({\frac {\hbar }{i}}{\frac {d}{dq}},q\right)\psi (-q)=HP\psi$
Under these conditions, if the wave function has a definite parity at a given initial instant of time, it conserves the same parity in the course of time.
This property is easily extended to a system having an arbitrary number of dimensions; in particular, it applies to systems of particles for which the parity operation amounts to a reflection in space $(\mathbf {r} _{i}\rightarrow -\mathbf {r} _{i})$ and for which the observable parity is defined by
$P\Psi (\mathbf {r} _{1},\mathbf {r} _{2}.\dots )=\Psi (-\mathbf {r} _{1},-\mathbf {r} _{2},\dots )$
# Open Pathology Project/Tuberculous granuloma

Tuberculous granuloma is a type of caseating granuloma that contains a caseating centre, surrounded by epithelioid cells, Langhan giant cells and lymphocytes.

## More information

* Category: Histopathology of tuberculous granuloma
* Structured data : depicts Caseous necrosis
* w:en:Caseous necrosis
# Open Pathology Project/Vesicular mole

Molar pregnancy is an abnormal pregmancy of a non-viable fertilized egg.

## More information

* Category: Histopathology of chorionic villi
* Structured data: depicts chorionic villi
* w:en:Molar pregnancy
* Libre Pathology: Chorionic villi
# Open Quantum Systems

Treating quantum systems as being completely isolated from its surroundings is an idealization that is not always justified. Moreover, the interaction with the environment can give rise to many interesting effects not found in closed quantum systems. I will introduce the theoretical tools required to analyze open quantum systems and discuss recent achievements in the preparation of strongly correlated many-body states by controlling the coupling to the environment.
# Open Pathology Project/Tuberculous lymphadenitis

Tuberculous lymphadenitis is the infection of the lymph nodes by tuberculosis bacteria.

## More information

* w:en:Tuberculous lymphadenitis
* Category: Histopathology of tuberculous lymphadenitis
* Structured data: Tuberculous lymphadenitis
# Open Pathology Project/Tumor giant cell

Tumor giant cell is found in malignant cancers.

## More information

* Category: Histopathology of giant cell tumors
* Structured data: depicts Tumor giant cell
# Python Concepts/Special Methods
## Further Reading or Review

* Previous Lesson: Interfacing with Unix
* This lesson: Special Methods
* Next Lesson: Custom Data Types
* Course Home Page
# Python Concepts/To Get You Started

* Objective
* Learn about comments.
* Review the shell and its prompt.
* Learn what a variable is and what it does.
* Review about Python 3.X and Python 2.X
* Review about a Python script.
* Learn some Python jargon.
Lesson
Note: This course is currently being written in American English: words like colour, centre, and maths are written color, center, and math.
This quick lesson will help you understand the basics of Python, while helping to explain some jargon that you'll hear later on.
The Basics
First off, a comment is a number sign, #, (also called a hashtag in social media) followed by some text. A comment is a human readable sentence that Python will ignore. For example, # This is a comment! will not be interpreted by Python.
Second, if you ever see three greater than signs (>>>) and three periods (...), then this implies that the Python shell is being used. Refer back to the previous lesson on how to access the shell.
Thirdly, a variable is a namespace , like var and somevar, and it can hold a Python data type, like a number or string that you'll learn about later.
Fourth, like the previous lesson stated, Python 3.X will be used in this course, so make sure you don't use Python 2.X or you'll definitely get an error message.
Fifth, a python script is just a plain text file with python code in it. Refer back to the previous lesson for more information.
Python Jargon
CPython: The original Python implementation written in C.
IDE: Stands for Integrated Development Environment. An IDE named IDLE is bundled with a CPython installation.
Namespace: The name of a variable.
PEP: Stands for Python Enhancement Proposals. These documents rationally govern Python.
PSF: Stands for Python Software Foundation. This non-profit organization helps maintain and advance Python.
Pythonic: Something that follows the idioms of Python.
Shell: A simple GUI that can print output and capture input.
Unpythonic: Opposite of Pythonic. Something that doesn't follow or goes against the idioms of Python.
Zen of Python: Formally known as PEP 20. A document that closely expresses Python's principles and philosophies. :
* Assignments
* Make sure you understand this lesson, as it will be assumed that you know this.
* Make sure you can open up Python from a shell or prompt.
* Double check to make sure that you have Python 3.X and if you don't, see Introduction and setup.
* Make sure you know how to create and run a Python script.
* Make sure you understand the Python vocabulary and jargon. If you're interested in learning more, see the Python Documentation Glossary.
Completion status: Ready for testing by learners and teachers. Please begin!
Previous Lesson
Next Lesson
Course Home Page :
# Python Concepts/Variable Scoping

This Programming Language resource is a "stub". Learn how you can help Wikiversity to develop it.

## Page Index

* Lesson
* Assignments
* Further Reading or Review
* References

## Further Reading or Review

Completion status: this resource is a stub, which means that pretty much nothing has been done yet.
Previous Lesson: Functions
This Lesson: Variable Scoping
Next Lesson: Built-in Functions
Course Home Page :
# Motivation and emotion/Book/2022/Relative deprivation and motivation

* Multimedia presentation (3 min)

# Overview

* Clearly state the relationship between Relative Deprivation RD and Motivational outcomes
* Key players: Walter Runcimen & Samuel Stouffer
* Context
* The concept of Relative Deprivation was divided into two clear conceptual distinctions, Egoistic Relative Deprivation and Fraternalisitic Deprivation ( Cite: Walter Runcimen)
John Stacey Adams - who is he? what was his role in equity theory.

## Relative Deprivation Theory

* DEFINITION:
* Context / history:
* The concept of Relative Deprivation was divided into two clear conceptual distinctions, Egoistic Relative Deprivation and Fraternalisitic Deprivation (Runciman, 1966).

### Egoistic Relative Deprivation

* Definition: the perceived discrepancy between an individual’s own current position and the comparison standard
* Characteristics
* Example

### Fraternalistic relative deprivation

* the perceived discrepancy between the position that the person’s ingroup actually has and the position the person thinks it ought to have
* Characteristics
* Example

## Motivation
### Relevant Theories of Motivation

* x
* x
* x
* Case

## Effects of Relative Deprivation
### Life Satisfaction

* x
* x

### Protest/ Social Movements

* Major para - well evidenced in the literature - RD is motivator for social mobility and change motivation
* Also is motivator; violence, riot, social deviance

### Mental Health

* Poorer physical and MH Outcomes (Mishra, & Carleton, 2015)
* Why?
* Implications

## Relative Satisfaction

* Offer opposing construct (define)
* Connect with effects of this on Self concept, attitudes, Motivation - outcomes; Wellbeing, social mobility/movement & 1
* Ways to mediate from a position of relative deprivation to a position of satisfaction?

## Conclusion

Notes to self answer:
* The effect of relative deprivation on motivation XXX
* On the basis of psych theory and research
* Answer focus Q.
* Take home message - Link (treat as thesis)
# PlanetPhysics/Category of Quantum Automata

Let us recall that as a Quantum Algebraic Topology object, a quantum automaton is defined by the quantum triple $Q_{A}=({\mathcal {G}},\mathbb {H} -\Re _{G},Aut({\mathcal {G}})$), where ${\mathcal {G}}$ is a (locally compact) quantum groupoid, $\mathbb {H} -\Re _{G}$ are the unitary representations of ${\mathcal {G}}$ on rigged Hilbert spaces $\Re _{G}$ of quantum states and quantum operators on the Hilbert space $\mathbb {H}$, and $Aut({\mathcal {G}})$ is the transformation, or automorphism equivalence relation of quantum transitions that represents all flip-flop quantum transitions of one cubit each between the permitted quantum states of the quantum automaton.
With the data from above definition we can now define also the category of quantum automata as follows.
The category of quantum automata ${\mathcal {\mathbb {Q} }}_{A}$ is defined as an algebraic category category of logic algebras whose objects are triples $(\mathbb {H} ,\Delta :\mathbb {H} \rightarrow \mathbb {H} ,\mu )$ (where $\mathbb {H}$ is either a Hilbert space or a rigged Hilbert space of quantum states and operators acting on $\mathbb {H}$, and $\mu$ is a measure related to the quantum logic, $LM$, and (quantum) transition probabilities of this quantum system), and whose morphisms are defined between such triples by homomorphisms of Hilbert spaces, $\emptyset :\mathbb {H} \rightarrow \mathbb {H}$, naturally compatible with the operators $\Delta$, and by homomorphisms between the associated Haar measure systems.
An alternative definition is also possible based on Quantum Algebraic Topology .
A quantum algebraic topology definition of the Cod of quantum algebraic automata involves the objects specified above in Definition 0.1 as quantum automaton triples $(Q_{A})$, and quantum automata homomorphisms defined between such triples; these $Q_{A}$ morphisms are defined by groupoid homomorphisms $h:{\mathcal {G}}\rightarrow {\mathcal {G}}^{*}$ and $\alpha :Aut({\mathcal {G}})\rightarrow Aut({\mathcal {G}}^{*})$, together with unitarity preserving mappings $u$ between unitary representations of ${\mathcal {G}}$ on rigged Hilbert spaces (or Hilbert space bundles).
# PlanetPhysics/Catenary

A chain or a homogeneous flexible thin wire takes a form resembling an arc of a parabola when suspended at its ends.\, The arc is not from a parabola but from the graph of the hyperbolic cosine function in a suitable coordinate system.
Let's derive the equation \,$y=y(x)$\, of this curve, called the catenary , in its plane with $x$-axis horizontal and $y$-axis vertical.\, We denote the line density of the weight of the wire by $\sigma$.
In any point \,$(x,\,y)$\, of the wire, the tangent line of the curve forms an angle $\varphi$ with the positive direction of $x$-axis.\, Then, $\tan \varphi ={\frac {dy}{dx}}=y'.$ In the point, a certain tension $T$ of the wire acts in the direction of the tangent; it has the horizontal component\, $T\cos \varphi$\, which has apparently a constant value $a$.\, Hence we may write $T={\frac {a}{\cos \varphi }},$ whence the vertical component of $T$ is $T\sin {\varphi }=a\tan {\varphi }$ and its differential $d(T\sin {\varphi })=a\,d\tan {\varphi }=a\,dy'.$ But this differential is the amount of the supporting force acting on an infinitesimal portion of the wire having the projection $dx$ on the $x$-axis.\, Because of the equilibrium, this force must be equal the weight\,
$\sigma {\sqrt {1+(y'(x))^{2}}}\,dx$ (see the arc length).\, Thus we obtain the differential equation
   ${\begin{matrix}\sigma {\sqrt {1+y'^{2}}}\,dx=a\,dy',\end{matrix}}$
which allows the separation of variables: $\int dx={\frac {a}{\sigma }}\int {\frac {dy'}{\sqrt {1+y'^{2}}}}$ This may be solved by using the substitution $y':=\sinh {t},\quad dy'=\cosh {t}\,dt,\quad {\sqrt {1+y'^{2}}}=\cosh {t}$ giving $x={\frac {a}{\sigma }}t+x_{0},$ i.e. $y'={\frac {dy}{dx}}=\sinh {\frac {\sigma (x-x_{0})}{a}}.$ This leads to the final solution $y={\frac {a}{\sigma }}\cosh {\frac {\sigma (x-x_{0})}{a}}+y_{0}$ of the equation (1).\, We have denoted the constants of integration by $x_{0}$ and $y_{0}$.\, They determine the position of the catenary in regard to the coordinate axes.\, By a suitable choice of the axes and the measure units one gets the simple equation
   ${\begin{matrix}y=a\cosh {\frac {x}{a}}\end{matrix}}$
of the catenary.
Some properties of catenary
* $\tan \varphi =\sinh {\frac {x}{a}},\quad \sin \varphi =\tanh {\frac {x}{a}}$
* The arc length of the catenary (2) from the apex\, $(0,\,a)$\, to the point\, $(x,\,y)$\, is\,\, $a\sinh {\frac {x}{a}}={\sqrt {y^{2}-a^{2}}}$.
* The radius of curvature of the catenary (2) is\, $a\cosh ^{2}{\frac {x}{a}}$, which is the same as length of the normal line of the catenary between the curve and the $x$-axis.
* The catenary is the catacaustic of the exponential curve reflecting the vertical rays.
* If a parabola rolls on a straight line, the focus draws a catenary.
* The involute (or evolvent) of the catenary is the tractrix.
# PlanetPhysics/Category of Representations

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }
# PlanetPhysics/Centre of Mass of Half Disc

Let $E$ be the upper half-disc of the disc\, $x^{2}+y^{2}\leqq R$\, in $\mathbb {R} ^{2}$ with a constant surface-density 1. By the symmetry, its centre of mass locates on its medium radius, and therefore we only have to calculate the ordinate $Y$ of the centre of mass. For doing that, one can use in this two-dimensional case instead a triple integral the double integral $Y={\frac {1}{\nu (E)}}\int \!\!\int _{E}y\,dx\,dy,$ where\, $\nu (E)={\frac {\pi R^{2}}{2}}$\, is the area (and the mass) of the half-disc. The region of integration is defined by $E=\{(x,\,y)\in \mathbb {R} ^{2}\,\vdots \;\;-R\leqq x\leqq R,\;0\leqq y\leqq {\sqrt {R^{2}-x^{2}}}\}.$ Accordingly, we may write Failed to parse (unknown function "\sijoitus"): {\displaystyle Y = \frac{2}{\pi R^2}\!\int_{-R}^R\!dx\int_0^{\sqrt{R^2-x^2}}\!y\,dy = \frac{2}{\pi R^2}\!\int_{-R}^R\frac{R^2\!-\!x^2}{2}\,dx = \frac{2}{\pi R^2}\sijoitus{-R}{\quad R}\left(\frac{R^2x}{2}-\frac{x^3}{6}\right) = \frac{4R}{3\pi}.}
Thus the centre of mass is the point\, $(0,\,{\frac {4R}{3\pi }})$.