%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: center of mass
%%% Primary Category Code: 45.40.-f
%%% Filename: CentreOfMass.tex
%%% Version: 12
%%% Owner: bci1
%%% Author(s): bci1, pbruin
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\begin{document}
The \emph{center of mass} (en: {\em centre of mass}) of an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} is a point in physical pace where, for computational purposes in theoretical \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}, the \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/Mass.html} of the object can be considered to be concentrated. This \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} is most often used for rigid bodies, but applies to any macroscopic arrangement of matter. The centre of mass can be used as a geometrical concept that allows in several ways to simplify an apparently complicated \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} or problem by treating an entire rigid body as a ``point particle''. For example, a rigid body hung from a string in a uniform gravitational \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} will have its centre of mass straight below the string if it is at rest. More generally, as long as the only forces acting are gravity and the tension in the string, in order to compute the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of the object in space (disregarding possible rotations) one can conceptually replace the solid object by a point particle located at the object's centre of mass.
The \htmladdnormallink{momentum}{http://planetphysics.us/encyclopedia/Momentum.html} of an object with total mass $M$ is
$$
\mathbf{p}=M\mathbf{v}_{\mathrm{cm}},
$$
and if a total force $\mathbf{F}$ acts on the object, Newton's second law may be formulated as
$$
\mathbf{F}=M\mathbf{a}_{\mathrm{cm}}.
$$
In these equations $\mathbf{v}_{\mathrm{cm}}$ and $\mathbf{a}_{\mathrm{cm}}$ are the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} and the \htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html} of its centre of mass, respectively; these are defined below.
The centre of mass of a solid object, rigid body, etc., occupying a region $V$ in space, is defined as the average \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} of all the points of $V$, each point $\mathbf{x}$ being weighted with the density of the object at that point, $\rho(\mathbf{x})$. This average is a \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html} integral over the region $V$:
$$
\mathbf{x}_{\mathrm{cm}}\equiv\frac{1}{M}\int_V
\rho(\mathbf{x})\mathbf{x}\,\mathrm{d}V.
$$
Here $M$ is the total mass of the object:
$$
M\equiv\int_V \rho(\mathbf{x})\,\mathrm{d}V.
$$
If the object consists of a number of point particles, these integrals become sums over the \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html}. The centre of mass of a collection of $N$ particles with masses $m_1,m_2,\ldots,m_N$ located at the positions $\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_N$ is
$$
\mathbf{x}_{\mathrm{cm}}\equiv\frac{1}{M}\sum_{i=1}^N m_i\mathbf{x}_i,
$$
where $M$ is now the sum of the masses of the particles:
$$
M\equiv\sum_{i=1}^N m_i.
$$
The velocity and acceleration of the centre of mass are now simply defined as the first and second time derivatives of $\mathbf{x}_{\mathrm{cm}}$:
$$
\mathbf{v}_{\mathrm{cm}}\equiv\mathbf{\dot x}_{\mathrm{cm}},\qquad
\mathbf{a}_{\mathrm{cm}}\equiv\mathbf{\ddot x}_{\mathrm{cm}}.
$$
\end{document}
