centre of massCentreOfMass2005-08-31 17:51:222009-01-17 19:37:29Definition% this is the default PlanetPhysics preamble. as your knowledge
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% define commands hereThe \emph{centre of mass} of an object is a point in space where, for many purposes, the mass of the object may be assumed to be concentrated. This concept is most often used for rigid bodies, but applies to any arrangement of matter. The centre of mass can be used in many ways to simplify a complicated system by treating it as a point particle. For example, a rigid body hung from a string in a uniform gravitational field 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 string, to describe the motion of the object in space (disregarding possible rotation) we may replace the solid object by a point particle located at the centre of mass.
The momentum 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 velocity and the acceleration of its centre of mass, respectively; these are defined below.
The centre of mass of a solid object occupying a region $V$ in space is defined as the average position 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 volume 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 particles. 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 $\mathbf{x}_{\mathrm{cm}}$:
$$
\mathbf{v}_{\mathrm{cm}}\equiv\mathbf{\dot x}_{\mathrm{cm}},\qquad
\mathbf{a}_{\mathrm{cm}}\equiv\mathbf{\ddot x}_{\mathrm{cm}}.
$$