centre of mass
CentreOfMass
2005-08-31 17:51:22
2006-03-24 06:29:54
Definition
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The \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. For example, an object hung from a string in a uniform gravitational field will have its centre of mass straight below the string. The centre of mass can be used in many other ways to simplify a complicated system by treating it as a point particle. 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 $\mathbf{x}_{\mathrm{cm}}$:
$$
\mathbf{v}_{\mathrm{cm}}\equiv\mathbf{\dot x}_{\mathrm{cm}},\qquad
\mathbf{a}_{\mathrm{cm}}\equiv\mathbf{\ddot x}_{\mathrm{cm}}.
$$