Definition:Center of Mass

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Definition

Let $B$ be a body of mass $M$.

Discrete

Let $B$ be made up of $n$ discrete particles each with:

mass $m_i$
position vector $\mathbf r_i$

where $i \in \set {1, 2, \ldots, n}$


The center of mass of $B$ is the point whose position vector $\bar {\mathbf r}$ is given by:

$\ds M \bar {\mathbf r} = \sum_{i = \mathop 1}^n m_i \mathbf r_i$


Continuous

Let $B$ be of density $\map \rho {\mathbf r}$ at the point with position vector $\mathbf r$.

The center of mass of $B$ is the point whose position vector $\bar {\mathbf r}$ is given by:

$\ds M \bar {\mathbf r} = \int_V \map \rho {\mathbf r} \mathbf r \rd V$

where:

$V$ is the volume of space occupied by $B$
$\d V$ is an infinitesimal volume element
$\mathbf r$ is the position vector of $\d V$.


Also known as

The center of mass of a body is also known as its mass center.

Also note that in UK English, center is spelt centre.

Some sources use the term barycenter, but that term has wider applications than applied mathematics, and is used a more general concept in affine geometry.


Examples

Uniform Lamina

Let $\LL$ be a uniform lamina embedded in a cartesian plane in the shape of the area between the curve $\map f x$, the straight lines $x = a$ and $x = b$, and the $x$-axis.

Let the area of $\LL$ be $A$.


Then the coordinates $\tuple {\bar x, \bar y}$ of the center of mass of $B$ are given by:

\(\ds A \bar x\) \(=\) \(\ds \int_a^b x y \rd x\)
\(\ds A \bar y\) \(=\) \(\ds \dfrac 1 2 \int_a^b y^2 \rd y\)


Also see

  • Results about centers of mass can be found here.


Sources

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