Definition:Center of Mass/Continuous

Definition

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

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 $B$

$\mathbf r$ is the position vector of $\d V$.

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.