Category:Centers of Mass

This category contains results about Centers of Mass.
Definitions specific to this category can be found in Definitions/Centers of Mass.

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 $\d V$.