# Definition:Moment of Inertia

## Definition

Let $B$ be a rigid body which is rotating in space about some axis $\LL$.

The moment of inertia of $B$ about $\LL$ is defined as follows:

### Discrete

Let $B$ be composed of a countable number of particles $P_1, P_2, \ldots$ such that:

each $P_i$ has mass $m_i$
each $P_i$ has perpendicular distance $r_i$ from $\LL$.

The moment of inertia of $B$ about $\LL$ is given by:

$I := \ds \sum m_i {r_i}^2$

### Continuous

Let each point in $B$ have:

a position vector $\mathbf r$ with respect to a given frame of reference.
a density $\map \rho {\mathbf r}$
a perpendicular distance $\map p {\mathbf r}$ from $\LL$

The moment of inertia of $B$ about $\LL$ is given by:

$I := \ds \int_B \paren {\map p {\mathbf r} }^2 \map \rho {\mathbf r} \rd v$

where $\d v$ is an infinitesimal volume element of $B$.

### Dimension

The dimension of measurement of moment of inertia is $\mathsf {M L}^2$.

## Also see

• Results about moment of inertia can be found here.