# Kinetics of rigid bodies

## Center of mass

Total mass of a body.

$m = \int_V \rho \,dV$

Center of mass $C$.

$\vec{r}_C = \frac{1}{m} \int_V \rho \vec{r} \,dV$

## Moment of inertia

Moment of inertia about axis $\hat{a}$ through point $P$.

$I_{P,\hat{a}} = \int_V \rho r^2 \,dV$

Here $r$ is the distance from the axis through $P$ in direction $\hat{a}$.

## Euler's equations

Rigid body equations for rotation about a principal axis $\hat{a}$.

\begin{aligned} \sum_i \vec{F}_i &= m \vec{a}_C \\ \sum_i \vec{M}_{C,i} &= I_{C,\hat{a}} \vec\alpha \\ \text{or } \sum_i \vec{M}_{O,i} &= I_{O,\hat{a}} \vec\alpha \qquad \text{if O is a fixed point} \end{aligned}

## Linear momentum

Linear momentum of a rigid body.

$\vec{p} = m \vec{v}_C$

Kinetics equation using linear momentum.

$\sum_i \vec{F}_i = \dot{\vec{p}}$

## Angular momentum

Angular momentum of a rigid body rotating about a principal axis $\hat{a}$ through point $P$.

$\vec{H}_P = I_{P,\hat{a}} \vec\omega$

Rotation equation using angular momentum.

\begin{aligned} \sum_i \vec{M}_{C,i} &= \dot{\vec{H}}_C \\ \text{or } \sum_i \vec{M}_{O,i} &= \dot{\vec{H}}_O \qquad \text{if O is a fixed point} \end{aligned}