Kinetics of rigid bodies #reg

Center of mass #reg‑sc

Total mass of a body.#reg‑em

$m = \int_V \rho \,dV$

Center of mass $C$ .#reg‑ec

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

Moment of inertia #reg‑si

Moment of inertia about axis $\hat{a}$ through point $P$ .#reg‑ei

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

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

Euler's equations #reg‑se

Rigid body equations for rotation about axis $\hat{a}$ .#reg‑ee

$\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 #reg‑sl

Linear momentum of a rigid body.#reg‑el

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

Kinetics equation using linear momentum.#reg‑el

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

Angular momentum #reg‑sa

Angular momentum of a rigid body rotating about axis $\hat{a}$ through point $P$ .#reg‑ea

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

Rotation equation using angular momentum.#reg‑er

$\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}$