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