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.

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

Force couples and pure moments