For a rigid body, the velocity of a point at position $\mathbf{r}$ relative to the center of mass is $\mathbf{v}=\mathbf{V}_{\mathrm{CM}}+\boldsymbol{\omega}\times\mathbf{r}$. This decomposition separates translational and rotational motion cleanly.

The rotational analog of mass is the inertia tensor $$I_{ij}=\int \rho(\mathbf{x})\left(r^2\delta_{ij}-x_ix_j\right)d^3x.$$ It depends on the chosen origin and encodes how the body resists angular acceleration about different axes.

In a body-fixed principal-axis frame, rigid-body rotation obeys Euler's equations $I_1\dot\omega_1-(I_2-I_3)\omega_2\omega_3=\tau_1$ and cyclic permutations. Free rotation exhibits rich phenomena such as torque-free precession and the intermediate-axis instability.