Orbital vs Spin Angular Momentum

When an extended object undergoes translation while also rotating about its own axis, its total angular momentum has two parts: orbital and spin.

Orbital angular momentum comes from the motion of the center of mass with respect to some pivot. Treat the object as a point particle at its center of mass:

$$L_{\mathrm{orbital}} = M v_{\mathrm{CM}} r_{\perp}$$

Here $M$ is the total mass of the object, $v_{\mathrm{CM}}$ is the center-of-mass speed, and $r_{\perp}$ is the perpendicular distance from the pivot to the velocity vector.

Spin angular momentum comes from rotation about the center of mass:

$$L_{\mathrm{spin}} = I_{\mathrm{CM}} \omega$$

Here $I_{\mathrm{CM}}$ is the rotational inertia about the center of mass and $\omega$ is the angular velocity of self-rotation.

The total angular momentum about the pivot is the sum:

$$L_{\mathrm{total}} = L_{\mathrm{orbital}} + L_{\mathrm{spin}}$$

For motion in 2D, use signs to indicate direction. Counterclockwise is positive; clockwise is negative. The two contributions may reinforce or oppose each other.