Rotational Collisions

Drop a still disk onto a spinning one and the stack slows down, but it doesn't stop. No outside torque acts on the two-disk system during the brief collision, since the only forces involved are internal friction forces between the disks, equal and opposite, acting through the same contact. Internal torques can't change the total angular momentum of the system. So whatever L the spinning disk had before, the combined stack has after, just spread over a bigger moment of inertia.

The bookkeeping works exactly like linear momentum, but with L = I omega instead of p = mv. Write L_initial as the sum of each piece's angular momentum about the shared axis, then write L_final as I_total times the common omega, and set them equal. If a point mass like a bullet or a glob of putty is involved, treat its angular momentum as m v r at the instant of impact, and remember that once it sticks it adds m r² to the system's moment of inertia. The motion is perfectly inelastic: the pieces end up rotating together at one omega, and kinetic energy is not conserved, because the internal forces that lock the pieces together do negative work, dissipating energy as heat and deformation.

This sits right next to the linear momentum collisions from earlier in the course, and the contrast is the point: a bullet embedding in a pivoted rod does not conserve linear momentum, because the pivot exerts an external force on the system, but it does conserve angular momentum about the pivot, since that same pivot force produces no torque about itself. Rotational collisions also connect forward to orbits: the same L = I omega bookkeeping, done continuously instead of in one instant, is what keeps a planet's angular momentum fixed as its distance from the sun changes.