Collisions That Start a Rotation

Throw a lump of clay at the center of a free disk and it just pushes the disk forward. Throw it at the rim instead, and the disk spins. The clay's momentum, applied at a distance from the center, carries angular momentum: L = m v r for a point mass moving in a straight line, measured about whatever axis you choose. That angular momentum does not vanish on impact. It gets transferred into the spinning clay-disk system, so the collision sets the whole thing rotating.

To solve these, conserve angular momentum about the pivot: L_initial = L_final. Before the hit, only the clay has angular momentum, m v r, where r is the distance from the axis to the line of the clay's velocity. After the hit, clay and disk move together as one rigid body, with total moment of inertia I_total = I_disk + m r² (the clay now orbits at radius r, so it adds m r² to the disk's own I). Set m v r equal to I_total omega and solve for omega.

This is the rotational twin of a perfectly inelastic linear collision: two objects collide, stick, and move off together, with kinetic energy lost to deformation and heat. Here the 'sticking together' means they end up with one shared angular velocity instead of one shared linear velocity. The same logic that makes you set m1 v1 = (m1+m2) v_f in a linear collision makes you set L_i = I_total omega_f here, just swap mass for moment of inertia and velocity for angular velocity.