A thin uniform rod of length L and mass M hangs horizontally from a light, frictionless hinge at its left end. The rod is initially at rest. A small steel ball of mass m travels horizontally with speed v along a line perpendicular to the rod and strikes the rod at its midpoint. The collision is perfectly elastic, and the ball reverses direction along the same line with speed v′. Immediately after the impact the rod begins to rotate about the hinge.
(a) Derive an expression that relates v and v′ based on appropriate conservation principles.
(b) Determine the angular speed ωe of the rod immediately after this elastic collision in terms of given quantities.
(c) The experiment is repeated, but the ball now sticks to the rod. Determine the angular speed ωi of the rod immediately after this perfectly inelastic collision.
(d) For the case m≪M, compare the magnitudes of ωe and ωi and justify your answer qualitatively.