A thin uniform square plate with side $l$ and mass $M$ can rotate freely about a stationary vertical

Question

A thin uniform square plate with side $l$ and mass $M$ can rotate freely about a stationary vertical axis coinciding with one of its sides. A small ball of mass $m$ flying with velocity $v$ at right angles to the plate strikes elastically the centre of it. Which of the following is/are correct?

Professor PC Thomas Classes · Answer

By conservation of angular momentum about the hinge: $mv(l/2) = mv'(l/2) + (Ml^2/3)\omega$. For elastic collision ($e=1$), velocity of approach equals velocity of separation: $v = \omega(l/2) - v'$. Solving these gives $\omega = \frac{12mv}{l(3m+4M)}$ and $v' = v\frac{3m-4M}{3m+4M}$. After impact, the CM of the plate moves in a circle of radius $l/2$. The horizontal hinge force provides the necessary centripetal force: $F = M\omega^2(l/2) = \frac{72m^2 M v^2}{l(3m+4M)^2}$.