$16\mathrm{.}98$ A ball suspended from a string that goes through a hole in the ceiling at $O$ moves with velocity $v_A$ in a horizontal circular path of radius $r_A.$ The string is then drawn through the hole until the ball moves with velocity $v_B$ in a horizontal circular path of radius $r_B.$ Use the principle of angular impulse and momentum to show that $r_A{v}_A=r_B{v}_B.$

Strategy: Let $e$ be a unit vector that is perpendicular to the ceiling. Although this is not a central$-$force problem$-$the ball's weight does not point toward $O-$you can show that $e*(rF)=0,$ so that $e*H_O$ is conserved.

Problem $16\mathrm{.}98$