$($III$)$ On a level billiards table a cue ball, initially at rest at point $O$ on the table, is struck so that it leaves the cue stick with a center$-$of$-$mass speed $v_0$ and a "reverse" spin of angular speed ${}_0($see Fig. $11-41).$ A kinetic friction force acts on the ball as it initially skids across the table. $($a$)$ Explain why the ball's angular momentum is conserved about point $O.($b$)$ Using conservation of angular momentum, find the critical angular speed ${}_C$ such that, if ${}_0={}_C,$ kinetic friction will bring the ball to a complete $($as opposed to momentary$)$ stop. $($c$)$ If ${}_0$ is $10\%$ smaller than ${}_C,$ i$.$e$.,{}_0=0\mathrm{.}90{}_C,$ determine the ball's CM velocity $v_{CM}$ when it starts to roll without slipping. $($d$)$ If ${}_0$ is $10\%$ larger than ${}_C,$ i$.$e$.,{}_0=1\mathrm{.}10{}_C,$ determine the ball's CM velocity $v_{CM}$ when it starts to roll without slipping. $[$Hint: The ball possesses two types of angular momentum, the first due to the linear speed $v_{CM}$ of its CM relative to point O $,$ the second due to the spin at angular velocity $$ about its own CM$.$ The ball's total $L$ about $O$ is the sum of these two angular momenta.$]$