A Yo$-$Yo is raised by pulling on the cord C $1$ wrapped

around the inner radius r$2.$ The Yo$-$Yo also has a cord C$2$

passing around its outer radius r$1,$ one end of which is

fastened to a fixed point A and the other to a mass M$.$

$($i$)$ If C$1$ is being pulled at velocity Vc and acceleration a c$,$ find

expressions for the corresponding quantities for the CM of the

pulley and M$.$ Evaluate the accelerations if r $1=6$cm$,$ r$2=2$cm and

ac $=0\mathrm{.}20$m$/$s$2.$

$($ii$)$ Assume a c is constant at $0\mathrm{.}2$m$/$s$2.$ At t $=0,$ Vc $=0\mathrm{.}50$ m$/$s $,$ the CM of the Yo$-$Yo is $0\mathrm{.}50$m

above the floor. At t $=2\mathrm{.}0$s$,$ what is the angular velocity of the CM of the Yo$-$Yo and how high is

it$?$