Homework H$\mathrm{.}5.$D

Given: An inextensible cable is wrapped around the inner radius of a solid, stepped drum $($having a mass of $m,$ inner radius $r,$ outer radius $R$ and radius of gyration about its center O of $k_O).$ One end of the cable is attached to ground, with a vertical force $F$ acting at the free end of the cable. A second inextensible cable is wrapped around the outer radius of the drum with block B $($of mass $m)$ attached at its free end. Neither cable slips on the drum as the system moves. The system is released from rest.

Find: On release:

$($a$)$ Determine the angular acceleration of the drum.

$($b$)$ Determine the acceleration of block B$.$

HINT: Since the drum does not slip on the cables, point C is the instant center for the drum.

Use the following parameters in your analysis: $m=8kg,R=0\mathrm{.}5m,r=0\mathrm{.}25m,k_O=0\mathrm{.}3m$ and $F=20N.$