A $4-$kg disk A is mounted on arm BC$,$which has a negligible mass. If a torque of M $=15$e$0\mathrm{.}5$t$2$ N # m$,$ where t is in seconds, is applied to the arm at C$,$ determine the angular velocity of BC in $2$ s starting from rest. Solve the problem assuming that $($a$)$ the disk is set in a smooth bearing at B so that it rotates with curvilinear translation, $($b$)$ the disk is fixed to the shaft BC$,$ and $($c$)$ the disk is given an initial freely spinning angular velocity of VD $=5-80$k$6$ rad$>$s prior to application of the torque