$[10](4)$ A cable is wrapped several times around a uniform solid cylinder with mass M $=50$ kg and radius R $=0\mathrm{.}10$ m and rotates freely without friction about a horizontal axis. The bucket $($mass m $=5$ kg$)$ must descend a height h $=5$ m to reach the water; it is suspended by a rope of negligible mass that wraps around the cylinder. The winch $($cylinder$)$ handle falls off, releasing the bucket to fall to the water, rotating the cylinder as it falls. Assuming that the cable unwinds without stretching or slipping:

$[5]($a$)$ Find the magnitude of the angular velocity of the cylinder just before the bucket hits the water.

$[5]($b$)$ Find the magnitude of the angular acceleration of the cylinder.