A solid uniform sphere of mass M and radius R can rotate about a fixed vertical axis. There is no frictional torque acting at the axis of rotation. A light string is wrapped around the equator of the sphere. The string has exactly 6 turns on the sphere. The string passes over a light pulley and carries a small mass m at its end (see figure). The string between the sphere and the pulley is always horizontal. The system is released from rest and the small mass falls down vertically. The string does not slip on the sphere till 5 turns get unwound. As soon as `5^(th)` turn gets unwound completely, the friction between the sphere and the string vanishes all of a sudden.
(a) Find the angular speed of the sphere as the string leaves it.
(b) Find the change in acceleration of the small mass `m` after 5 turns get unwound from the sphere.

Answer

(a) `omega = sqrt((100 pi mg)/(R (5m + 2M))`
(b) `g ((2m)/(5m + 2M))`