A uniform sphere of radius R has a spherical cavity of radius `(R)/(2)` (see figure). Mass of the sphere with cavity is M. The sphere is rolling without sliding on a rough horizontal floor [the line joining the centre of sphere to the centre of the cavity remains in vertical plane]. When the centre of the cavity is at lowest position, the centre of the sphere has horizontal velocity V. Find:
(a) The kinetic energy of the sphere at this moment.
(b) The velocity of the centre of mass at this moment.
(c) The maximum permissible value of V ( in the position shown ) which allows the sphere to roll without bouncing

Answer

(a) `(31)/(40) MV^(2)`
(b) `(15V)/(14)`
(c) `V le sqrt(14/15 gR)`