Consider a hollow sphere rolling with slipping with velocities as shown. Find the translational velocity after the hollow sphere starts pure rolling.
(`a`)
(`b`)
(`c` )

Explanation

(`a`) Since `v_(0)gtRomega_(0)`, hence the velocity of hollow sphere at `A` is not zero, i.e. impure rolling. After some time the body will start pure rolling so that `v_(c.m.)=Romega`. For this linear velocity should decrease and angular velocity should increase. Therefore, friction will act in the backward direction, it will provide linear retardation and angular acceleration. Let `v_(c.m.)=v`, when the body starts pure rolling.

Since all forces passing through `A`, hence torque about `A` is zero, hence by the conservation of angular momentum about `A`.
`mv_(0)R+I_(c.m)omega_(0)=mvR+I_(c.m.)omega`
`mv_(0)R+(2)/(3)mR^(2)(v_(0))/(2R)=mvR+(2)/(3)mR^(2)(v)/(R)`
`(4)/(3)mv_(0)R=(5)/(3)mvRimpliesv=(4v_(0))/(5)`
(`b`) `(4)/(3)mv_(0)R-(2)/(3)mR^(2)(v_(0))/(2R)=(5)/(3)mvR`
`(2)/(3)mv_(0)R=(5)/(3)mvRimpliesv=(2v_(0))/(5)`
(c) `m(v_(0))/(2)R+(2)/(3)mR^(2)(v_(0))/(R)=(5)/(3)mvR`
`v=(7v_(0))/(10)`