A ball (solid sphere) is thrown down the valley in such a way that it slides with a speed ` v_(0)` initially without rolling. Prove that it will roll without any sliding when its speed falls to `(5/7)v_(0)`. The transition from pure sliding to pure rolling is gradual, so that both sliding and rolling take place during this interval.

Explanation

We can see that the frictional force f, weight mg and normal contact force N pass through the point of contact P. Therefore, their torque about P will be equal to zero.
`rArr` Angular momentum of the sphere remains constant before and after start of pure rolling about P.
Conseving the angular momentum of the sphere at position 1 & 2, about the instantaneous point of contact. We obtain
`mv_(0) r = mvr + ((2)/(5)mr^(2)) omega`
Putting `r omega = v` for pure rolling, we obtain `v = 5 v_(0)//7`.