A solid sphere of mass `m` and radius`r` is given an initial angular velocity `omega_(0)` and a linear velocity `v_(0)=lambda r omega_(0)` from a point `A` on a rough horizontal surface .It is observed that the ball turns back and returens to the point `A` after some time if `lambda` is less than a certain maximum value `lambda_(0)` . Find `lambda_(0)`.

Answer

`lambda_(0)=2//5`

Explanation

Ball will comes back to the initial position if its angular velocity is greater than zero is the same direction (in which it was released) at the moment its's linear velocity becomes zero. In his condition ball would returen back.

`0=v_(0)+(mug)t `
`t=(v_(0))/(mug)` (time when ball stop)
For rotation motion

`tau =I alpha`
`rArr alpha=(tau)/(I)=overset(mugm xxR)(bar((2)/(5)MR^(2)))=(5mug)/(2R^(2))`
using `omega_(f)=omega_(0)-alpha t`
`omega _(f)gt 0`
`rArr (v_(0))/(alpha R )gt(5mug)/(2R).(v_(0))/(mug)` for limiting condition
`rArr lambda_(0)=(2)/(5)`