A plank of mass `M` is moving on a smooth horizontal surface with speed `'v_(0)'`. At `t=0` a sphere of mass `'m'` and radius `'r'` is gently placed on it and simultaneously a constant horizontal force `'F'` is applied on the plank in the opposite direction of `v_(0)`. Find the time at which sphere starts rolling on the plank. The coefficient of friction between the plank and the sphere is `mu`.

Explanation

Till rolling starts kinetic friction acts on the bodies.
`F.B.D.` of sphere for no vertical acceleration `N=mg`….(`1`)
`rArrf_(k)=mu mg`(as `f_(k)=muN`)
`rArr a= mug`
`v=mu"gt"`
Also `alpha=(mu mgr)/(I)rArromega=(mu mg rt)/(I)`
The velocity of the lower point of the sphere at time `t` is
`v_(s)=v+omegar`
`v_(s)=mu "gt"+(mu mgr^(2))/(I)t`
The `F.B.D.` of the plank
`rArr a_(P)=-((F+f_(k))/(M))=-((F+mu mg)/(M))`
`v_(P)=v_(0)-((F+mu mg)/(M))t`
For no slipping `v_(s)=v_(P)`
`rArr mu"gt"+(mu mgr^(2))/(I)t=v_(0)-((F+mu mg)/(M))t`
`rArr t= (v_(0))/([(7)/(2)mug+((F+mu mg)/(M))])`