A horizontal plank having mass `m` lies on a smooth horizontal surface. A sphere of same mass and radius `r` is spined to angular frequency `omega_(0)` and gently placed on the plank as shown in the figure. If coefficient of friction between the plank and the sphere is `mu`. Find the distance moved by the plank till sphere starts pure rolling on the plank. the plank is long enough.

Explanation

Let t be the time after which slipping between the sphere and plank disappears.
For the sphere,
N = mg and `mu N = ma_(s)`
`rArr" "a_(s) = mu g`
and, `tau = I alpha`
`rArr" "mu m gr = (2)/(5) mr^(2)alpha`
`rArr" "alpha = (5 mu g)/(2r)`

For the plank,
`mu N = Ma_(P) rArr a_(p) = (mu mg)/(M)`
After time t,
Velocity of plank, `v_(p) = v - (mu m g)/(M)t`
and Velocity of sphere, `v_(s) = mug t`
Angular velocity of shpere, `omega = (5 mu g)/(2r) t`
For no slipping, the point of contact of sphere should have the same speed as that of plank,
`rArr" "v_(s) + omega r = v_(p)`
Substituting and solving,
`mu g t + ((5 mu g)/(2r) t) r = v - (mu m g)/(M)t`
`rArr" "t = (2v)/(g(2 mu + 5 mu + 2 mu m/M))=(2v)/(mu g [ 7 + (2m//M)])`