A man is standing on a plank which is placed on smooth horizontal surface. There is sufficient friction between feet of man and plank. Now man starts running over plank, correct statement is/are

Options

A. The cylinder is performing impure rolling motion initially on the plank and finally performs pure rolling motion
B. The direction of frictional force on the cylinder while rolling on plank is initially in backward direction and finally it attains zero value
C. The velocity of the centre of mass of the plank + cylinder system, after the cylinder starts pure rolling is `sqrt((gh)/2)`
D. The minimum length of the plank required for pure rolling of the cylinder over plank is `3v_(0)^(2)//16 mu g`, where `v_(0) = sqrt(2gh)`.

Answer

A, B, C, D

Explanation

As the curve is smooth, there is no gain in angular motion. `rArr mgh =1/2 m u^(2) rArr u = sqrt(2gh)`. i.e., velocity with which it gets onto the plank.
`Nmu = 1/2 MR^(2)alpha rArr alpha = (2gmu)/R`

Let V and be linear velocity of plank and centre of cylinder when rolling beings.
`rArr V = gmu , v=u - gmu t, omega = (2gmu)/Rt` And when pure rolling begings.
`v-R omega = V rArr u -gmu t, omega = (2gmu)/R t` And when pure rolling begins.
`v-Romega =rArr mu - gmut = gmut rArr t=mu/(4gmu) rArr v=(3mu)/4` and `V=mu/4`
`rArr` Velocity of centre of mass `=(m(3mu//4)+ m(mu//4))/(m+m) = mu/2`
For cylinder w.r.t. plank, `a_(r) = -2gmu, mu_(r) =mu`
`v_(r) = (3u)/4 -mu/4 = mu/2 rArr ((mu)/2)^(2) = mu^(2) +2(-2gmu)l rArr l=(3mu^(2))/(16 gmu`.