Question

A Solid sphere of mass m and radius R is placed a plank of equal mass, which lies on a smooth horizontal surface. The sphere is given a sharp impulses in the horizontal direction so that it starts sliding with a speed of Vo . find the time taken by the sphere to start pure rolling on the plank . The coefficient of friction between plank and sphere is myu.

Answer 1

velocity of center of mass is v & angular velocity of sphere is 0 initially... friction will act in backward direction ....now   f  = kmg       (f = friction force) ma = kmg  a=-kmg       (a is retardation of center of mass)

now , torque = fR = I(alfa)                        (alfa = angular accleratio

alfa = fR/I  

 = 5kg/2R                              (Isphere = 2MR2/5)

now after time t let velocity of ceneter of mass  is V then

V = U + at                             (initial linear velocity is v)

V = v - kgt         ............1

let at this time angular velocity os W then

W = Wo + (alfa)t                                (initial angular velocity is 0)

W = 5kgt/2R                 ............2                

now if pure rolling has started then V = WR

so , v - kgt = 5kgt/2

      t = 2v/7kg    

Answer 2

Kinetic friction acts on the sphere towards right until pure rolling starts. This provides a torque as shown. The plank and the sphere move in opposite directions W.R.T. each other.