A solid sphere of mass m and radius R is placed on a plank of equal mass, which lies in a smoth horizontal surface.The sphere is given a sharp impulse as a result it stats slidding with velocity v.Find the time taken by the sphere to start pure roliing on the plank.the coefficient of friction between plank and sphere is k.

							
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 accleration)
         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                     
						

							
Initially velocity of plank is 0 and of disc is v.  friction will act in backward direction ....nowf = kmg                           (f = friction force) ma = kmga = - kg            (a is retardation of center of mass)now , torque = fR = I(alfa)                (alfa = angular accleration)alfa = fR/I alfa = 5gk/2R                        (I sphere = 2MR2/5)On plank friction will act on forward direction hence acceleration is in forward direction. f = makmg = ma `a` of plank is kg Let velocity of plank at time t is V1 v = u + atV1 = kgtSimilarly velocity of disc at time t is V2 = v - kgtAnd W (angular velocity) at time t is W = 2kgt/rSo velocity of contact point is  = (v - kgt) - (2kgt/r)×r = v - 3kgtFor pure rolling kgt = v - 3kgt4kgt = vt = v/4kg


The above answer become so conjusted so I am sending this again.                                                                 f = kmg (f =  friction force).                                             ma = kmg.                a = - kg.           (a is retardation of center of mass).    torque = fR = I(alfa).   (alfa = angular acceleration)Alfa = fR/I                          alfa = 5gk/2R              (I sphere = 2MR2/5).                                                                  On plank  friction will act on forward direction hence acceleration is in forward  direction.                                                f = ma.             kmg = ma.                       a = kg                                     Let velocity of plank at time t is V1.          v = u + at.                        V1 = kgt.                             Similarly  velocity of disc at time t is              V2 = v - kgt.                      And W (angular velocity) at time t is.                W = 2kgt/r.                     So velocity of contact point is = (v - kgt) - (2kgt) =  v - 3kgt.                For pure rolling                                                                         kgt = v - 3kgt.                              4kgt = v.                                     t = v/4kg