a plank of mass M1 with a uniform sphere of mass M2 placed on it rests on a smooth horizontal plane . Aconstant horizontal force F is applied to the plank .With what aceleration will the plank and the centre of the sphere move provided there is no sliding b/w the plank and sphere  

Let f denotes the force of friction . Then for the linear motion of th plank

F-f= M1a--------------(1)

and considering the motion of the sphere standing on the plank ( a non-inertial reference system ), because of no sliding , its angular accn= a1/R ( R = radius of the sphere ). where a1= accn of the  centre of the sphere wrt the plank .Then introducing pseudo force we have : For angular motion of the sphere ::

fR = (2/5 M2 R^2) (a1/R) .........(2)

and for the linear motion of the sphere

M2a- f = M2a1 ........(3)

solving (1) and (2)  and (3)I get the reqd accn .as

a=accn  of the plank =F/( M1 + 2/7 M2)

and a1 = accn of the sphere wrt plank = 5/7 a ( directed opposite to the applied force )

Now accn of the sphere wrt ground = ( a-a1 ) = 2/7 a= 2/7 F/( M1 + 2/7 M2)

sir both sphere and plank has different acceleration

Dear , if the bottom most point of the sphere and the plank has got different accelaration then there would be sliding .

and the centre of the sphere and the plank has obviously got different accn .

sir i m unable to understand why centre of sphere and plank got different acceleration when there is no sliding b/w them            help me