A block of mass m lies on wedge of mass M, which lies on fixed horizontal surface. The wedge is free to move on the horizontal surface. A horizontal force of magnitude F is applied on block as shown, neglecting friction at all surfaces, the value of force F such that block has no relative motion w.r.t. wedge will be: (where g is acceleration due to gravity)
For no relative motion between wedge and block, let the acceleration of both block and wedge be 'a' towards left.
From FBD of block; ...(i)
and ...(ii)
From FBD of wedge; ...(iii)
from equations (i), (ii) and (iii)
solving we get