A block of mass m is placed on a wedge of mass M and a force F is acting on its as shown . If all surfaces are smooth , find the force F such that, there is no relative motion between m and M.

Explanation

The force acting on the bodies m and M are shown in the figure along with free body diagrams of mass m and M. As the two bodies move together, we can find the acceleration of system towards right directly as
`a=(F)/(m+M)`

Here the condition is, the small block of mass .m. should remain at rest on the incline surface of the wedge block. Look at the FBD of m, the force acting on it towards left is .ma. . It is the pseudo force on it as its reference frame is the wedge. As wedge is moving with an acceleration, we consider m relative to it. Now with respect to wedge, block m is at rest or in equilibrium, we can balance all the forces along the tendency of motion of body (i.e., inclined plane)and perpendicular to it shown in FBD of it.
For m to be at rest, from FBD of m, along the plane
`mg sin theta = ma cos theta or a =g tan theta or (F)/(m+M)= g tan theta , F=(m + M) g tan theta `