The block $A$ is of mass $m$ and block $B$ is mass $M$ . Block $B$ is resting on a smooth horizontal surface, while friction coefficient between blocks $A$ and $B$ is $\mu$ . What maximum force $F$ can be applied so that block $A$ does not slip over the block $B$ .

The maximum friction that can available for block $A$ $=\mu mg$ . If $a$ is the maximum acceleration that this friction provides to block $A$ , then by Newtown's second law

                                      $\mu mg = ma$

$\therefore$                                       $a = \mu g$ .

For both the blocks move together with acceleration $a$ ,

                                 $F = (M+m)a$

                                     $=(M+m)\mu g$ .