A block of mass $m$ is placed on a wedge of mass $M$ and angle $\theta$. The wedge is placed on a smooth horizontal surface. A horizontal force $F$ is applied to the wedge such that the block remains stationary relative to the wedge. Find the value of $F$.

Solution

1. For the block to remain stationary relative to the wedge, its acceleration $a$ must be horizontal.
2. The forces on the block are gravity $mg$ downwards and normal force $N$ perpendicular to the wedge.
3. Resolving forces on the block gives $N\sin (\theta )=ma$ and $N\cos (\theta )=mg$, so $a=g\tan (\theta )$.
4. Applying Newton's second law to the entire system of mass $(M+m)$, we get $F=(M+m)a$.
5. Hence the answer is (A).