A block of mass m is placed on a smooth wedge of mass M. The value of force F such that m does not slide down relative to wedge is

To solve this problem, we need to consider the forces acting on the block and the wedge.

Step 1: Identify the forces acting on the block The block of mass m is acted upon by its weight (mg) downwards and the normal force (N) from the surface of the wedge.

Step 2: Resolve the forces The weight of the block can be resolved into two components: one parallel to the surface of the wedge (mg sin θ) and one perpendicular to the surface of the wedge (mg cos θ). Here, θ is the angle of the wedge.

Step 3: Apply Newton's second law According to Newton's second law, the net force acting on the block should be equal to the mass of the block times its acceleration. Since the block is not sliding down relative to the wedge, its acceleration is zero. Therefore, the net force acting on the block should be zero.

Step 4: Set up the equation Setting the net force equal to zero gives us: N - mg cos θ = 0 and F - mg sin θ = 0.

Step 5: Solve for F Solving the second equation for F gives us: F = mg sin θ.

So, the value of the force F such that the block does not slide down relative to the wedge is mg sin θ.