Question

All the surfaces shown in the figure. are assumed to be frictionless. The block of mass m slides on the prism which in turn slides backward on the horizontal surface. Find the acceleration of the smaller block with respect to the prism.

Answer 1

Draw the FBD of both the blocks. In the reference frame of block M, the block m will experience pseudo force. Apply Newton’s second law on the block m along the inclined plane and Newton’s first law along the perpendicular to the inclined plane. For block M apply Newton’s second law along the horizontal.

Let the acceleration of the prism be in the backward direction. Consider the motion of the smaller block from the frame of the prism. The forces on the block are

(i) N normal force,

(ii) mg downward (gravity), and

(iii) m a₀ forward(pseudo).

The block slides down the plane. Components of the forces parallel to the incline give

ma₀cosθ +  mgsinθ =  ma

or a = a₀cosθ + gsinθ   ... (i)

Components of the force perpendicular to the incline give

N + ma₀sinθ = mgcosθ . ...(ii)

Now, consider the motion of the prism from the lab frame. No pseudo force is needed as the frame used is inertial. The forces acting now are

(i) Mg downward,

(ii) N normal to the incline (by the block), and

(iii) N′ upward (by the horizontal surface).

Horizontal components give