Question

The figure shows a block of mass $M=2m$ having a spherical smooth cavity of radius $R$ replaced on a smooth horizontal surface. There is a small ball of mass $m$ moving at an instant vertically downward with a velocity $v$ with respect to the block. At this instant:

• A. the normal reaction on the ball by the block is $\frac{m{v}^2}{R}$
• B. the normal reaction on the ball by the block is $\frac{2}{3}\frac{m{v}^2}{R}$
• C. the acceleration of the block with respect to the ground is $\frac{v^2}{3R}$
• D. the acceleration of the block with respect to the ground is $\frac{v^2}{2R}$

Answer 1

Answer: (A), (D)

The correct options are
A the normal reaction on the ball by the block is $\frac{m{v}^2}{R}$
D the acceleration of the block with respect to the ground is $\frac{v^2}{2R}$

The ball tends to move in a circular path of radius $R$ with velocity $v$ at the instant.

Thus normal reaction on the ball by the block at the instant$=$Centripetal force experienced by the ball$=\frac{m{v}^2}{R}$

A reaction pair of the same acts on the block. This provides an acceleration to the block$=\frac{F}{M}=\frac{\frac{m{v}^2}{R}}{M}=\frac{v^2}{2R}$