Question

The figures shows a block of mass $M=2m$ having a spherical smooth cavity of radius $R$ is placed on a smooth horizontal surface. There is a small ball of mass m moving vertically downwards at an instant with 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 in with respect to the ground is $\frac{v^2}{2R}$

Answer 1

Answer: (B), (C)

The correct options are
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 $\frac{v^2}{3R}$


For Block For Ball

Let, a be the acceleration of block in ground frame.
Let,, the normal force between the block and the ball be $N$.

For the block, from Newton's second law, we have
$N=Ma=2ma$

For ball (with respect to the block), from Newton's second law, we have $N+ma=\frac{m{v}^2}{R}$

After solving both the equations, we get
$N=\frac{2}{3}\frac{m{v}^2}{R}$ and $a=\frac{v^2}{3R}$