Question

A point mass m is released from rest at a distance of $3$R from the centre of a thin-walled hollow sphere of radius R and mass M as shown. The hollow sphere is fixed in position and the only force on the point mass is the gravitational attraction of the hollow sphere. There is a very small hole in the hollow sphere through which the point mass falls as shown. The velocity of a point mass when it passes through point P at a distance $R/2$ from the centre of the sphere is?

• A. $\sqrt{\frac{2GM}{3R}}$
• B. $\sqrt{\frac{5GM}{3R}}$
• C. $\sqrt{\frac{25GM}{24R}}$
• D. None of these

Answer 1

The correct option is D None of these

Gravitational potential inside hollow sphere $=-\frac{GM}{R}$

Gravitational potential at distance $3R=-\frac{GM}{3R}$

$\therefore$ Conservation of energy

$\Rightarrow \frac{1}{2}m{v}^2-\frac{GMm}{R}=-\frac{GMm}{3R}\Rightarrow 12m{v}^2=\frac{GMm}{R}(1-\frac{1}{3})\Rightarrow \frac{v^2}{2}=\frac{2GM}{3R}\Rightarrow v^2=\frac{4GM}{3R}v=\sqrt{\frac{4GM}{3R}}$