Question

A point mass $m$ is released from rest at a distance of $3R$ 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 ass 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 C $\sqrt{\frac{25GM}{24R}}$

Using conservation of mechanical energy :

$U_i+K_i=U_f+K_f$

$-\frac{GMm}{3R}+0=-\frac{GMm}{2R^3}(3R^2-(\frac{R}{2}{)}^2)+\frac{1}{2}m{v}^2$

$-\frac{GMm}{3R}+0=-\frac{11GMm}{8R}+\frac{1}{2}m{v}^2$

$-\frac{GMm}{3R}+0=-\frac{11GMm}{8R}+\frac{1}{2}m{v}^2$

$v=\sqrt{\frac{25GM}{24R}}$