Question

A ring of mass $m_1$ and radius $R$ is fixed in space at some location. An external agent brings a point mass $m_2$ from infinity to the centre of the ring. Work done by the external agent will be

• A. $-\frac{G{m}_1{m}_2}{R}$
• B. $\frac{G{m}_1{m}_2}{R}$
• C. $-\frac{G{m}_1{m}_2}{2R}$
• D. $\frac{G{m}_1{m}_2}{2R}$

Answer 1

The correct option is A $-\frac{G{m}_1{m}_2}{R}$

Work done will be equal to the change in the potential energy of the ring and mass system.

When mass $m_2$ is at infinity,
Initial potential energy, $U_i=0$

When mass $m_2$ is at centre of ring,
Final potential energy, $U_f=-\frac{G{m}_1{m}_2}{R}$

So, external work done,
$W=\Delta U=U_f-U_i=-\frac{G{m}_1{m}_2}{R}-0=-\frac{G{m}_1{m}_2}{R}$

Hence, option (a) is the correct answer.
​​​​

Why this question? The gravitational potential of a ring of radius $R$ and mass $M$ at a distance $r$ from its centre on the axis is given by $V=-\frac{GM}{\sqrt{R^2+r^2}};0\le r\le \infty$