Question

Two identical thin rings each of radius R are coaxially placed at a distance R. If the rings have a uniform mass distribution and each has masses 2m and 4m respectively, then the work done in moving a mass m from centre of one ring to that of the other is

Moderate

Solution

Work done, $W = U_{2} - U_{1} = m v_{2} - m v_{1}$
$= m [(- \frac{G m_{2}}{R} - \frac{G m_{1}}{\sqrt{2} R}) - (- \frac{G m_{1}}{R} - \frac{G m_{2}}{\sqrt{2} R})]$
$= \frac{G m (m_{1} - m_{2})}{\sqrt{2} R} (\sqrt{2} - 1) = \frac{\sqrt{2} G m^{2}}{R} (1 - \sqrt{2})$

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