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A body of mass $m$ is moving in a circular orbit of radius $R$ about a planet of mass $M$. At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius $\frac{R}{2}$ , and the other mass, in a circular orbit of radius $\frac{3R}{2}$. The difference between the final and initial total energies is
$ - \frac{{GMm}}{{2R}}$
$ + \frac{{GMm}}{{6R}}$
$ - \frac{{GMm}}{{6R}}$
$ \frac{{GMm}}{{2R}}$
Solution
Initial gravitational potential energy,
${E_i} = – \frac{{GMm}}{{2R}}$
Final gravitational potential energy,
${E_f} = – \frac{{GMm/2}}{{2\left( {\frac{R}{2}} \right)}}\frac{{GMm/2}}{{2\left( {\frac{{3R}}{2}} \right)}}$
$ = – \frac{{GMm}}{{2R}} – \frac{{GMm}}{{6R}}$
$ = – \frac{{4GMm}}{{6R}} = – \frac{{2GMm}}{{3R}}$
$\therefore $ Difference between initial and final energy,
${E_f} – {E_i} = \frac{{GMm}}{R}\left( { – \frac{2}{3} + \frac{1}{2}} \right) = – \frac{{GMm}}{{6R}}$
