A body of mass $m$ is moving in a circular orbit of radius $R$ about a planet of mass $M$. At some instant,…
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 $\frac{3 R}{2}$. The difference between the final and initial total energies is:
$-\frac{G M m}{2 R}$
$+\frac{G M m}{6 R}$
$-\frac{G M m}{6 R}$
$\frac{G M m}{2 R}$
Solution
Initial gravitational potential energy,
$
E_i=-\frac{G M m}{2 R}
$
Final gravitational potential energy,
$
\begin{aligned}
E_f &=-\frac{G M m / 2}{2\left(\frac{R}{2}\right)}-\frac{G M m / 2}{2\left(\frac{3 R}{2}\right)} \\
&=-\frac{G M m}{2 R}-\frac{G M m}{6 R} \\
&=-\frac{4 G M m}{6 R}=-\frac{2 G M m}{3 R}
\end{aligned}
$
$\therefore$ Difference between initial and final energy,
$
E_f-E_i=\frac{G M m}{R}\left(-\frac{2}{3}+\frac{1}{2}\right)=-\frac{G M m}{6 R}
$