Question

Find the proper potential energy of gravitational interaction of matter forming a uniform sphere of mass $m$ and radius $R$.

Answer 1

Let the sphere be formed till radius x and mass m at time t and a shell of thickness dx is created with mass dm.

Thus, $dU=-\frac{Gmdm}{x}$

$dm=4\rho \pi x^{2}dx$

$dU=-\frac{G}{x}\frac{4}{3}\rho \pi x^3\left(4\pi x^2\rho dx\right)$

$\rho =\frac{m}{\frac{4}{3}\pi R^3}$

Integrating, we get $U=-\frac{3}{5}\frac{G{M}^2}{R}$

Answer is $-\frac{3}{5}\frac{G{M}^2}{R}$