Question

Find the potential energy of the gravitational interaction of two mass points of masses $m_1$ and $m_2$ located at a distance $r$ from each other.

Answer 1

The gravitational potential due to $m_1$ at the point of location of $m_2$,
$V_2={\int }_r^{\infty }\overrightarrow{G}\cdot d\overrightarrow{r}={\int }_r^{\infty }-\frac{\gamma m_1}{x^2}dx=-\frac{\gamma m_1}{r}$
So, $U_{21}=m_2{V}_2=-\frac{\gamma m_1{m}_2}{r}$
Similarly $U_{12}=-\frac{\gamma m_1{m}_2}{r}$
Hence, $U_{12}=U_{21}=U=-\frac{\gamma m_1{m}_2}{r}$