Question

The gravitational potential energy of a two-particle system is derived in this chapter as $U=\frac{G{m}_1{m}_2}{r}$. Does it follow from this equation that the potential energy for $r=\infty$ must be zero? Can we choose the potential energy for $r=\infty$ to be 20 J and still use this formula? If no, what formula should be used to calculate the gravitational potential energy at separation r?

Answer 1

The gravitational potential energy of a two-particle system is given by $U=-\frac{G{m}_1{m}_2}{r}$.
This relation does not tell that the gravitational potential energy is zero at infinity. For our convenience, we choose the potential energies of the two particles to be zero when the separation between them is infinity.

No, if we suppose that the potential energy for $r=\infty$ is 20 J, then we need to modify the formula.

Now, potential energy of the two-particle system separated by a distance r is given by

$$\begin{gathered} U\left(r^{,}\right)=U\left(r\right)-U(\infty ) \\ \mathrm{Given}:U(\infty )=20\mathrm{J} \\ \therefore \mathit{}U\mathit{\left(}r\mathit{\right)}\mathit{=}\mathit{-}G\frac{m_{\mathit{1}}{m}_{\mathit{2}}}{{\mathrm{r}}^2}-20 \end{gathered}$$

This formula should be used to calculate the gravitational potential energy at separation r.