Gravitational Potential Energy

Since the physical meaning of potential energy was already introduced for springs and for gravity-near-the-surface in an earlier unit, and since the gravitational force is a conservative force, we can define a gravitational potential energy for any pair of masses.

Building the formula. Hold $m_1$ fixed, and consider moving $m_2$ from an initial separation $r'$ to a final separation $r$. Moving $m_2$ through a tiny step $dr$ requires an infinitesimal external work $dW_{ext}=\vec{F}_{ext}\cdot d\vec{r}$, and since this work is done against the attractive gravitational force $F_G = Gm_1m_2/r^2$, integrating from $r'$ to $r$ gives the total work

where $U(r) = -Gm_1m_2/r$. In other words, this work done is the change in gravitational potential energy between the two separations.

Sign conventions, worked out physically. If $r<r'$ (the masses move closer together), gravity itself pulls $m_2$ inward, doing the work using the system's own stored energy, so the work done by an external agent is negative. If $r>r'$ (the masses move apart), an external agent must do positive work against the attraction to separate them.

Choosing the reference point. Taking $r'\to\infty$ (where $U(\infty)=0$ by convention) gives the standard definition:

$U(r) = -\frac{Gm_1m_2}{r}. \qquad (6.29)$ …