Gravitational Potential Energy

Near the Earth's surface, over small height changes $h$, the gravitational potential energy gained by a mass $m$ is given by the familiar formula $U = mgh$ (measured relative to the ground, $U=0$ at $h=0$), because $g$ itself can safely be treated as constant over such small height changes. This formula breaks down completely, however, over the much larger distances relevant to satellites and planets, where $g$ itself varies significantly (Section 7.6) -- a genuinely general definition of gravitational potential energy, valid at ANY separation, is needed instead.

The general definition is built the same way potential energy is always built: as (minus) the work done by the gravitational force in bringing the mass $m$ from a chosen reference point to its actual position, at distance $r$ from a mass $M$. The reference point is conventionally chosen to be infinity -- a distance so large that the gravitational force there is effectively zero -- and the potential energy AT infinity is defined to be exactly zero. Since the gravitational force is attractive, bringing $m$ IN from infinity toward $M$ means the gravitational force does POSITIVE work on it at every step; the potential energy of the system therefore DECREASES as $m$ is brought closer, becoming steadily more negative the closer $m$ gets to $M$. Formally,

(the force is written with a leading minus sign inside the integral because it acts inward, toward decreasing $r$, while the integration variable $r'$ increases outward). This gives the general result

which is negative for every finite $r$, increases (becomes LESS negative) as $r$ increases, and correctly reaches its reference value of exactly zero only as $r \to \infty$ -- exactly as the choice of reference point requires. The negative sign is not a mathematical inconvenience to be explained away; it carries real physical meaning: it says the system (mass $m$ near mass $M$) is in a lower-energy, gravitationally BOUND state than it would be if the two masses were infinitely far apart, and it is exactly this negative binding energy that must be supplied, from outside, to pull the two masses apart to infinity (Section 7.10 returns to this directly, in finding the escape velocity). …