Gravitational Potential and Energy

Gravitational potential (V_g) — the work done per unit mass in bringing a small test mass from infinity to a point in the field:

V_g = − G M / r

measured in J kg⁻¹. It is a scalar, so potentials from several masses simply add without needing directions.

Why it is negative — the zero of potential is defined at infinity, and gravity is attractive, so a mass moving in from infinity has work done on it by the field. Its potential energy therefore falls below zero. Every gravitational potential is negative, approaching zero only at infinite distance.

Example: the negative sign is not a bookkeeping convention to be ignored. It says a mass is bound: to escape to infinity it must be given energy equal to the magnitude of its potential energy. A positive total energy means it is not bound and will escape.

Gravitational potential energy — for a mass m at potential V_g:

E = m V_g = − G M m / r

Relation to ΔE_p = mgΔh — over small height changes near a surface the field is effectively uniform, and the change in the −GMm/r expression reduces to mgΔh. The two are consistent; one is the general case and the other its local approximation.

Potential gradient — the field strength is the negative gradient of the potential:

g = − ΔV_g / Δr

so a graph of V_g against r has a gradient whose magnitude gives g at any point. Equipotential surfaces — spheres around a point mass — are always perpendicular to field lines.

No work is done moving along an equipotential — the potential is unchanged, so ΔE = 0. This is why a satellite in a circular orbit needs no energy input to maintain it: it stays on one equipotential the whole way round.

Graph shapes worth knowing — beyond the surface, V_g varies as −1/r, rising towards zero with distance, while g varies as 1/r². The potential curve is the shallower of the two, which is another way of saying that potential falls off more slowly than field strength.