Gravitational potential energy

Gravity is a conservative force: the work it does depends only on the start and end points, so a potential energy can be defined.

Near the surface the force is nearly constant (mg), and lifting m from h₁ to h₂ takes work mg(h₂ − h₁). This gives W(h) = mgh + W₀, where W₀ is the value at the surface.

Far from the surface the force changes with distance; the work to lift m from r₁ to r₂ is G M_E m (1/r₁ − 1/r₂), so W(r) = −G M_E m / r + W₁ for r > R_E.

Only differences of potential energy have physical meaning. The usual choice is zero at infinity, so the potential energy of two masses is V = −G m₁ m₂ / r, always negative.

The potential energy at a point is then the work done in bringing the body in from infinity to that point.

Gravitational potential is the potential energy per unit mass at a point: −G M / r for a point outside a mass M.

For many particles, the total potential energy is the sum of −G mᵢ mⱼ / rᵢⱼ over every pair. Example: four masses m on a square of side l make 4 pairs at distance l and 2 pairs at √2 l, so the total is −(4 + √2) G m²/l = −5.41 G m²/l.

mgh is only an approximation, valid for heights small compared with R_E, to the exact difference −G M_E m (1/r₂ − 1/r₁).