Result

Gravitational potential energy

(M7.2.2) M7.2 · Gravitational Field and Potential Energy →

U_{g} = - \frac{GM m}{r}

Statement

The potential energy of two masses separated by r, with U = 0 taken at infinite separation.

Symbols and units

$U_{g}$	gravitational potential energy	J
$G$	gravitational constant	6.674×10⁻¹¹ N·m²/kg²
$M, m$	the two masses	kg
$r$	separation between the masses	m

When it applies

Two point masses (or uniform spheres), with U = 0 at infinite separation.

Don't use when

The near-surface approximation U = mgh should be used instead for small height changes close to a planet's surface, where this exact form is unnecessarily complex.

Derivation

Integrate the gravitational force from infinity in to separation r, following the same procedure used for electric potential energy:

U_{g} = - \int_{\infty}^{r} F_{g} d r^{'} = - \frac{GM m}{r}

Worked examples

Escape velocity

Setting total mechanical energy to zero at infinity gives the minimum launch speed to escape:

\frac{1}{2} m v_{esc}^{2} - \frac{GM m}{R} = 0 \Rightarrow v_{esc} = \frac{2 GM}{R}

For Earth:

v_{esc} = \frac{2 ( 6.674 \times 1 0 ^{- 11} ) ( 5.97 \times 1 0 ^{24} )}{6.37 \times 1 0 ^{6}} \approx 11.2 km/s

Watch out for

This is the exact form; U = mgh (M3.2) is only its near-surface approximation, valid for small height changes where g is nearly constant.

Connects to

Energy of a pair