Gravitational Potential Energy Formula
Learn the gravitational potential energy formula U = -GMm/r for calculating energy in gravitational fields.
The Formula
Gravitational potential energy describes the energy stored in the configuration of two masses separated by a distance r. The negative sign is a deliberate convention: it indicates that work must be done against gravity to separate the masses, and that the system is in a bound state. As the distance r increases toward infinity, the potential energy approaches zero, which represents two completely separated, non-interacting masses.
This formula gives the exact gravitational potential energy for any two masses at any separation distance. It is more general than the simplified approximation U = mgh that is commonly taught in introductory physics, which only works near Earth's surface where the gravitational field is approximately uniform. The full formula accounts for the fact that gravitational strength decreases with the square of the distance.
The relationship between gravitational potential energy and gravitational force is that force equals the negative derivative of potential energy with respect to distance: F = -dU/dr. Differentiating U = -GMm/r gives F = -GMm/r2, which is Newton's law of gravitation (the negative sign indicates the force is attractive, pointing inward).
Gravitational potential energy plays a central role in orbital mechanics. The total mechanical energy of an orbiting body is the sum of its kinetic and gravitational potential energy. For a circular orbit, the total energy equals -GMm/(2r), exactly half of the potential energy. A negative total energy means the orbit is bound (elliptical or circular), while zero total energy corresponds to a parabolic escape trajectory, and positive total energy means a hyperbolic trajectory where the object escapes to infinity with kinetic energy to spare.
This concept is essential for calculating how much energy is needed to move satellites between orbits (orbital transfer maneuvers), understanding tidal forces, and analyzing the energy budget of astrophysical systems from binary star systems to galaxy clusters.
Variables
| Symbol | Meaning |
|---|---|
| U | Gravitational potential energy (Joules) |
| G | Gravitational constant (6.674 × 10-11 N m2/kg2) |
| M | Mass of the larger body (kg) |
| m | Mass of the smaller body (kg) |
| r | Distance between centers of the two masses (m) |
Example 1
Problem: Calculate the gravitational potential energy of a 1,000 kg satellite orbiting Earth at the surface level. MEarth = 5.972 × 1024 kg, r = 6.371 × 106 m.
U = -(6.674 × 10-11)(5.972 × 1024)(1000) / (6.371 × 106)
U = -(3.986 × 1017) / (6.371 × 106)
U ≈ -6.26 × 1010 J ≈ -62.6 GJ
Example 2
Problem: Find the gravitational potential energy between Earth and the Moon. MEarth = 5.972 × 1024 kg, MMoon = 7.342 × 1022 kg, r = 3.844 × 108 m.
U = -(6.674 × 10-11)(5.972 × 1024)(7.342 × 1022) / (3.844 × 108)
U = -(2.925 × 1037) / (3.844 × 108)
U ≈ -7.61 × 1028 J
When to Use It
Gravitational potential energy is used extensively in astrophysics and engineering.
- Calculating the energy required for satellite orbital transfers (Hohmann transfers)
- Determining escape energy for spacecraft leaving a planet
- Analyzing the binding energy of gravitational systems (binary stars, galaxies)
- Computing tidal heating in moons orbiting giant planets
- Understanding the energy balance in gravitational collapse and star formation