Gravitational Potential Energy (General)
Potential energy of two masses at distance r.
Formula
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Example
Zero at infinite separation
How it works
This formula gives the gravitational potential energy of a two-body system when the zero of energy is chosen at infinite separation. The negative sign appears because gravity is attractive: as two masses move closer together, the system’s potential energy decreases, and energy must be supplied to pull them apart. It comes from integrating the gravitational force F = GMm/r^2 from infinity to distance r.
When to use it
Use this in problems involving the interaction energy of two separated masses, especially in orbital mechanics, escape-energy calculations, or comparing energies at different separations. It is most useful when the masses can be treated as point masses or as spherically symmetric bodies outside their radii.
Worked example, step by step
Find the gravitational potential energy of a 1200 kg satellite 6.80 × 10^6 m from Earth’s center. Use G = 6.674 × 10^-11 N·m^2/kg^2, M = 5.972 × 10^24 kg, m = 1200 kg, r = 6.80 × 10^6 m. U = -GMm/r U = -(6.674 × 10^-11)(5.972 × 10^24)(1200) / (6.80 × 10^6) U = -8.87 × 10^10 J Final answer: U ≈ -8.9 × 10^10 J
Common mistakes to avoid
- Forgetting the negative sign and reporting gravitational potential energy as positive
- Using the distance from a planet’s surface instead of the center when r is required
- Mixing this total two-body potential energy with the near-Earth approximation U = mgh
Did You Know?
Gravitational potential energy is why a raised hammer, a full reservoir, and a drawn bow all store usable power.
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