Gravitational Potential Energy

Gravitational PE is the work done against gravity to bring a mass from a reference (typically infinity) to its position.

Concept

General (point masses or sphere of mass $M$): $U(r) = -\frac{GMm}{r}$. Negative because gravity is attractive and the reference $U(\infty)=0$ is chosen at infinity.

Near Earth's surface (uniform g): $U=mgh$, where $h$ is height above the reference level.

Difference of PE between two points: $\Delta U = -GMm(1/r_2 - 1/r_1)$.

Derivation

$U(r) - U(\infty) = -\int_{\infty}^r \vec F\cdot d\vec r = -\int_{\infty}^r -\frac{GMm}{r'^2}\,dr' = -\frac{GMm}{r}$.

Worked Example

Energy to lift 100 kg from Earth's surface to infinity = $|U(R)|=GMm/R = mgR \approx 6.26\times 10^9\,\mathrm{J}$.

Common Confusions

• $U=mgh$ is only valid near the surface where g is nearly constant.
• $U$ is intrinsically negative when reference is infinity.
• Always take the difference, never absolute value at infinity for problems.

Key Takeaways

• $U=-GMm/r$ globally; $U=mgh$ locally.
• Negative sign indicates bound state.
• Going to infinity costs $GMm/R$ of energy.