Gravitational Potential and Potential Energy: Concepts Clear

Lifting a stone to a rooftop stores energy in it — drop it and that energy becomes motion. That stored energy is gravitational potential energy, and the bookkeeping device behind it is gravitational potential: the potential energy that one kilogram would have at that point. Separating "per kilogram" (potential) from "for this mass" (potential energy) makes gravity problems much simpler.

This matters wherever energy is counted: roller coasters, hydroelectric dams, satellite launches and space-mission budgets all run on potential energy. The potential picture explains escape velocity in one line and tells you exactly how much work it takes to climb out of Earth's gravity well.

Key Concepts

• Gravitational potential V = work per unit mass to bring 1 kg from infinity to a point, without acceleration. Unit: J/kg.
• Potential energy U = mV: the energy stored in a mass m at that point. Unit: joule (J).
• Both are negative: V = −GM/r. Zero is defined at infinity; gravity does positive work as a mass falls inward, so bound values sit below zero.
• Near Earth's surface this becomes the familiar U = mgh — the school formula is the small-height approximation of −GMm/r.

Formulas

• Potential of mass M at distance r: V = −GM/r (J/kg).
• Potential energy of mass m: U = −GMm/r (J). More negative means more tightly bound.
• Work moving m from r₁ to r₂: W = U₂ − U₁ = GMm(1/r₁ − 1/r₂).
• At Earth's surface: V = −GM/R ≈ −6.25 × 10⁷ J/kg.

Worked Example

A 1000 kg satellite sits 400 km above Earth. What is its potential energy, and how much work frees it to infinity?

1. r = 6.371 × 10⁶ + 0.4 × 10⁶ = 6.771 × 10⁶ m.
2. U = −GMm/r = −(6.674 × 10⁻¹¹ × 5.972 × 10²⁴ × 1000)/6.771 × 10⁶.
3. Numerator = 3.986 × 10¹⁷; U = −3.986 × 10¹⁷/6.771 × 10⁶ = −5.89 × 10¹⁰ J.
4. At infinity U = 0, so freeing it needs +5.89 × 10¹⁰ J of work (ignoring its orbital kinetic energy).

That is roughly 16,400 kWh — a neighbourhood's daily electricity — just to lift one tonne out of Earth's gravity well from low orbit. This is why rockets are mostly fuel.

Common Misconceptions

• "Energy can never be negative." Potential energy can — negative simply means "bound". Zero is an arbitrary choice at infinity.
• "Potential and potential energy are the same." Potential (J/kg) belongs to the location; potential energy (J) belongs to the mass there. Multiply by m to convert.
• "mgh works at any height." Only near the surface. For satellites you must use −GMm/r; mgh gives wildly wrong answers in orbit.

Real-World Applications

• Space missions: every trajectory is an energy budget in U; transfer orbits minimise the potential-energy climb.
• Hydroelectric power: a dam stores mgh per kilogram of water; the height difference sets the power output.
• Geophysics: potential variations reveal hidden mass — magma chambers, aquifers, ore bodies.

Gravitational potential V = −GM/r is gravity's energy ledger: negative because masses are bound, zero at infinity, multiplied by m for real stored energy. Keep the per-kilogram and total versions straight, switch from mgh to −GMm/r off the ground, and energy problems become simple subtraction.