Why Is Gravitational Potential Energy Negative?

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Gravitational Potential Energy: Why U Is Negative and What the Well Depth Costs

Gravitational Potential Energy: Why U Is Negative and What the Well Depth Costs

DRAG THE MASS AWAY FROM THE PLANET
U VERSUS r
GRAVITATIONAL POTENTIAL ENERGY  U = −GMm/r —
WELL DEPTH — ENERGY TO REACH U = 0 —
KINETIC ENERGY KE = ½mv² —
TOTAL ENERGY E = U + KE —
ΔU EXACT  A → B (r₂ = — RE) —
CONTROLS & PARAMETERS
CENTRAL MASS Mkg
CENTRAL RADIUS REm
TEST MASS mkg
MARKER A  r×RE
h = —
MARKER B  r₂×RE
Δr = —

U is negative because zero is defined at infinite separation. Gravity only ever pulls inward, so every outward move costs energy. Sitting at any finite r means you are already below that zero line — and the closer in you sit, the further below it you are.

The well depth is the energy bill to climb out, +GMm/r. It falls as 1/r: double r and the bill halves exactly. The tan line on the graph is that bill drawn to scale — drag the mass outward and watch it shrink, but never reach zero.

m·gsurf·Δr is the flat-field shortcut, and it assumes g never changes with height — which only holds close to the surface. Move marker B away from marker A: near the surface it would track ΔU exact closely, but far apart gravity has weakened enough that the shortcut is simply wrong — which is the entire reason U = −GMm/r exists.

U = −GMm/r · g = GM/r² · E = U + KE © The Science Cube

Why is gravitational potential energy negative?

In short: We choose U = 0 at infinite separation, where the masses no longer attract each other. Gravity is attractive, so bringing a mass in from infinity releases energy and U drops below zero: U = −GMm ÷ r. The closer the mass is to the planet, the more negative U becomes. The depth of this well, |U|, is the energy you must supply to lift the mass back up to U = 0.

Common doubts

  • “Does negative U mean the energy is a debt?” No. It only means the energy is below the zero we chose. What matters physically is the change ΔU. Moving the mass away from the planet raises U toward zero.
  • “How does −GMm ÷ r relate to mgh?” Near a planet’s surface, the change in U over a small height h is approximately mgh. The formula −GMm ÷ r is the general version that works at any distance.
  • “What does the total energy E = U + KE tell me?” If E is negative, the mass is bound to the planet and cannot reach infinity. If E is zero or positive, it can escape.

Drag the mass away from the planet in the simulation above, or press RELEASE, and watch U, KE and the total energy respond.

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