Gravitational Potential Energy Calculator

Calculate gravitational potential energy near Earth's surface or between two masses.

Near surface: height above ground; Universal: distance between centers
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When to Use mgh and When to Use Universal Gravitational Energy

The familiar expression mgh is a near-surface approximation for changes in gravitational potential energy, while U = −GMm/r describes the universal two-body gravitational potential with zero chosen at infinity. Near Earth's surface, gravitational acceleration changes very little over ordinary height differences, so ΔU ≈ mgΔh is accurate and convenient.

For satellites, planets, or large altitude changes, g is not constant and the universal expression is more appropriate. The separation r is measured between the centers of the two masses, not from the surface. Its negative sign comes from the conventional choice U = 0 at infinite separation: bound objects at finite r have lower energy than that reference. A change in potential energy is found from Ufinal − Uinitial.

ΔU ≈ mgΔh     U(r) = −GMm/r
SymbolMeaningWhy it appears / units
mObject masskg; gravitational energy scales directly with mass
gLocal gravitational acceleration≈ 9.81 m/s2 near Earth's surface
hVertical height change in the near-surface modelm; valid when g is approximately constant
rCenter-to-center separationm; required by the universal expression
G, MGravitational constant and central massSet the strength of the universal gravitational interaction

Raising an object increases its gravitational potential energy. In the universal convention, that means U becomes less negative as r increases. Escape corresponds to supplying enough total mechanical energy for the object to reach infinitely large r with zero remaining speed in the ideal minimum-energy case.

Worked Examples

Example 1: 10 kg object at 5 m height
PE = 10×9.81×5
Result: 490.5 J
Energy released if dropped
Example 2: Satellite at r=6.7×10⁶ m from Earth center
U = −GMm/r, m=1000 kg
Result: −5.95×10¹⁰ J
Must add this energy to escape Earth
Example 3: Near-surface height change
m = 2.0 kg, Δh = 12 m → ΔU = mgΔh = 2.0 × 9.81 × 12
Result: ΔU ≈ +235 J
The positive change means work must be supplied to raise the object, neglecting other losses.
Example 4: Raising a 1000 kg spacecraft by 400 km
RE = 6.371 × 106 m, r2 = 6.771 × 106 m; ΔU = GMm(1/RE − 1/r2)
Result: ΔU ≈ +3.70 × 109 J
For hundreds of kilometers, using the changing 1/r gravitational potential is more accurate than assuming constant g.

Common Mistakes

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Using altitude instead of center-to-center radius

In U = −GMm/r, r is measured from Earth's center. Add Earth's radius to altitude above the surface.

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Interpreting negative U as negative energy stored in an object

The sign comes from the chosen zero at infinity and indicates a gravitationally bound configuration.

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Using mgh for very large altitude changes

The approximation assumes nearly constant g. For orbital distances, use the universal expression or calculate the exact potential-energy difference.

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Comparing absolute U with an mgh value without matching reference levels

Potential energy depends on the chosen zero. Compare energy changes unless both quantities use the same reference convention.

Frequently Asked Questions

Why is universal GPE negative?
The zero reference is set at infinity (r→∞). Since gravity is attractive, work must be done to move objects apart, making potential energy negative at any finite distance.
What is escape velocity?
The velocity needed for KE = |U|: v_esc = √(2GM/r). For Earth: 11.2 km/s. This is where an object has enough kinetic energy to escape to infinity.
Why is mgh only an approximation?
The formula treats g as constant over the height interval. Earth's gravitational field actually decreases with distance from the center. For heights small compared with Earth's radius, that change is tiny, making mgh an excellent approximation for ordinary laboratory and building-scale problems.
Why does gravitational potential energy increase when U becomes less negative?
With zero chosen at infinity, a bound object starts with negative U. Moving it farther from the attracting mass requires positive work, so its value moves upward toward zero. For example, changing from −60 GJ to −55 GJ is an increase of +5 GJ.
What radius should I use for an object above Earth?
Use the distance from Earth's center. If the object is at altitude h above the surface, r = RE + h, where Earth's mean radius is about 6.371 × 106 m. Using altitude alone can produce a severely incorrect universal gravitational-energy value.
Does an orbiting satellite have only gravitational potential energy?
No. A satellite also has kinetic energy. For a circular orbit, gravity supplies centripetal acceleration and the total mechanical energy is the sum K + U. The satellite can have very negative gravitational potential energy while still moving rapidly with substantial positive kinetic energy.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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