Gravitational Potential Energy Calculator
Calculate gravitational potential energy in two modes: near-surface (U = m·g·h) and universal (U = −G·M·m/r). Each mode has a 4-way solver, energy unit conversion (J, kJ, MJ, Wh, kWh, cal), celestial body presets, and solution steps.
Example: m = 1 kg, h = 10 m on Earth → U = 1 × 9.80665 × 10 = 98.0665 J = 0.0980665 kJ = 0.0000272407 kWh
Example: Earth–Moon system → U = −G·Mearth·mmoon/r ≈ -7.613 × 10²⁸ J (negative — see explanation below)
Applies when height h is small compared to the celestial body’s radius, so g is treated as constant. You may freely choose the h = 0 reference (e.g. the floor); h below the reference gives negative U.
General formula valid at any distance from the central mass M. The zero of energy is at infinite distance, so the result U is always negative (or zero) — that is normal and explained in the FAQ.
Verified examples
1 kg at 10 m height (Earth)
U = m·g·h = 1 × 9.80665 × 10 = 98.0665 J = 0.0980665 kJ = 0.0000272407 kWh. This is this calculator’s verification anchor.
Earth–Moon system
U = −G·M·m/r = −6.67430×10⁻¹¹ × 5.97237×10²⁴ × 7.342×10²² / 3.844×10⁸ ≈ -7.613 × 10²⁸ J. This very large negative value is normal: separating the Moon to infinity takes that much energy.
Lifting 1 kg by 1 m: mgh vs universal formula
mgh formula: ΔU = 1 × 9.80665 × 1 = 9.80665 J. Universal formula (U difference between r = R and r = R+1): ΔU = 9.8206 J. The 0.14% difference is not because the formulas differ, but because the nominal Earth mass and radius are rounded values; g = 9.80665 m/s² itself is a defined value. For h small compared to the radius, both formulas are practically identical.
1 kg in LEO 400 km orbit
U = −G·M·m/r = -58,870,756.30 J (≈ −58.87 MJ). The circular orbital velocity there is v = √(GM/r) = 7,672.73 m/s, so its orbital kinetic energy is K = −U/2 and total energy E = U/2 (virial theorem). Escape velocity from Earth’s surface: v = √(2GM/R) = 11,186.32 m/s ≈ 11.19 km/s.
1 kg on Earth’s surface (universal formula)
U = −G·M·m/R = -62,566,926.84 J (≈ −62.57 MJ). Compare with the Earth–Sun system: U ≈ -5.298 × 10³³ J.
Scope & related tools
Not spring energy: elastic potential energy (U = ½kx²) is computed at elastic-potential-energy-calculator — a totally different formula (spring vs gravity).
Not escape velocity: for the speed needed to escape gravity (v = √(2GM/r), in m/s), use escape-velocity — escape velocity is derived from the zero-total-energy condition (K + U = 0).
The U = mgh formula is an approximation for small h; for distances comparable to a planet’s radius use the universal mode. This calculator uses classical Newtonian physics (sufficient for all practical purposes in the solar system).
Frequently asked questions (FAQ)
Why is gravitational potential energy (universal formula) negative?
Because the zero of gravitational potential energy is set at infinite distance (there U = 0). Bringing an object closer to mass M releases energy, so at finite distance U is always negative: U = −G·M·m/r. Example: the Earth–Moon system has U ≈ -7.613 × 10²⁸ J — meaning that much energy is needed to separate the Moon to infinite distance. What is physically meaningful is the energy difference between two positions, not its absolute value.
When to use U = mgh vs U = −GMm/r?
Use U = m·g·h when height h is small compared to the celestial body’s radius so g is practically constant (e.g. buildings, hills, jumps). Use U = −G·M·m/r for distances comparable to or larger than a planet’s radius — satellite orbits, interplanetary missions, or the Earth–Moon system energy. As a bridge: lifting 1 kg by 1 m gives ΔU = 9.80665 J via mgh, while the universal-formula difference between r = R and r = R+1 gives 9.8206 J — practically identical for small h.
How does this calculator differ from gravitational-force-calculator?
This calculator computes ENERGY (joules, a scalar quantity): how much energy is stored in the two-mass configuration. Meanwhile gravitational-force-calculator computes gravitational FORCE (F = G·M·m/r², in newtons, a vector quantity): how strong the attraction is. The two are linked mathematically — force is the slope of the energy curve with respect to distance, F = −dU/dr.
How does it differ from elastic-potential-energy-calculator?
The formula is totally different. The elastic-potential-energy-calculator tool computes SPRING potential energy, U = ½·k·x² — the energy stored when a spring is stretched or compressed. This tool computes GRAVITATIONAL potential energy (U = m·g·h or U = −G·M·m/r) — the energy due to an object’s position in a gravitational field. Both are potential energy, but their physical sources differ.
How does it differ from escape-velocity?
Escape-velocity computes ESCAPE VELOCITY (v = √(2GM/r), in m/s) — the minimum speed for an object to escape gravity. This calculator computes its ENERGY. The two are linked: escape velocity is derived from the zero-total-energy condition, i.e. kinetic energy ½mv² exactly balancing the negative potential energy −GMm/r. Example: the escape velocity from Earth’s surface is 11,186.32 m/s.
What is the difference between G and g? (gravitational constant vs gravitational acceleration)
G = 6.67430 × 10⁻¹¹ m³·kg⁻¹·s⁻² (CODATA 2018) is the UNIVERSAL gravitational constant — the same throughout the universe, used in U = −G·M·m/r. While g is the LOCAL gravitational acceleration at a place, g = G·M/R² — on Earth’s surface g = 9.80665 m/s², on the Moon 1.62 m/s², on Mars 3.71 m/s². Do not confuse them: uppercase G is one value, lowercase g depends on location.
Where is the zero of gravitational potential energy?
For the universal formula U = −G·M·m/r, the zero is at infinite distance — the only choice that keeps the formula simple at all distances. For U = m·g·h, you may freely choose the zero (h = 0): the floor, sea level, or a building roof. In both cases, only the DIFFERENCE in energy between two positions is physically meaningful; shifting the zero does not change that difference.
How is potential energy related to orbital kinetic energy?
For circular orbits the virial theorem holds: orbital kinetic energy K = +G·M·m/(2r) = −U/2, exactly half the magnitude of the potential energy (with opposite sign). Its total energy is E = K + U = U/2 — always negative, meaning the object is bound. Example: 1 kg in a 400 km LEO orbit has U ≈ −58.87 MJ, so K ≈ +29.44 MJ and E ≈ −29.44 MJ, with an orbital velocity of 7,672.73 m/s.
Which units does this calculator support?
Energy: joule (J), kilojoule (kJ), megajoule (MJ), watt-hour (Wh), kilowatt-hour (kWh), and calorie (cal). Mass: kg, gram, ton. Distance/height: m, cm, km. All internal calculations use SI units; the constant G = 6.67430 × 10⁻¹¹ uses the CODATA 2018 value and standard g = 9.80665 m/s².
Does U = mgh apply on the Moon or Mars?
Yes — as long as you use that celestial body’s local g. This calculator provides presets: Moon 1.62 m/s², Mars 3.71 m/s², Jupiter 24.79 m/s², Sun 274 m/s². Example: a 1 kg mass at h = 10 m stores 16.2 J on the Moon and 37.1 J on Mars, compared to 98.0665 J on Earth. For heights comparable to a planet’s radius, switch to the universal mode.
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