AkCalculators

Gravitational Potential Energy — concepts, units and practical use

Gravitational potential energy (GPE) quantifies the energy an object possesses because of its position in a gravitational field. For everyday near-Earth applications, GPE is well-approximated by the simple formula E = m × g × h, where m is mass, g the local gravitational acceleration, and h the height relative to a reference level. This article explains the physics behind the formula, unit conversions, when the approximation is valid, measurement tips and practical examples.

Derivation and assumptions

The E = m g h expression emerges from integrating gravitational force over a small vertical displacement when g is approximately constant. It assumes a uniform gravitational field (good near Earth's surface) and negligible changes in g with altitude. For large height ranges (kilometres) or planetary-scale calculations, use the full Newtonian potential ΔU = G M m (1/r₁ − 1/r₂) where G is the gravitational constant and M the planetary mass.

Units and common conversions

SI units: mass in kilograms (kg), height in metres (m), g in metres per second squared (m/s²). Energy in SI is joules (J). This tool supports common alternate units and converts internally to SI before calculation:

• Energy: 1 kJ = 1000 J; 1 cal ≈ 4.184 J; 1 kcal = 4184 J.
• Mass: 1 g = 0.001 kg; 1 lb ≈ 0.45359237 kg.
• Height: 1 cm = 0.01 m; 1 mm = 0.001 m; 1 ft = 0.3048 m.

Choosing g

Standard gravity g0 = 9.80665 m/s² is commonly used. For higher fidelity: use local measured g (varies with latitude and altitude) or planetary gravity (Moon ≈ 1.62 m/s², Mars ≈ 3.71 m/s²). This calculator allows selecting standard values or entering a custom g.

When E = m g h is valid

The formula is valid when g can be treated as constant over the height in question and relativistic/gravitational potential corrections are negligible. For objects moved a few metres to a few kilometres near Earth's surface, E = m g h is an excellent approximation. For satellites and interplanetary scales, use the full gravitational potential formulas.

Measurement and experimental tips

Masses should be measured on calibrated scales; heights relative to a defined datum (floor, ground, sea-level) measured with tape, laser rangefinder or survey equipment depending on precision. For precise energy accounting (e.g., lab experiments), use SI units and account for measurement uncertainty. When converting energy units (calories ↔ joules), confirm whether dietary calories (kcal) or small calories (cal) are intended.

Worked examples

Example 1 — Energy from mass and height: A 70 kg person standing on a 2 m tall platform: E = m g h = 70 × 9.80665 × 2 ≈ 1,372.931 J ≈ 1.373 kJ.

Example 2 — Height from energy: A 500 J lift for a 2 kg object at g = 9.81 m/s²: h = E / (m g) = 500 / (2 × 9.81) ≈ 25.48 m.

Example 3 — Mass from energy: If a system stores 10 kcal (≈ 41840 J) of potential energy at h = 1 m on Earth, equivalent mass m = E / (g h) ≈ 41840 / (9.80665 × 1) ≈ 4,268 kg (illustrates that calories are small compared to mechanical energy at human scales).

Limitations

This calculator does not account for non-uniform gravity, rotational potential, elastic or chemical energy, or relativistic effects. It is intended for classical, near-Earth applications and educational use.

Using this tool effectively

Pick units that match your measurements. Enter two independent known values and leave the target field blank. Use the step-by-step option to show conversions and derivations for documentation. Export results as CSV for lab logs.