Gravitational Potential Energy Calculator
Calculate gravitational potential energy with PE = mgh. Enter mass and height above a reference point to find stored energy in joules.
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Definition
Gravitational potential energy is the energy an object possesses because of its position above a reference level. Raising a book to a higher shelf does work against gravity, and that work is stored as potential energy in the book-Earth system, ready to be released when the book falls.
Potential energy is energy of position, not energy of motion. A boulder at the edge of a cliff has large potential energy even when it is completely still. The moment it falls, that stored energy converts into kinetic energy.
The formula: PE = mgh
Gravitational potential energy near Earth's surface is calculated with a single equation:
PE = m × g × h
| Symbol | Quantity | SI unit |
|---|---|---|
| PE | Gravitational potential energy | Joule (J) |
| m | Mass of the object | Kilogram (kg) |
| g | Standard gravitational acceleration | 9.80665 m/s² |
| h | Height above the reference plane | Metre (m) |
The formula follows directly from the definition of work: raising a mass m by height h against gravity requires a force mg over distance h, so work done = mgh. That work is stored as potential energy.
The standard value of g
The constant 9.80665 m/s² is the standard acceleration of gravity adopted by the 3rd General Conference on Weights and Measures (CGPM) in 1901. It represents the mean gravitational acceleration at sea level at a geodetic latitude of approximately 45°.
In practice, surface gravity varies slightly:
- Equator: ~9.764 m/s² (weaker — farther from Earth's centre, centrifugal effect)
- Poles: ~9.834 m/s² (stronger — closer to Earth's centre)
For everyday calculations, laboratories, and engineering work, 9.80665 m/s² is the universally accepted standard, and it is what this calculator uses.
Contrast with kinetic energy
Kinetic energy is the energy of motion: KE = ½mv², where v is speed. Potential and kinetic energy are two sides of the same coin in classical mechanics:
| Property | Potential energy | Kinetic energy |
|---|---|---|
| Source | Position | Motion |
| Formula | PE = mgh | KE = ½mv² |
| Zero when | h = 0 (reference level) | v = 0 (at rest) |
| Unit | Joule (J) | Joule (J) |
The total mechanical energy of a system is E = PE + KE. When no non-conservative forces (friction, air resistance) act, E is constant — a principle called conservation of mechanical energy.
Energy conservation: PE converts to KE
When an object falls freely from rest at height h, its potential energy converts entirely into kinetic energy. Setting PE = KE:
mgh = ½mv²
The mass cancels, giving the impact speed:
For h = 30 m: m/s (about 87 km/h).
Roller coaster example
A roller coaster car at the top of a 40 m hill (relative to the lowest point of the track) has:
PE = m × 9.80665 × 40 = 392.3 m joules
where m is the mass of the car and passengers. At the bottom of the hill, ignoring friction, all of that has become kinetic energy. The car reaches m/s (about 100 km/h).
In reality, friction and air resistance dissipate some energy as heat, so the actual speed is slightly lower. Engineers account for this with efficiency factors, but PE = mgh gives the theoretical maximum.
Pendulum example
A pendulum swings between two extreme positions where it is momentarily at rest (KE = 0, all energy is PE) and a lowest point where PE = 0 (taking the lowest point as the reference) and all energy is kinetic. At every intermediate angle, PE + KE equals the total energy at the release point. This is why a pendulum swings to the same height on both sides (if undamped) — energy conservation demands it.
The reference plane
PE = mgh only gives a meaningful absolute value once the reference plane (h = 0) is fixed. Physics only ever requires differences in PE, so the choice of reference cancels out.
Example: A 2 kg book sits on a 1 m high table. The table stands on a floor that is 5 m above sea level.
- Relative to the table: PE = 0 (h = 0 by definition)
- Relative to the floor: PE = 2 × 9.80665 × 1 = 19.6 J
- Relative to sea level: PE = 2 × 9.80665 × 6 = 117.7 J
Each value is correct for its chosen reference. When the book falls off the table onto the floor, it loses exactly 19.6 J of PE regardless of which reference was used, and that energy appears as kinetic energy (and then sound and heat on impact).
A common convention is to set h = 0 at the lowest point in the problem, so all PE values are positive and easy to track.
Worked example
A 5 kg object is lifted from the floor to a shelf 2.4 m above the floor. How much gravitational PE has it gained?
PE = m × g × h = 5 × 9.80665 × 2.4 = 117.68 J
That is also the minimum work done against gravity to lift it. If the object were released and fell freely back to the floor, it would arrive with a kinetic energy of 117.68 J and a speed of:
6.86 m/s
Limitations of PE = mgh
The formula assumes g is constant. This holds accurately when the height change is small compared to Earth's radius (~6371 km):
- Heights up to a few kilometres: error < 0.1 % — use PE = mgh freely.
- Satellite orbits or spacecraft trajectories: g decreases significantly with altitude; use the full gravitational potential energy formula PE = −GMm/r, where G is the gravitational constant, M is Earth's mass, and r is the distance from Earth's centre.
For everything from buildings and bridges to sports biomechanics and roller coaster design, PE = mgh is the right tool.
Frequently Asked Questions (FAQ)
What is the formula for gravitational potential energy?
Gravitational potential energy is PE = mgh, where m is the mass in kilograms, g is the standard gravitational acceleration (9.80665 m/s²), and h is the height above the reference point in meters. The result is in joules (J). For example, a 5 kg object raised 2 m has PE = 5 × 9.80665 × 2 ≈ 98.07 J.
Why is g = 9.80665 m/s²?
The value 9.80665 m/s² is the standard acceleration of gravity defined by the 3rd General Conference on Weights and Measures (CGPM) in 1901. It represents the mean gravitational acceleration at sea level at a latitude of approximately 45°. Actual surface gravity varies from about 9.764 m/s² at the equator to 9.834 m/s² at the poles due to Earth's rotation and flattening, but 9.80665 m/s² is the universally accepted standard for calculations.
How does potential energy convert to kinetic energy?
By conservation of energy, gravitational PE converts entirely into kinetic energy (KE = ½mv²) when an object falls freely in the absence of friction and air resistance. A roller coaster car at the top of a 30 m hill has PE = mgh = m × 9.80665 × 30 ≈ 294 m joules. At the bottom, that same energy appears as kinetic energy, giving a speed of v = √(2gh) = √(2 × 9.80665 × 30) ≈ 24.3 m/s (about 87 km/h).
Does potential energy depend on the reference point?
Yes. Gravitational PE is always measured relative to a chosen reference plane (where h = 0). Only differences in PE matter physically — the absolute value depends on where the zero is placed. For a ball on a table, h = 0 can be set at the floor (PE relative to floor) or at the table surface (PE = 0 on the table, negative below it). Physics problems always use the same reference for initial and final states, so the choice cancels out.