The Potential Energy Calculator finds stored energy from position or spring stretch. Gravitational mode uses PE = mgh with standard gravity 9.80665 m/s². Elastic mode uses U = ½kΔx² from spring constant and displacement. Enter the known values and the calculator returns joules with the substituted arithmetic shown, including planet presets when surface gravity changes.
Stored energy is relative to a chosen zero. Only differences in potential energy do work or convert to motion, which is why the reference height you pick matters less than the change between two heights.
Calculate gravitational potential energy
Gravitational potential energy near Earth's surface is PE = mgh: mass times field strength times height above a chosen reference. With m in kilograms, g in metres per second squared and h in metres, PE is in joules. The Potential Energy Calculator defaults g to exactly 9.80665 m/s² and lets you override it for other bodies or a local value.
| Symbol | Quantity | SI unit |
|---|---|---|
| PE | gravitational potential energy | joules (J) |
| m | mass | kilograms (kg) |
| g | gravitational field strength | metres per second squared (m/s²) |
| h | height above reference | metres (m) |
Near Earth's surface, g is treated as constant. That approximation holds for heights that are small compared with Earth's radius. Orbital mechanics and escape-energy problems need the inverse-square form instead; those are outside this tool.
Mass and height must be consistent with the reference you choose. A book on a 0.8 m desk has PE = m × 9.80665 × 0.8 relative to the floor, and PE = 0 relative to the desk top. Both statements are correct; they use different zeros.
Scientific notation works in the 3.45e9 form on every numeric field. Unit selectors convert mass and height before multiplication so a problem stated in grams and centimetres does not need hand conversion.
Calculate elastic potential energy
A spring (or any Hookean elastic element) stores U = ½kΔx² when stretched or compressed by Δx from its unstressed length. Spring constant k is in newtons per metre; displacement Δx is in metres; energy is in joules. The Potential Energy Calculator switches to this formula in elastic mode and does not mix g into the result.
| Symbol | Quantity | SI unit |
|---|---|---|
| U | elastic potential energy | joules (J) |
| k | spring constant | newtons per metre (N/m) |
| Δx | extension or compression | metres (m) |
Displacement is squared, so stretching 0.2 m stores four times the energy of stretching 0.1 m at the same k. Compression and extension contribute the same energy for the same |Δx|; the sign of Δx drops out after squaring.
Hooke's law F = −kx must still hold. Beyond the elastic limit the force is no longer proportional to displacement and ½kΔx² stops being valid. Stiffness k itself is owned in force terms by the Hooke's Law Calculator; this page owns the stored-energy side only.
Series and parallel spring combinations change the effective k before energy is computed. Enter the effective constant for the system you actually have, not the label on one coil if several coils share the load.
Choose a reference point for height
Gravitational potential energy is defined only relative to a chosen zero of height. The floor, a desk, or the lowest point of a track are all valid references. The Potential Energy Calculator treats the height you type as the h in mgh relative to whatever zero you have in mind, and it does not invent a reference on its own.
Only changes matter for mechanics problems. Raising a 3 kg mass from 1 m to 4 m above the floor changes PE by m g (4 − 1) = 3 × 9.80665 × 3 = 88.26 J, whether you call the floor h = 0 or call some other level zero and adjust both heights. The difference Δh is what enters the energy budget.
Exam traps often set the reference at the top of a ramp or at the bottom of a pendulum swing. State the reference in one line before substituting. If two students pick different zeros but compute the same ΔPE, both can be right.
Negative height is allowed when the object sits below the reference (a mass in a hole with the surface as zero). Negative PE then means "below the chosen zero," not that energy is somehow unphysical. Absolute PE values depend on the zero; energy differences and conversions to kinetic energy do not.
Calculate PE for a 5 kg object at 10 m
A 5 kg object held 10 m above the reference, with g = 9.80665 m/s², stores 490.3325 J of gravitational potential energy. That triple is the gravitational worked fixture for this page, and the steps below show the same product the calculator prints when those three values are entered.
1. List what is known. m = 5 kg, h = 10 m, g = 9.80665 m/s².
2. Write the formula. PE = mgh
3. Substitute. PE = 5 × 9.80665 × 10
4. Multiply. 5 × 9.80665 = 49.03325, then × 10 = 490.3325 J
Textbooks that round g to 9.81 get PE = 5 × 9.81 × 10 = 490.5 J. Those that use 9.8 get 490 J. The calculator uses the exact defined standard gravity 9.80665 so results match engineering tables and the site-wide constant.
If height falls from 10 m to 0 with no non-conservative losses, that 490.3325 J becomes kinetic energy at the bottom: ½mv² = 490.3325 gives v = √(2 × 490.3325 / 5) = √196.133 = about 14.0 m/s. The Potential Energy Calculator reports PE; pair it with the Kinetic Energy Calculator when you need the speed after the fall.
Calculate elastic PE for k = 200, Δx = 0.1
A spring with k = 200 N/m stretched by 0.1 m stores 1 J. That pair is the elastic worked fixture for this page and must not be confused with the Hooke's Law force example on the sibling page, which owns restoring force rather than stored energy.
1. List what is known. k = 200 N/m, Δx = 0.1 m.
2. Write the formula. U = ½kΔx²
3. Square the displacement. (0.1)² = 0.01
4. Substitute and multiply. U = ½ × 200 × 0.01 = 100 × 0.01 = 1 J
Doubling the stretch to 0.2 m gives U = ½ × 200 × 0.04 = 4 J, four times as much. Halving k to 100 N/m at the original 0.1 m stretch gives 0.5 J. Energy is linear in k and quadratic in Δx.
The restoring force at 0.1 m would be |F| = kx = 200 × 0.1 = 20 N. Force and energy answer different questions; do not substitute one formula for the other.
Calculate potential energy on other planets
Surface gravity changes by body, so the same mass at the same height stores different gravitational PE. The Potential Energy Calculator can swap g for published surface values while leaving m and h fixed. Figures below come from the site physics reference.
| Body | g (m/s²) | PE for m = 5 kg, h = 10 m |
|---|---|---|
| Earth | 9.80665 | 490.3325 J |
| Moon | 1.62 | 81 J |
| Mars | 3.72 | 186 J |
| Venus | 8.87 | 443.5 J |
| Jupiter | 24.79 | 1,239.5 J |
| Sun | 274 | 13,700 J |
On the Moon, the same 5 kg at 10 m holds only 5 × 1.62 × 10 = 81 J, about one sixth of the Earth value. On Jupiter it holds 5 × 24.79 × 10 = 1,239.5 J. Mass did not change; the field strength did.
These surface values are reference averages. Local geology, rotation and altitude shift g slightly on any real world, just as Earth g ranges roughly from 9.764 to 9.834 m/s². Coursework almost always wants the single tabulated number.
Weight mg also scales with g, which the Mass Calculator discusses when distinguishing mass from weight. Here the output is energy, not force.
Understand energy conservation
Without friction, drag and other non-conservative work, mechanical energy is conserved: PE + KE stays constant while forms trade. A falling object loses gravitational PE and gains KE. A pendulum at the top of its swing is mostly potential; at the bottom it is mostly kinetic. The Potential Energy Calculator supplies the PE term for that ledger.
For a drop from height h starting at rest, mgh at the top equals ½mv² at the bottom if the reference is the bottom and losses are neglected. Cancel m (if m ≠ 0) to get v = √(2gh). With g = 9.80665 and h = 10 m, v = √(196.133) ≈ 14.0 m/s, matching the conversion from the 490.3325 J example.
Elastic systems trade ½kΔx² with kinetic energy during oscillation while total energy stays constant in the ideal case. Real springs dissipate a little heat each cycle; ideal conservation is the first model, then damping is added if the problem demands it.
When friction does work, mechanical energy is not conserved. The missing energy left the mechanical account as thermal energy. The Work Calculator frames that as negative work removing kinetic energy.
Frequently asked questions
What is the formula for gravitational potential energy?
Near Earth's surface, PE = mgh with mass in kilograms, g in m/s² and height in metres above a chosen reference. Standard g on this site is 9.80665 m/s². The result is in joules. Large altitudes relative to Earth's radius need a different gravitational formula.
What is the formula for elastic potential energy?
Elastic potential energy in a Hookean spring is U = ½kΔx², with k in N/m and Δx the stretch or compression from the unstressed length in metres. The result is in joules. The formula holds only inside the elastic limit where force stays proportional to displacement.
Why does the choice of reference height matter?
Potential energy is measured from a zero you choose. Different zeros change the absolute PE number but not the change ΔPE between two positions. Mechanics problems care about that change when energy converts to kinetic energy or work.
What g value should coursework use?
This calculator uses 9.80665 m/s² by default. Many textbooks round to 9.81 or 9.8. Follow the value your assignment specifies; state it in the working. Mixing 9.8 in one step and 9.81 in another creates avoidable rounding disputes.
Can potential energy be negative?
Yes, if the object is below your chosen reference height. Negative PE means "below zero on this scale," not that the concept failed. Energy differences and conservation statements remain well defined.
How does potential energy relate to kinetic energy?
When non-conservative work is negligible, loss in PE equals gain in KE and vice versa. Total mechanical energy PE + KE stays constant. Falling objects, pendulums and ideal spring oscillators are the standard illustrations.
Is elastic energy the same as Hooke's law?
Hooke's law gives force F = −kx. Elastic potential energy integrates that force to U = ½kΔx². Related, not identical. Force problems belong on the Hooke's Law Calculator; stored energy belongs here.
Does mass change on the Moon while PE changes?
Mass is the same everywhere. PE = mgh changes on the Moon because g is about 1.62 m/s² instead of 9.80665. Weight mg changes for the same reason. The Mass Calculator owns the mass-versus-weight wording in full.
What happens to PE when an object falls?
Gravitational PE decreases as height decreases. If drag and friction are neglected, that loss appears as kinetic energy. With drag, some of the energy leaves the mechanical account as heat, so the speed at the bottom is lower than √(2gh).
Summary
The Potential Energy Calculator evaluates gravitational PE = mgh with g = 9.80665 m/s² by default and elastic U = ½kΔx² for springs. Height is always relative to a chosen reference, so only changes in PE enter energy accounts.
The gravitational fixture m = 5 kg, h = 10 m yields 490.3325 J; the elastic fixture k = 200 N/m, Δx = 0.1 m yields 1 J.
Planet presets swap g so the same mass and height show how stored energy scales with surface gravity, and conservation links PE to kinetic energy when non-conservative work is absent.