Potential Energy Calculator: PE = mgh
Work out gravitational potential energy from mass, gravity and height, or find any one of those from the other three. Kilograms, pounds, metres, feet.
Potential Energy Calculator
Gravitational potential energy is the energy a body has because of how high it sits. Fill in three of the four fields, leave the fourth blank, and the blank one is worked out.
Documentation
Potential Energy Calculator
Gravitational potential energy is the energy a body holds because of how high it sits above a chosen level. This calculator works out that energy from a body's mass, the strength of gravity where it is, and its height. It also runs the other way: given the energy and two of the other three figures, it returns the fourth.
What the calculator does
The page has four fields: potential energy in joules, mass, gravity, and height. Mass and height each have a unit menu. Three fields are filled in, one is left blank, and the blank one is the answer. All four figures are then shown together, so the three that were entered can be checked beside the result.
Mass can be entered in kilograms, grams, milligrams, metric tonnes, pounds, ounces, stone, US short tons or long tons. Height can be entered in metres, kilometres, centimetres, millimetres, micrometres, nanometres, miles, yards, feet, inches or nautical miles. Energy has no unit menu: it is read and shown in joules. Gravity has no unit menu either. It is read in metres per second squared, so a figure given in feet per second squared has to be converted before it is typed in.
Before any arithmetic runs, the mass is converted to kilograms and the height to metres by the site's shared unit registry. An answer is converted back into the unit chosen for that field. Results appear to eight significant figures. Only the display is rounded, and the arithmetic runs on the values as typed. A result smaller than 0.0001, or 1,000,000,000 and above, is shown in scientific notation.
Every field opens blank, gravity included. The address bar keeps in step with the fields, and a blank field cannot be written into a link, so a shared link carries exactly the three figures that were filled in and leaves the fourth blank for the next reader.
The potential energy formula
where is the potential energy in joules (J), is the mass in kilograms (kg), is the strength of gravity in metres per second squared (m/s²), and is the height in metres (m).
The same relation, rearranged, gives the other three answers:
The energy grows in step with each of the three figures. Doubling the mass doubles the energy. So does doubling the height, or moving the body somewhere gravity pulls twice as hard. Nothing is squared here, which is the main difference from kinetic energy.
Solving for gravity is the unusual direction. It answers a question of the form "what pull would give this energy for this mass at this height", which is one way to compare one body with another from a measured energy.
Height is measured from a level the reader picks
Potential energy has no natural zero. The formula gives the energy relative to some level, and the reader chooses that level: a floor, the ground, sea level, or anything else that suits the problem. Heights are then measured from there.
Below that level the height is negative, and the energy is negative with it. The page keeps the sign rather than clamping it to zero. A basement floor 3 m below the chosen zero really does hold less energy than the zero level, and reporting it as zero would be wrong rather than tidy.
Which constants are exact
The unit conversions on this page are definitions agreed by standards bodies, not measurements, so no better experiment will revise one.
| Constant | Value | Exact? | Source |
|---|---|---|---|
| Standard gravity | 9.80665 m/s² | exact by definition | 3rd CGPM (1901); NIST SP 811, Appendix B |
| Joule | 1 kg⋅m²/s² | exact by definition of the SI derived unit | SI Brochure, 9th edition |
| Pound | 0.45359237 kg | exact by definition | 1959 International Yard and Pound Agreement |
| Ounce, stone, short ton, long ton | 1/16 lb, 14 lb, 2,000 lb, 2,240 lb | exact; each follows from the pound | NIST SP 811, Appendix B |
| Metric tonne | 1,000 kg | exact by definition | SI Brochure, 9th edition |
| Foot | 0.3048 m | exact by definition | 1959 Agreement; NIST SP 811, Appendix B |
| Inch, yard, mile | 0.0254 m, 0.9144 m, 1,609.344 m | exact; each follows from the same agreement | 1959 Agreement; NIST SP 811, Appendix B |
| Nautical mile | 1,852 m | exact by definition | International Hydrographic Conference, Monaco, 1929 |
| Foot pound-force | 1.3558179483314004 J | exact; the pound times standard gravity times the foot | NIST SP 811, Appendix B |
Standard gravity is a convention, not a reading taken at any particular place. Real surface gravity on Earth varies with latitude and altitude, roughly between 9.76 and 9.83 m/s². The figures for other bodies are measurements and are rounded: about 1.62 m/s² on the Moon and about 3.71 m/s² on Mars.
The page does not fill the gravity field in with 9.80665. The figure is quoted in the wording beside the field instead, and a test ties that wording to the registry's constant so the two cannot drift apart.
Worked example
A 70 kg person climbing 2 m of stairs on Earth, with standard gravity.
Multiplying 70 by 9.80665 gives 686.4655. Multiplying that by 2 gives 1,372.931, so the climb stores 1,372.931 J, or about 1.37 kilojoules.
The same figures run backwards. Leave the mass blank and enter 1,372.931 J with 9.80665 m/s² and 2 m: dividing by the gravity gives 70 exactly, and 70 kg is what the page shows. Leave the gravity blank instead, and 1,372.931 J divided by 70 and then by 2 comes back to 9.80665 m/s².
The same 70 kg lifted 2 m on the Moon, where gravity is about 1.62 m/s², stores only 226.8 J, roughly a sixth as much.
An imperial case works from two of the defining constants above. One pound is 0.45359237 kg and one foot is 0.3048 m. Multiplying 0.45359237 by 9.80665 gives 4.4482216152605, which is one pound-force in newtons. Multiplying that by 0.3048 gives 1.3558179483314004 J, one foot pound-force. The page shows 1.3558179 J after rounding to eight significant figures.
A case that does not come out even: a shelf 0.1 m above a bench that is itself 0.2 m above the floor. A computer stores those two numbers in binary, and their sum is 0.30000000000000004 rather than 0.3. For a 1 kg book at that height the energy is 2.9419950000000004 J, which the page shows as 2.941995 J. Feeding that energy back in with the same mass and gravity returns the height it started from, digit for digit.
How the arithmetic is arranged
Eight significant figures is fine enough to show rounding dust, so each rearrangement is written to cancel exactly at round values.
Each answer divides once by each factor rather than once by their product. Dividing by the gravity and then by the height carries a pair of figures whose product would overflow to infinity or fall to zero on its own. One kilogram lifted one metre under standard gravity stores exactly 9.80665 J, and every rearrangement of that row divides back out to exactly 1.
What the calculator refuses
An entry box passes through any number typed or pasted into it, so the checks that matter are the ones in the calculation code.
| Situation | Why there is no answer |
|---|---|
| Mass of zero or less | A body has mass above zero. |
| Gravity of zero or less | A body with no pull on it stores no energy at any height. |
| Working out a mass or a gravity at a height of zero | The energy at the chosen zero level is zero whatever the body weighs, so nothing can be recovered from it. Dividing by that zero would report infinity, which is a confident wrong answer. |
| A mass or a gravity that works out to zero or less | An energy and a height with opposite signs would need a negative one. An energy of zero away from the reference level needs none at all. An energy tiny beside the other two figures also divides down to zero although the true answer is above it. |
| An answer too large to hold | The figures are outside the range the page's numbers carry, so the result is not an energy. |
| A unit the page does not have | A shared or hand-edited link can name any unit, including one belonging to another quantity. The page says so rather than breaking. |
A negative height is not refused, and neither is a negative energy, because both are real answers below the chosen zero level.
Two or more blank fields, or none, are treated as guidance rather than as a fault. The page asks for exactly one blank field and works that one out.
Where this formula stops working
The relation assumes gravity is the same strength at every point of the climb. That holds close to a surface, where a few metres change nothing measurable. It fails over distances where gravity itself weakens, such as a satellite orbit, which needs the inverse-square form of gravitational potential energy instead.
The page also covers gravitational potential energy only. A stretched spring, a compressed gas and a chemical bond all store energy, and each follows its own rule.
Frequently asked questions
What is gravitational potential energy? It is the energy a body holds because of its height above a chosen level. Lifting the body puts energy in. Letting it fall gives that energy back, usually as motion. The unit is the joule.
How is potential energy calculated? Multiply the mass in kilograms by gravity in metres per second squared, then by the height in metres. For a 70 kg person climbing 2 m on Earth, 70 times 9.80665 is 686.4655, and 686.4655 times 2 is 1,372.931 J.
What value of g should be used? Standard gravity, 9.80665 m/s², suits any everyday problem on Earth, and it is exact by definition rather than measured. Surface gravity varies a little by place, from about 9.76 to 9.83 m/s². For other bodies the field takes any figure: about 1.62 m/s² for the Moon, about 3.71 m/s² for Mars.
Can potential energy be negative? Yes. Height is measured from a level the reader picks, and a body below that level has a negative height and a negative energy. The page keeps that sign. Kinetic energy behaves differently, because a squared speed is never below zero.
How is the height found from a known energy? Divide the energy by the mass, then by the gravity. For 1,372.931 J with a mass of 70 kg on Earth, dividing by 70 gives 19.6133, and dividing that by 9.80665 gives 2 m.
Why will the page not work out a mass when the height is zero? At the chosen zero level the stored energy is zero whatever the body weighs, so the energy says nothing about the mass. There is nothing to recover, and the division needed would have zero on the bottom.