How To Calculate Potential Energy

Gravitational potential energy is the energy an object has because of its position in a gravitational field. Lift something up and you have done work against gravity; that work is stored and can be released later by letting the object fall. The higher it goes, the more is stored.

The formula is mass times gravity times height, usually written PE equals m g h. All three factors must be present. Doubling any one of them doubles the energy, which makes the relationship unusually easy to reason about: this is a linear formula, unlike kinetic energy, which depends on the square of speed.

The height is measured from a reference point you choose, and that choice is arbitrary as long as you are consistent. Energy is only ever meaningful as a difference, so measuring from the floor, from the table or from sea level all give the same answer for the same change in height. What you cannot do is measure the top from one reference and the bottom from another.

Gravity near the Earth's surface is about 9.80665 metres per second squared, and this single number is what makes the formula usable without any calculus. It is not truly constant; it falls off with distance from the centre of the Earth, but over the heights a building or a crane deals with, the change is far too small to matter.

The unit is the joule, which is one kilogram metre squared per second squared. Multiplying kilograms by metres per second squared by metres produces exactly those dimensions, so the units come out right without any fudge factors. A joule is a small amount of energy, roughly what it takes to lift an apple one metre.

Releasing the object converts the stored potential energy into kinetic energy. Because the total is conserved, the potential energy at the top equals the kinetic energy at the bottom, provided nothing is lost to air resistance. That equality is the fastest route to the impact speed: set m g h equal to half m v squared and the mass cancels, leaving v equals the square root of two g h.

The mass cancelling out of the fall is a striking result. A heavy object and a light one dropped from the same height hit the ground at the same speed, because the extra energy stored by the heavier object is exactly offset by the extra energy needed to move its greater mass. Air resistance is the only reason everyday objects behave differently.

Because the height appears to the first power, potential energy grows steadily with height rather than explosively. Raising a load to twice the height stores twice the energy. This is why the energy needed to lift something is so predictable, and why cranes and lifts can be specified with simple arithmetic.

The reference point is not always the ground. In problems involving a table, the natural zero is the table top; in problems involving a pendulum, it is the lowest point of the swing. Choosing the reference so that the motion you care about happens at positive heights keeps the arithmetic positive and the signs easy to follow.

The formula assumes gravity is the only force doing work on the object. If something else is also lifting it, such as a motor or a hand, the energy stored is still m g h but the energy supplied is larger, because some is lost to friction and some goes into kinetic energy if the object is still moving. Separating stored energy from supplied energy is the step most often skipped.