Potential Energy Calculator

Definition in a uniform field

Near Earth’s surface the weight of a body of constant mass can be treated as a downward force mg, with a constant acceleration g. The potential energy assigned to that field is Ep = m·g·h, where h is the height relative to a chosen zero. It is not a separate kind of energy. It is the energy of position in a field that, over a short distance, is treated as uniform. The SI unit is the joule: 1 J = 1 N·m = 1 kg·m²/s². The name honors James Prescott Joule. The unit is defined in the SI Brochure (BIPM, 9th edition).

For 8 kg at 1.8 m the steps are short. Step 1: the weight mg = 8 × 9.81 = 78.48 N. Step 2: the product with the height, 78.48 × 1.8 = 141.264 J. The panel rounds that result to 141.26 J. A mass of 60 kg at 2 m has 1,177.2 J, or 1.177 kJ. One kilogram at one meter has 9.81 J, because only the constant g remains.

Work of a lift

For a particle of constant mass, the net work equals the change in kinetic energy. When a body rises at constant speed, the change in kinetic energy is zero, so the work of the lifting force and the work of the weight add to zero. The lifting force then equals the weight, and its work along the height h is m·g·h. Potential energy is defined so that the work of the weight equals minus the change in Ep. The increase in Ep between two levels is therefore m·g·h.

The calculator reports that increase. It has no force field, so it does not replace the work page. For an 8 kg backpack the same 141.264 J would appear on the W = F·s page if you entered 78.48 N and 1.8 m. This form takes mass and height, and it computes the weight itself.

Reference level and sign

The zero of potential energy is a choice, provided the same choice is kept through one problem. Only the difference matters. Shifting the zero by a constant adds the constant m·g·C to Ep and does not change the difference between two levels. A floor, a counter, and the bottom of a slope are equally valid zeros. Height above the sea is not an argument of the formula unless the problem itself chooses the sea as the zero.

A negative height means a level below the chosen zero. The energy is then negative, because the product m·g·h changes sign with h. Mass cannot be negative. Two levels with the same difference h give the same change in energy, even when both sit high above the ground, as long as g can still be treated as constant.

The constant 9.81 and standard gravity

Standard gravity gn = 9.80665 m/s² was adopted by the 3rd General Conference on Weights and Measures in 1901 as a conventional value, not as the local g at every place. The calculator uses 9.81 m/s², the value common in school problems. For 8 kg and 1.8 m the gap 9.81 − 9.80665 produces 0.048 J. The page has no second mode for gn.

Local gravity changes with latitude and with height. Textbooks quote a range from about 9.78 m/s² at the equator to about 9.83 m/s² at the poles. The calculator has no latitude field and does not correct the result for that spread. It also does not substitute lunar gravity.

Kinetic energy, the spring, and the point-mass potential

Doubling the height doubles Ep. Doubling the speed in kinetic energy quadruples Ek, because speed is squared there. Those are different fields and a different formula. In free fall without drag, a loss of Ep can become a gain of Ek, but this calculator does not compute a speed. Height in a vertical throw is on a separate page, and that page starts from an initial speed rather than from an entered h.

Spring energy ½kx² depends on the extension and on the spring constant. It is not equal to m·g·h. The gravitational potential of a point mass has the form −GMm/r. The difference of that potential between radius R and radius R + h is close to m·g·h when g = GM/R² and h is small compared with R. For a vertical drop of 300 m and a mean Earth radius of 6,371 km, the ratio h/R is about 4.7×10⁻⁵. The calculator does not apply that correction. It keeps the product at g = 9.81 m/s².

Outside this formula

The formula does not include friction, air resistance, or heat. It also does not include a change of shape, and it is not the energy budget of a roller coaster, because the track has losses. The 117.72 kJ of the car is the energy of a 30 m difference, not the speed at the bottom. The formula is not orbital energy and does not use the virial theorem.

When the entered height is at least 10 km, the note beside the result recalls that a constant g is an approximation. The number on the panel stays the product m·9.81·h. For a backpack, a bucket, or a skier, that limit does not apply.

Sources

The definition, the unit, and the link between work and potential energy follow mechanics textbooks and metrology sources: D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, chapters on work and potential energy; H. D. Young and R. A. Freedman, University Physics, the chapter on potential energy; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume I, chapters 4 and 13; BIPM, SI Brochure, 9th edition; Resolution of the 3rd General Conference on Weights and Measures (1901) on standard gravity; NIST Special Publication 330, the value of gn. The page is educational.