Elastic Potential Energy Calculator
Work with the spring energy formula U = ½·k·x² in any direction. Solve for the stored elastic potential energy, the spring constant, or the displacement, switch between metric and imperial units, and add a mass to see how fast the spring would launch it.
Formula
Worked example
A spring with k = 80 N/m stretched x = 0.2 m: U = ½ × 80 × 0.2² = ½ × 80 × 0.04 = 1.6 J. Working backward, if you know U = 1.6 J and x = 0.2 m, then k = 2 × 1.6 / 0.2² = 80 N/m.
What elastic potential energy is
Elastic potential energy is the energy stored in an object when it is deformed elastically, stretched, compressed, or bent, and is ready to spring back to its original shape. For an ideal spring that obeys Hooke's law, the stored energy depends only on the spring constant k and the displacement x from the relaxed position. Because the restoring force grows linearly with displacement, the energy is the area under that force-versus-distance line, which gives the familiar one-half factor in U = ½·k·x².
Solving in any direction
The same formula rearranges three ways, and this calculator handles all of them. Solve for energy with U = ½·k·x² when you know the stiffness and the stretch. Solve for the spring constant with k = 2U / x² when you have measured the energy a spring stores at a known displacement, which is a quick way to characterise an unknown spring. Solve for displacement with x = sqrt(2U / k) when you need to know how far to pull a spring of known stiffness to store a target amount of energy. Pick the quantity to find from the "Solve for" menu and the remaining two fields become your inputs.
Why displacement is squared
The restoring force of a spring is F = k·x, so pushing the spring a little further at the start costs little energy but each additional millimetre near full stretch costs much more. Adding up all that work from zero to x produces a quadratic relationship: the stored energy is proportional to x². This is why a spring compressed to twice the distance holds four times the energy, and why springs, bows, and trampolines can release dramatic amounts of energy after only a modest extra pull. The same squaring applies whether the spring is stretched or compressed, since squaring removes the sign of x.
From stored energy to launch speed
When a stretched or compressed spring is released against a mass, its stored elastic energy converts into the kinetic energy of motion. Ignoring friction, all of U becomes ½·m·v², so the launch speed is v = sqrt(2U / m). Turn on the launch option and enter a mass to see this speed. It is the principle behind spring-loaded toys, pinball plungers, valve springs and catapults: store energy slowly by deforming the spring, then release it quickly into a moving object. The launch speed is an upper limit, since real springs lose a little energy to internal friction and the spring's own moving mass.
Using consistent units
To get an answer in joules, every quantity must end up in SI base units: the spring constant in newtons per metre (N/m) and the displacement in metres (m). This calculator converts for you, so you can enter the spring constant in N/m, N/cm, N/mm or lbf/in and the displacement in metres, centimetres, millimetres or inches. Mixing units by hand, for example using a spring constant in N/m with a displacement in centimetres, is one of the most common mistakes and gives an answer off by a large factor. Let the unit switches do the conversion to avoid it.
Energy stored for k = 100 N/m at different displacements
| Displacement x (m) | Energy U (J) | Restoring force (N) |
|---|---|---|
| 0.05 | 0.125 | 5 |
| 0.1 | 0.5 | 10 |
| 0.2 | 2 | 20 |
| 0.4 | 8 | 40 |
Notice how doubling the displacement quadruples the stored energy (U = ½·k·x²).
Frequently asked questions
What is the formula for elastic potential energy?
The elastic potential energy of an ideal spring is U = ½·k·x², where k is the spring constant in newtons per metre and x is the displacement from the spring's natural length in metres. The result is in joules.
How do I solve for the spring constant or the displacement?
Rearrange U = ½·k·x². To find stiffness, use k = 2U / x². To find how far to displace the spring, use x = sqrt(2U / k). Just choose the quantity in the "Solve for" menu and enter the two known values, and the calculator does the rearranged math for you.
Does it matter whether the spring is stretched or compressed?
No. Because the displacement x is squared in U = ½·k·x², stretching and compressing by the same distance store exactly the same amount of energy. Only the magnitude of the displacement matters, not its direction.
How fast will a spring launch a mass?
If all the stored energy converts to motion with no losses, the launch speed is v = sqrt(2U / m), where U is the elastic potential energy and m is the mass. Turn on the launch option and enter a mass to see this speed. Real springs reach slightly less because of friction and the spring's own mass.
How is elastic potential energy different from kinetic energy?
Elastic potential energy is stored energy held in a deformed spring at rest, while kinetic energy is the energy of motion. When a stretched spring is released, its elastic potential energy converts into kinetic energy as the attached mass speeds up.