Elastic Potential Energy Calculator

1. Enter any two of energy, spring constant and displacement.
2. Leave the one you want to find blank.
3. Read the result.
4. See which value was solved for and the formula used.

Here is what each result means:

Elastic potential energy is the energy stored in a spring or elastic object when it is stretched or compressed. Do work to deform the spring and that work is stored, ready to be released when the spring springs back. It is the energy that launches a catapult, powers a wind-up toy and cushions a car over a bump.

The stored energy equals one half the spring constant times the square of the displacement. Because the displacement is squared, stretching a spring twice as far stores four times the energy, not twice. The spring constant sets how stiff the spring is, so a stiffer spring stores more energy for the same stretch. In SI units the energy is measured in joules.

This energy is the area under the force-displacement line of Hooke's law, which is why the one half appears. As long as the spring stays within its elastic limit and obeys Hooke's law, all the work you put in is stored and can be fully recovered. Beyond that limit some energy is lost to permanent deformation and the neat formula no longer holds.

It rearranges one relation to solve for whichever value you leave blank.

1. For energy, multiply one half by the spring constant by the displacement squared.
2. For the spring constant, divide twice the energy by the displacement squared.
3. For displacement, take the square root of twice the energy divided by the spring constant.

This builds on Hooke's law; see the Hookes law calculator and potential energy calculator.

A spring of stiffness 200 newtons per meter is stretched 0.1 meters.

Calculation: PE = 0.5 x 200 x 0.1 squared = 1 joule. Stretching it twice as far, to 0.2 meters, would store four times as much: 4 joules.

One relation, rearranged for whatever you need to find.

All three come from PE equals one half k x squared, so any two of the values give the third. Solving for displacement takes a square root because the displacement appears squared, so a given amount of stored energy corresponds to a single positive stretch for a given spring.

Because the displacement is squared, the stored energy grows fast as you stretch further.

This square-law growth is why a slightly longer draw on a bow or catapult stores much more energy. It also means most of the energy is stored in the last part of the stretch, close to the elastic limit, where the spring is working hardest.

The Spring Constant

A stiffer spring stores more energy for the same stretch, in direct proportion to k.

The Displacement

Stretching further stores much more energy, growing with the square of the displacement.

The Elastic Limit

Beyond the elastic limit the spring deforms and not all the energy can be recovered.

Physics Homework

Solve for stored energy, stiffness or stretch in an elasticity problem.

Engineering and Design

Work out the energy a spring stores in a mechanism or shock absorber.

Sports and Toys

Estimate the energy in a bow, catapult or trampoline before release.

1. Forgetting the One Half

The energy is one half k x squared, not k x squared. Dropping the half doubles the answer.

2. Forgetting to Square

The displacement is squared. Leaving it unsquared badly understates the energy.

3. Confusing with Force

Hooke's law force is k x; the stored energy is one half k x squared. They are different quantities.

4. Mixing Units

Use newtons per meter and meters to get an energy in joules.

5. Exceeding the Elastic Limit

The formula only holds while the spring obeys Hooke's law.

The relation is exact; only the displayed decimals are rounded, and very large or small results are shown in scientific notation.

It is the energy stored in a stretched or compressed spring. It equals one half the spring constant times the displacement squared. A 200 newton per meter spring stretched 0.1 meters stores 1 joule.

Multiply one half by the spring constant by the square of the displacement. For a 200 newton per meter spring stretched 0.1 meters, that is 0.5 times 200 times 0.01, which is 1 joule.

Because the energy is the area under the force-displacement line of Hooke's law, which forms a triangle. The area of that triangle introduces the factor of one half.

Take the square root of twice the energy divided by the spring constant. Leave the displacement field blank and enter energy and spring constant to solve for it.

Hooke's law gives the spring force, k times x. Elastic potential energy gives the energy stored, one half k x squared. One is a force, the other is energy.

No. Because the displacement is squared, stretching twice as far stores four times the energy. Three times the stretch stores nine times the energy.

Joules for energy, newtons per meter for the spring constant and meters for displacement. Keep the units consistent for a correct result.

Yes. The same formula applies whether the spring is stretched or compressed, since the displacement is squared and the stored energy is always positive.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Elastic energy (Wikipedia).
• Hooke's law (Wikipedia).
• Springs and energy (Maths Is Fun).