Elastic Potential Energy Calculator

When a spring is stretched or compressed it stores energy. This stored energy is called elastic potential energy and remains in the spring until it is released.

For an ideal spring the energy stored follows a simple formula. This calculator allows you to enter two of three values - energy, spring constant or displacement - and it will calculate the third value for you.

Formula for elastic potential energy:

The elastic potential energy of a spring is half the product of the spring constant and the square of the displacement.

In this formula U is the elastic potential energy, k is the spring constant, and x is the displacement - that is, the amount by which the spring has been stretched or compressed from its natural length. In SI units, the unit of k is newtons per meter, the unit of x is meters, and the calculated energy is in joules.

Elastic potential energy is always positive. Because it depends on the square of the displacement, for a given amount of compression or extension, the result will be the same.

Example calculation:

A spring with a spring constant of 200 newtons per meter is stretched by 0.3 meters. How much energy is stored in the spring?

First square the amount of stretch: 0.3 x 0.3 = 0.09. Then multiply by half the spring constant: 0.5 x 200 x 0.09 = 9. This spring stores 9 joules of energy. The softer the spring or the less it is stretched, the less energy will be stored.

How to use this calculator:

Enter two of the three values (spring constant, stretch, energy) in their respective fields. The calculator will then fill in the third value. If the field for energy is blank, you can calculate the energy from the spring constant and the stretch. If the field for either the spring constant or the stretch is blank, you can work backwards to find these values given the known energy.

Each field has a separate option to switch units. The stretch can be entered in centimeters, the spring constant in Newtons per millimeter and energy can be shown in Joules or Calories. The calculator will do all conversions.

Determine spring constant or deflection

These three quantities are related by a single equation so that if two of them are known the third can be found. By rearranging the equation U = ½kx2, we obtain two alternative forms.

If you know the energy and displacement and want to determine the stiffness (e.g., to calculate the spring constant from a measured energy), use the first form. If you know the energy and the spring constant and want to determine the displacement (e.g., to calculate how far a spring must be stretched to store a certain amount of energy), use the second form. The calculator will automatically select the correct rearrangement if you leave the field for the value you wish to calculate blank.

Where does Hooke's law and factor of 1/2 come from?

An ideal spring obeys Hooke's law, where the restoring force exerted by the spring is proportional to the amount of stretch.

When a spring is stretched, the force exerted increases from zero to kx at a constant rate. Therefore, the average force during the stretching process is half of kx. Since work is the product of force and distance, the stored energy is equal to the average force multiplied by the displacement, which is one-half of kx2. This explains the factor 1/2 in the formula for energy.

This calculator also calculates the force of the spring, F = k x, depending on the extension and displays it along with the energy. You can check both the acting force as well as the stored energy at the same time.

The energy increases proportionally to the square of the extension.

The elastic potential energy is not directly proportional to the extension. As the extension is squared, the stored energy increases much faster than the extension itself.

If the extension is doubled, then the stored energy quadruples. If the extension is tripled, then the stored energy increases ninefold. This square increase explains why so much energy can be stored by stretching a bowstring just a little or compressing a spring in its final centimetres.

Elastic potential energy of everyday objects.

The energy stored depends on both the stiffness and the amount of stretch. A very stiff spring stretched only a small amount can sometimes store as much energy as a soft spring that is stretched a lot.

Please note that a small spring and a hard spring can store the same amount of energy. The stiffness of a hard spring is four times as high but its extension is only half as much. As the extension is squared, these two effects exactly cancel out.

Where does elastic potential energy occur?

Springs, rubber bands, elastic cords and bows store elastic potential energy when they are deformed. A stretched bowstring, a pulled rubber band or the compressed spring of a toy car all contain stored energy that can be released at any time.

When an object regains its original shape the stored energy is released and usually converted into kinetic energy. Trampolines, pole vaults, shock absorbers and spring mechanisms all work on this principle. The energy is stored when inputted and released when outputted.

Elastic Potential Energy and Gravitational Potential Energy

Elastic potential energy is stored through the deformation of an object and depends on the stiffness of the object as well as the amount it has been stretched or compressed. When an object regains its original shape, elastic potential energy is released.

Gravitational potential energy is different and is created by lifting an object. It depends on the mass of the object, acceleration due to gravity, and height. Both are forms of stored energy, or potential energy, but their origins are different.

Energy units:

The energy can be expressed in any units. The joule is the SI unit but there are other common units as well.

Feather combinations:

When two springs are working together, the effective stiffness changes so that the combined spring constant can be used to calculate energy. Springs placed side by side are in parallel and their stiffnesses add up. The resulting stiffness is k_eq = k1 + k2 which means a higher stiffness.

Springs connected end to end are said to be in series. This reduces the stiffness of the springs. The relationship for combined stiffness is 1/k_eq = 1/k1 + 1/k2. First calculate the combined spring constant and then enter it along with the extension into this calculator to find the stored energy.

Common mistakes and limitations:

The most common mistakes are forgetting to square the stretch or forgetting to multiply by half. Both of these can be easily checked. The energy must increase proportionally with the square of the stretch, not the stretch itself.

Please make sure the units match or use the conversion options. Also this formula is based on the assumption that it's an ideal linear spring which obeys Hooke's law. In reality actual springs lose their linearity when stretched too far and the U = 1/2 k x² formula no longer holds true.

This tool is for general educational purposes and solving common problems. For engineering applications, safety issues or other high risk activities you must verify the values against standards and data required by your project.