# Elastic Potential Energy Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Dec 22, 2022

This elastic potential energy calculator makes it easy to determine the potential energy of a spring when stretched or compressed. Read on to get a better understanding of this concept, including an elastic potential energy definition and an example of calculations. Make sure to check out our potential energy calculator, too!

## Elastic potential energy definition

Imagine a simple helical spring. You can compress or stretch it (to some extent, of course). To do it, though, you need to perform some work - or, in other words, to provide it with some energy. This energy is then stored in the spring and released when it comes back to its equilibrium state (the initial shape and length). Remember that the elastic potential energy is always positive.

Why exactly is this called 'potential energy'? You can think of it like this: the spring doesn't spend the energy at once (in contrary to the energy we described in the kinetic energy calculator), but has the potential to do so.

Don't forget that you cannot compress or stretch a spring to infinity and expect it to return to its original shape. After you reach its elasticity limit, it will get deformed permanently.

## Spring potential energy equation

Our elastic potential energy calculator uses the following formula:

$U = \frac{1}{2} k \Delta x^2$

where:

• $k$ is the spring constant. It is a proportionality constant that describes the relationship between the strain (deformation) in the spring and the force that causes it. However, in the case of rotational forces, the rotational stiffness is of interest - you can find more details about it in the rotational stiffness calculator. The value of the spring constant is always real and positive. The units are Newtons per meter;
• $\Delta x$ is the deformation (stretch or compression) of the spring, expressed in meters; and
• $U$ is the elastic potential energy in Joules.

Try the Hooke's law calculator if you want to calculate the force in the spring as well.

## How to calculate the potential energy of a spring

Follow these steps to find its value in no time!

1. Determine the spring constant $k$. We can assume a spring of $k = 80 \ \mathrm{N/m}$.
2. Decide how far you want to stretch or compress your spring. Let's say that we compress it by $x = 0.15 \ \mathrm m$. Note that the initial length of the spring is not essential here.
3. Substitute these values to the spring potential energy formula: $U = \frac{1}{2} k \Delta x^2$.
4. Calculate the energy. In our example it will be equal to $U = 0.5 \times 80 \times 0.15^2 = 0.9 \ \mathrm J$.
5. You can also type the values directly into the elastic potential energy calculator and save yourself some time :)

Potential energy is inherently related to work. Both physical quantities even have the same units. You can explore the latter with our work calculator. Be sure to give it a try!

## FAQ

### How do I estimate the elastic potential energy stored in a stretched wire?

The elastic potential energy stored in a stretched wire is half of the product of the stretching force (F) and the elongation (Δx):

• U = (1/2) FΔx

### Why is elastic potential energy always positive?

The compression or stretching of any string involves storing supplied energy in the form of potential energy. Hence, this results in an increase in the elastic potential energy.

### What the formula for elastic potential energy per unit volume?

If a force, F, stretches a wire of length x and cross-sectional area A, to an elongation of Δx, the elastic potential energy per unit volume is:

• u = (1/2) × (F/A) × (Δx/x)

But, (F/A) is stress, and (Δx/x) is the strain. Thus,

• u = (1/2) × stress × strain

### Does elastic potential energy depend on mass?

No, elastic potential energy is due to the deformation of an object's shape and so does not depend upon the object's mass.

### What is the elongation of a string with a constant 15 N/m, if the strain energy is 98J?

The elongation of a stretched string with a constant k and a strain energy, or elastic potential energy, of 98 J is:

• Δx = √(2 × U / k)

Thus, the given string has an elongation of:
√(2 × 98 J/ 15 Nm-1) = 3.6 m.

Bogna Szyk
Spring force constant (k)
N/m
Spring stretch length (Δx)
ft
Spring potential energy (U)
J
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