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.

Spring potential energy equation

Our elastic potential energy calculator uses the following formula:

The formula for elastic potential energy is:

\[ U = \frac{1}{2} k (\Delta x)^2 \]

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. The spring constant is always real and positive, with units of Newtons per meter (N/m).
\(\Delta x\) is the deformation (stretch or compression) of the spring, expressed in meters (m).
U is the elastic potential energy, measured in Joules (J).

How to calculate the potential energy of a spring

Follow these steps to find its value in no time!

Let's determine the elastic potential energy given the spring constant and deformation.

The formula for elastic potential energy is:

\[ U = \frac{1}{2} k (\Delta x)^2 \]

Given Values:

Spring constant (k): \( 80 \, \text{N/m} \)
Deformation (x): \( 0.15 \, \text{m} \)

Steps to Calculate Energy:

Write the formula for elastic potential energy:
Substitute the given values into the formula:
Perform the calculation:
The elastic potential energy is \( 0.9 \, \text{J} \).

You can also type the values directly into the Elastic Potential Energy Calculator to save time.

Note: Potential energy is inherently related to work, and both quantities share the same units.

Example

Calculating Elastic Potential Energy

Elastic potential energy is the energy stored in an object when it is stretched or compressed. This type of energy is most commonly associated with springs and other elastic materials. The energy is stored when the object is deformed and released when the object returns to its original shape.

The general approach to calculating elastic potential energy includes:

Identifying the spring constant (\( k \)) and the displacement (\( x \)) from the equilibrium position.
Using the formula for elastic potential energy to calculate the result.

Elastic Potential Energy Formula

The general formula for elastic potential energy is:

\[ E = \frac{1}{2} k x^2 \]

Where:

k is the spring constant (in Newtons per meter, N/m).
x is the displacement from the equilibrium position (in meters, m).

Example:

If a spring with a spring constant of \( 200 \, \text{N/m} \) is compressed by \( 0.1 \, \text{m} \), the elastic potential energy is:

Step 1: Square the displacement: \( x^2 = (0.1)^2 = 0.01 \, \text{m}^2 \).
Step 2: Multiply by the spring constant and divide by 2: \( E = \frac{1}{2} \times 200 \times 0.01 = 1 \, \text{J}. \)

Elastic Potential Energy with Extension

Elastic potential energy can also be calculated when the spring is stretched instead of compressed. The formula remains the same, but the displacement (\( x \)) represents how much the spring is extended from its original length.

Example:

If a spring with a spring constant of \( 300 \, \text{N/m} \) is stretched by \( 0.2 \, \text{m} \), the elastic potential energy is:

Step 1: Square the displacement: \( x^2 = (0.2)^2 = 0.04 \, \text{m}^2 \).
Step 2: Multiply by the spring constant and divide by 2: \( E = \frac{1}{2} \times 300 \times 0.04 = 6 \, \text{J}. \)

Real-life Applications of Elastic Potential Energy

Calculating elastic potential energy has many practical applications, such as:

Determining the energy stored in bungee cords during jumps.
Calculating the energy in mechanical systems like spring-loaded doors or suspension systems in vehicles.
Analyzing the energy stored in elastic materials used in engineering and design (e.g., shock absorbers, trampolines).

Common Units of Elastic Potential Energy

SI Unit: The standard unit of elastic potential energy is the Joule (J).

The elastic potential energy can also be expressed in other units such as kilojoules (kJ), but the SI unit is commonly used for most calculations.

Common Operations with Elastic Potential Energy

Elastic Potential Energy in Compression: The energy stored in a spring when it is compressed from its equilibrium position.

Elastic Potential Energy in Extension: The energy stored in a spring when it is stretched beyond its equilibrium position.

Energy Release: The energy stored in the spring is released when the spring returns to its equilibrium position (e.g., when a compressed spring is released or a stretched spring contracts).

Calculating Elastic Potential Energy Examples Table
Problem Type	Description	Steps to Solve	Example
Calculating Elastic Potential Energy from Spring Compression	Finding the elastic potential energy stored in a spring when it is compressed or stretched.	Identify the spring constant \( k \) and the displacement \( x \) from the spring's equilibrium position. Use the elastic potential energy formula: \( E = \frac{1}{2} k x^2 \).	If a spring with a spring constant of \( 200 \, \text{N/m} \) is compressed by \( 0.1 \, \text{m} \), the elastic potential energy is \( E = \frac{1}{2} \times 200 \times (0.1)^2 = 1 \, \text{J}. \)
Elastic Potential Energy with Extension	Calculating the energy stored when a spring is stretched or extended.	Identify the spring constant \( k \) and the extension \( x \). Use the formula \( E = \frac{1}{2} k x^2 \), where \( x \) is the extension of the spring.	If a spring with a spring constant of \( 300 \, \text{N/m} \) is stretched by \( 0.2 \, \text{m} \), the elastic potential energy is \( E = \frac{1}{2} \times 300 \times (0.2)^2 = 6 \, \text{J}. \)
Elastic Potential Energy with Variable Spring Constant	Finding the energy stored in a spring with varying spring constant values over different sections of its stretch or compression.	Identify the varying spring constant \( k \) over the displacement \( x \). Use integration or average \( k \) over the stretch/compression to calculate the total energy.	If a spring has a variable spring constant \( k(x) \), integrate to find the total energy over the displacement \( x \).
Real-life Applications	Using the elastic potential energy formula in real-world scenarios like spring-loaded devices or bungee cords.	To calculate energy stored in bungee cords during jumps. To find the energy in a loaded spring used in mechanical systems.	If a bungee cord with a spring constant of \( 500 \, \text{N/m} \) is stretched by \( 2 \, \text{m} \), the elastic potential energy is \( E = \frac{1}{2} \times 500 \times (2)^2 = 1000 \, \text{J}. \)