# 6.2.2 Elastic Potential Energy

### Area under a Force-Extension Graph

• The work done in stretching a material is equal to the force multiplied by the distance moved
• Therefore, the area under a force-extension graph is equal to the work done to stretch the material
• The work done is also equal to the elastic potential energy stored in the material Work done is the area under the force – extension graph

• This is true for whether the material obeys Hooke’s law or not
• For the region where the material obeys Hooke’s law, the work done is the area of a right angled triangle under the graph
• For the region where the material doesn’t obey Hooke’s law, the area is the full region under the graph. To calculate this area, split the graph into separate segments and add up the individual areas of each

• The force-extension curve for stretching and contraction of a material that has exceeded its elastic limit is shown below • The curve for contraction is always below the curve for stretching
• The area X represents the net work done or the thermal energy dissipated in the material
• The area X + Y is the minimum energy required to stretch the material to extension e

#### Exam Tip

Make sure to be familiar with the formula for the area of common 2D shapes such as a right angled triangle, trapezium, square and rectangles.

### Elastic Potential Energy

• Elastic potential energy is defined as the energy stored within a material (e.g. in a spring) when it is stretched or compressed
• It can be found from the area under the force-extension graph for a material deformed within its limit of proportionality

### Calculating Elastic Potential Energy

• A material within it’s limit of proportionality obeys Hooke’s law. Therefore, for a material obeying Hooke’s Law, elastic potential energy can be calculated using: Elastic potential energy can be derived from Hooke’s law

• Where k is the spring constant (N m-1) and x is the extension (m)

#### Exam Tip

The formula for EPE = ½ kx2 is only the area under the force-extension graph when it is a straight line i.e. when the material obeys Hooke’s law and is within its elastic limit. ### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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