AQA · GCSE · Combined Science
Course version: 8464 · Version from Sept 2018
Changes in energy: elastic potential energy = 0.5 × spring constant × extension ²
This equation gives the energy transferred to the elastic potential store of a spring when it is stretched.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
This equation gives the energy transferred to the elastic potential store of a spring when it is stretched. In symbols, Ee = 0.5 × k × e², where Ee is measured in joules (J), k is the spring constant in newtons per metre (N/m) and e is the extension in metres (m). The extension is the increase in length from the spring's natural length. The squared term means that doubling the extension quadruples the energy stored, because 2² = 4.
For example: A spring with k = 100 N/m stretched by 0.20 m stores 0.5 × 100 × (0.20)² = 2.0 J. The equation is valid while the spring obeys Hooke's law, so the force–extension graph is a straight line through the origin and the spring has not passed its limit of proportionality.
What earns the marks
- Write the equation in words or symbols as elastic potential energy = 0.5 × spring constant × extension², with Ee = 0.5 × k × e².
- Use the correct units: Ee in joules (J), k in newtons per metre (N/m) and e in metres (m).
- Calculate extension as the difference between the stretched length and the original length before substituting.
- Square the extension first, then multiply by 0.5 and by the spring constant, following the order of operations.
- Explain that doubling the extension increases the stored energy by a factor of four because extension is squared.
Common mistakes and how to put them right
Multiplying by 2 instead of 0.5, or omitting the 0.5 entirely.
Put it right: The factor is one half because the force rises steadily from zero.
Squaring the spring constant or the energy instead of the extension.
Put it right: Only the extension is squared in this equation.
Using the extension in centimetres without converting to metres.
Put it right: Divide centimetres by 100 to obtain metres before squaring.
Exam tips
- Show the substitution line with numbers and units so the examiner can follow your method clearly.
- If the question asks for extension, rearrange the equation carefully and take the square root as the final step.
- Sanity-check the answer: a stiffer spring or a larger extension should give a larger stored energy.
Try the lesson
The start of the Changes in energy: elastic potential energy = 0.5 × spring constant × extension ² lesson, made from checked teaching notes for this specification point.
Have a go first
A spring has a spring constant of 200 N/m and is stretched by 0.1 m. Using elastic potential energy = 0.5 × spring constant × extension², what is the elastic potential energy stored in the spring?
Listen
This equation gives the energy transferred to the elastic potential store of a spring when it is stretched. In symbols, Ee = 0.5 × k × e², where Ee is measured in joules (J), k is the spring constant in newtons per metre (N/m) and e is the extension in metres (m). The extension is the increase in length from the spring's natural length. The squared term means that doubling the extension quadruples the energy stored, because 2² = 4. For example, a spring with k = 100 N/m stretched by 0.20 m stores 0.5 × 100 × (0.20)² = 2.0 J. The equation is valid while the spring obeys Hooke's law, so the force–extension graph is a straight line through the origin and the spring has not passed its limit of proportionality. A strong answer earns marks for these points. Write the equation in words or symbols as elastic potential energy = 0.5 × spring constant × extension², with Ee = 0.5 × k × e². Use the correct units: Ee in joules (J), k in newtons per metre (N/m) and e in metres (m). Calculate extension as the difference between the stretched length and the original length before substituting. Square the extension first, then multiply by 0.5 and by the spring constant, following the order of operations. A common slip: Multiplying by 2 instead of 0.5, or omitting the 0.5 entirely. Instead, the factor is one half because the force rises steadily from zero. Exam tip: Show the substitution line with numbers and units so the examiner can follow your method clearly.
Key words
Say what each word means before you turn the card.
Elastic potential energy
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Spring constant
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Extension
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Hooke's law
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See every definition
- Elastic potential energy
- The energy transferred to the elastic potential store of a spring when it is stretched, measured in joules (J).
- Spring constant
- A quantity measured in newtons per metre (N/m) that appears in the equation for the energy stored in a stretched spring.
- Extension
- The increase in length from the spring's natural length, measured in metres (m).
- Hooke's law
- The condition under which the equation is valid, so the force–extension graph is a straight line through the origin and the spring has not passed its limit of proportionality.
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