AQA · GCSE · Combined Science
Course version: 8464 · Version from Sept 2018
Forces and elasticity: Eₑ = ½ke²
This equation gives the elastic potential energy stored in a stretched or compressed spring, provided the spring has not been stretched beyond its limit of proportionality.
Checked against the AQA 8464 specification (opens in a new tab) on .
- Point 17 of 19
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
This equation gives the elastic potential energy stored in a stretched or compressed spring, provided the spring has not been stretched beyond its limit of proportionality. Eₑ is the energy in joules (J), k is the spring constant in newtons per metre (N/m), and e is the extension or compression in metres (m). The factor ½ arises because the force needed grows steadily from zero to ke as the spring stretches, so the average force is ½ke and the work done is ½ke × e.
For example: A spring with k = 200 N/m stretched by e = 0.10 m stores Eₑ = ½ × 200 × 0.10² = 1.0 J. Always convert centimetres to metres and grams to kilograms before substituting, and square the extension before multiplying by k.
What earns the marks
- State the equation as Eₑ = ½ke² and identify Eₑ as elastic potential energy in joules (J).
- Identify k as the spring constant in newtons per metre (N/m) and e as the extension or compression in metres (m).
- Convert all quantities to SI units before substitution, for example 15 cm becomes 0.15 m.
- Substitute values correctly, square the extension first, then multiply by k and by ½.
- Explain that the equation applies only while the spring obeys Hooke's law, up to the limit of proportionality.
- Interpret the result as the energy stored or transferred when the spring is stretched or released.
Common mistakes and how to put them right
Forgetting to square the extension.
Put it right: Apply the index to e before any multiplication.
Using the total length instead of the extension.
Put it right: Subtract the original length to find e.
Leaving the extension in centimetres.
Put it right: Divide by 100 to convert to metres.
Exam tips
- Write the equation, then the substituted values with units, then the answer with the correct unit; this makes method marks easier to award.
- Check whether the question gives extension or total length, and convert units before substituting.
- If asked for energy transferred when the spring is released, state that the stored elastic potential energy is transferred to kinetic or thermal energy stores.
Try the lesson
The start of the Forces and elasticity: Eₑ = ½ke² lesson, made from checked teaching notes for this specification point.
Have a go first
A spring has a spring constant k of 200 N/m and is stretched by an extension e of 0.10 m. Using Eₑ = ½ke², what is the elastic potential energy stored in the spring?
Listen
This equation gives the elastic potential energy stored in a stretched or compressed spring, provided the spring has not been stretched beyond its limit of proportionality. Eₑ is the energy in joules (J), k is the spring constant in newtons per metre (N/m), and e is the extension or compression in metres (m). The factor ½ arises because the force needed grows steadily from zero to ke as the spring stretches, so the average force is ½ke and the work done is ½ke × e. For example, a spring with k = 200 N/m stretched by e = 0.10 m stores Eₑ = ½ × 200 × 0.10² = 1.0 J. Always convert centimetres to metres and grams to kilograms before substituting, and square the extension before multiplying by k. A strong answer earns marks for these points. State the equation as Eₑ = ½ke² and identify Eₑ as elastic potential energy in joules (J). Identify k as the spring constant in newtons per metre (N/m) and e as the extension or compression in metres (m). Convert all quantities to SI units before substitution, for example 15 cm becomes 0.15 m. Substitute values correctly, square the extension first, then multiply by k and by ½. A common slip: Forgetting to square the extension: the error is calculating ½ke instead of ½ke². Instead, apply the index to e before any multiplication. Exam tip: Write the equation, then the substituted values with units, then the answer with the correct unit; this makes method marks easier to award.
See it
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 or compression
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Limit of proportionality
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See every definition
- Elastic potential energy
- The energy stored in a stretched or compressed spring, given by Eₑ = ½ke², provided the spring has not been stretched beyond its limit of proportionality.
- Spring constant
- A quantity in newtons per metre (N/m), represented by k in the equation Eₑ = ½ke².
- Extension or compression
- The change in length of a spring in metres (m), represented by e in the equation Eₑ = ½ke².
- Limit of proportionality
- The point beyond which a spring must not be stretched for the equation Eₑ = ½ke² to apply.
Label the diagram
Some boxes have lost their names. Pick the right name for each number.
1Box 1
2Box 2
3Box 3
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