AQA · GCSE · Combined Science
Course version: 8464 · Version from Sept 2018
Forces and elasticity: F = k e
This is the symbolic form of Hooke's law: F is the force in newtons (N), k is the spring constant in newtons per metre (N/m) and e is the extension in metres (m).
Checked against the AQA 8464 specification (opens in a new tab) on .
- Point 6 of 19
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
This is the symbolic form of Hooke's law: F is the force in newtons (N), k is the spring constant in newtons per metre (N/m) and e is the extension in metres (m). The equation says that, for a spring or elastic object obeying Hooke's law, force and extension are directly proportional, so doubling the force doubles the extension. The spring constant k is the force needed per metre of extension; a spring with k = 50 N/m needs 50 N to stretch it by 1 m, or 3.0 N to stretch it by 0.06 m. Rearranged, k = F ÷ e and e = F ÷ k. The relationship is linear only up to the limit of proportionality; beyond that the force–extension graph curves and the equation no longer predicts behaviour. On such a graph, the gradient of the straight section equals k.
What earns the marks
- Recall and use F = k e, identifying F as force in N, k as spring constant in N/m and e as extension in m.
- Rearrange the equation to k = F ÷ e or e = F ÷ k and substitute values correctly.
- Convert extension to metres before calculating, since the spring constant is expressed per metre.
- Recognise that F and e are directly proportional only up to the limit of proportionality, shown by a straight graph through the origin.
- Use the gradient of a force–extension graph to find the spring constant, choosing a large triangle on the linear region.
- Explain that a larger k indicates a stiffer spring, because more force is needed to produce the same extension.
Common mistakes and how to put them right
Confusing k with the gradient's reciprocal.
Put it right: Gradient = ΔF ÷ Δe, which equals k in N/m.
Treating e as the total length of the spring.
Put it right: e is the increase in length, found by subtracting the original length.
Mixing units, such as using newtons with centimetres.
Put it right: Convert all lengths to metres so that k comes out in N/m.
Exam tips
- Quote the equation in symbols first, then substitute numbers with units to make your method clear to the examiner.
- If the answer looks unrealistic, such as a spring constant of 0.5 N/m for a stiff spring, recheck whether you divided by extension in metres.
- For graph questions, state that the straight line through the origin shows direct proportionality and that the gradient gives k.
Try the lesson
The start of the Forces and elasticity: F = k e lesson, made from checked teaching notes for this specification point.
Have a go first
A spring has a spring constant k of 20 N/m. It is stretched by an extension e of 0.5 m. Using F = k e, what is the force F applied to the spring?
Listen
This is the symbolic form of Hooke's law: F is the force in newtons (N), k is the spring constant in newtons per metre (N/m) and e is the extension in metres (m). The equation says that, for a spring or elastic object obeying Hooke's law, force and extension are directly proportional, so doubling the force doubles the extension. The spring constant k is the force needed per metre of extension; a spring with k = 50 N/m needs 50 N to stretch it by 1 m, or 3.0 N to stretch it by 0.06 m. Rearranged, k = F ÷ e and e = F ÷ k. The relationship is linear only up to the limit of proportionality; beyond that the force–extension graph curves and the equation no longer predicts behaviour. On such a graph, the gradient of the straight section equals k. A strong answer earns marks for these points. Recall and use F = k e, identifying F as force in N, k as spring constant in N/m and e as extension in m. Rearrange the equation to k = F ÷ e or e = F ÷ k and substitute values correctly. Convert extension to metres before calculating, since the spring constant is expressed per metre. Recognise that F and e are directly proportional only up to the limit of proportionality, shown by a straight graph through the origin. A common slip: Confusing k with the gradient's reciprocal. Instead, gradient = ΔF ÷ Δe, which equals k in N/m. Exam tip: Quote the equation in symbols first, then substitute numbers with units to make your method clear to the examiner.
See it
Key words
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Spring constant
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Extension
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Directly proportional
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Limit of proportionality
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See every definition
- Spring constant
- The force needed per metre of extension, measured in newtons per metre (N/m).
- Extension
- The increase in length, in metres, found by subtracting the original length from the total length.
- Directly proportional
- A relationship where doubling the force doubles the extension, shown by a straight graph through the origin up to the limit of proportionality.
- Limit of proportionality
- The point beyond which the force–extension graph curves and the equation no longer predicts behaviour.
Label the diagram
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