Apply Hooke’s law and spring constant
Use proportional force-extension data to calculate a spring constant and predict extension within the linear region.
AS Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 6. The wire investigation is an original study guide and data-analysis practice, not a supervised laboratory activity or proof of practical competence. Full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use force, area, length, extension and graph gradient with consistent SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does spring constant k measure?
What you will learn
- Recall and use Hooke's law
- Calculate spring constant k = F/x
- Check that the proportionality limit is not exceeded
Hooke's law links load and extension
For a spring or wire within its proportional region, extension is proportional to applied load. This is Hooke's law. In magnitude form, F = kx, where F is load, x is extension and k is the spring constant.
The spring constant is the force per unit extension and is measured in N m⁻¹. A larger k means a stiffer spring: more force is needed for the same extension. Use extension in metres for SI results.
Check the model range before predicting
Rearrange Hooke's law to k = F/x or x = F/k. The relationship is valid only while the sample remains in the proportional region. If a force-extension graph bends, a single k no longer describes the whole response.
On a force-against-extension graph, the gradient is k in the linear region. On an extension-against-force graph, the gradient is 1/k, so check the axes before interpreting a gradient.
Pause and explain
What is the SI unit of spring constant?
Worked example
A spring extends by 0.020 m under a 6.0 N load, within its proportional region. Find k and the extension under 9.0 N if it stays in that region.
Show the worked solution
- k = F/x = 6.0/0.020 = 300 N m⁻¹.
- For the second load, x = F/k = 9.0/300.
- Check the question states the response remains proportional.
Answer k = 300 N m⁻¹; extension = 0.030 m under 9.0 N.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A spring extends 4.0 mm under a force of 2.0 N while it obeys Hooke's law. Find its spring constant.
Give me a hint
Convert extension to metres before dividing force by extension.
Compare my reasoning
- x = 4.0 mm = 0.0040 m.
- k = F/x = 2.0/0.0040.
- Give the unit N m⁻¹.
k = 500 N m⁻¹.
Look for these in your work
- Millimetres converted
- Correct ratio formed
- SI spring-constant unit
A wire has spring constant 2.5 × 10³ N m⁻¹. Find the extension under 5.0 N, assuming it remains within its proportional limit.
Give me a hint
Use x = F/k and substitute the stated load and spring constant.
Compare my reasoning
- x = 5.0/(2.5 × 10³) m.
- x = 0.0020 m.
- Convert to millimetres if helpful.
0.0020 m = 2.0 mm.
Look for these in your work
- Equation rearranged
- Extension calculated
- Unit conversion correct
A force-extension graph has gradient 400 N m⁻¹ until 5.0 mm, then curves. Can you predict the extension at 8.0 N using x = F/k if the linear region ended at 5.0 mm? Explain.
Give me a hint
Compare the predicted extension with the range over which the gradient is constant.
Compare my reasoning
- Linear extrapolation predicts x = 8.0/400 = 0.020 m = 20 mm.
- The measured proportional region ends at 5.0 mm, far below 20 mm.
- So Hooke's-law prediction is not valid for the 8.0 N case without further data.
No reliable prediction can be made with that constant k: 8.0 N would imply 20 mm by linear extrapolation, beyond the 5.0 mm proportional range, where the graph curves.
Look for these in your work
- Linear estimate calculated
- Range checked
- Extrapolation limitation stated
Common mistakes
- Using total length instead of extension
- Forgetting to convert millimetres to metres
- Extrapolating Hooke's law beyond the proportional limit
What can you explain now?
What does spring constant k measure?
Compare with the explanation
Force per unit extension
k = F/x in the proportional region.
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Make it stick
Why should you check a force-extension graph before using one spring constant for a new load?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.