Calculate spring constant using force and extension
Use force per unit extension with converted length units and a stated proportional range.
Cross-referenced to the 2028–2029 0653 section using original teaching from the related single-science courses. A 0653-specific objective-by-objective and Core/Supplement review is in progress. Written practice is self-checked and does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Measure extension rather than total length.
- Convert centimetres or millimetres to metres.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How many metres is 1.5 cm?
What you will learn
- Define spring constant and use k = F/x.
- Rearrange the spring equation with consistent units.
Force per unit extension describes stiffness
For the proportional spring model, spring constant k is force per unit extension: k = F/x, or F = kx. If force is in N and extension in m, k has unit N/m. A load of 3.0 N extending a spring by 1.5 cm gives x = 0.015 m and k = 200 N/m.
The original straight example graph has load vertically in N and extension horizontally in cm. Its gradient is 2 N/cm, which equals 200 N/m. Do not call that same numerical gradient 2 N/m without converting the extension unit. Use the extension from the unloaded reference, not the total loaded length.
Rearrange inside the supported range
For a spring with k = 200 N/m, force 2 N gives x = F/k = 0.010 m or 1.0 cm while the proportional model applies. Another spring requiring 3 N for 0.010 m extension has k = 300 N/m: it needs more force for the same extension and is stiffer in this comparison.
A single constant k is justified by a through-origin straight load–extension trend over the selected range. Beyond that range, a quotient at a point is not permission to apply the old constant everywhere. At the origin F/x is undefined as 0/0; use a non-zero proportional pair or the line's gradient rather than dividing zero by zero.
Sources for this lesson
Pause and explain
The load–extension gradient is 2 N/cm. What is the spring constant in N/m?
Worked example
A spring extends 1.5 cm under 3.0 N in its proportional range. Calculate k in N/m.
Show the worked solution
- Convert 1.5 cm to 0.015 m.
- k = F/x = 3.0/0.015.
- k = 200 N/m.
Answer 200 N/m.
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.
Find spring constant for 3 N producing 1.5 cm extension in the proportional range.
Give me a hint
Use the change in length in metres before dividing.
Compare my reasoning
- 1.5 cm = 0.015 m.
- k = 3/0.015.
- k = 200 N/m.
200 N/m.
Look for these in your work
- Convert the extension.
- Give force-per-length units.
A spring extends 2.0 cm under 4 N. Calculate k, then extension at 2 N in the same proportional range.
Give me a hint
Find the constant with the first pair and rearrange for the second.
Compare my reasoning
- k = 4/0.020 = 200 N/m.
- x = 2/200 = 0.010 m.
- That extension is 1.0 cm.
200 N/m; 0.010 m or 1.0 cm.
Look for these in your work
- Keep a valid common range.
- Rearrange for extension.
Spring A has k = 200 N/m. Spring B needs 3 N for 0.010 m extension in its proportional range. Compare stiffness.
Give me a hint
Compare force per identical unit extension rather than total spring length.
Compare my reasoning
- For B, k = 3/0.010 = 300 N/m.
- At the same extension, B requires the larger force.
- B is stiffer in the stated proportional models.
B has 300 N/m and is stiffer than A.
Look for these in your work
- Calculate B's constant.
- State the matched-extension comparison.
Common mistakes
- Extension in cm needs conversion for N/m.
- Total spring length is not x.
- Do not use 0/0 or extend a constant beyond its valid range.
What can you explain now?
What extension does 2 N produce for k = 200 N/m in the proportional range?
Compare with the explanation
0.010 m or 1.0 cm.
Divide force by spring constant and retain a length unit.
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Make it stick
Convert a graph gradient from N/cm to N/m and explain when a constant spring value is valid.
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.