Spring force equation: F = k e
Force on a spring = spring constant × extension, written F = k e by AQA and F = k x by Edexcel and OCR. F is the force in newtons (N), k is the spring constant in newtons per metre (N/m) and the extension is in metres (m). All three boards include it, and the equation sheet in your exam paper shows it.
Equation
F = k e
| Symbol | Quantity | Unit |
|---|---|---|
F | force | N |
k | spring constant | N/m |
e | extension | m |
What each specification says
| Exam board | Status | Equation |
|---|---|---|
| AQA Physics (8463) | Recall | F = k e |
| AQA Combined Science (8464) | Recall | F = k e |
| Edexcel Physics (1PH0) | Recall | F = k x |
| Edexcel Combined Science (1SC0) | Recall | F = k x |
| OCR Gateway Physics A (J249) | Recall | F = k x |
| OCR Twenty First Century Physics B (J259) | Recall | F = k x |
Calculator: F = k e
Stretching a spring
Hang a mass on a spring and the spring gets longer. The extra length is the extension: the new length minus the length it had with nothing on it. Pull twice as hard and, up to a point, you get twice the extension. This pattern is called Hooke's law.
The spring constant k measures how stiff the spring is. A soft spring in a pen might have k of about 50 N/m. A car suspension spring can be tens of thousands of newtons per metre. A big k means you need a big force for a small stretch.
The straight-line pattern stops at the limit of proportionality. Past that point, the spring stretches more than the equation predicts, and it may not go back to its first length when you let go.
Which letter your board uses
- AQA (8463 and 8464): F = k e, where e is the extension.
- Edexcel (1PH0 and 1SC0): F = k x, where x is the extension.
- OCR Gateway (J249) and OCR Twenty First Century (J259): F = k x.
Only the letter changes. Every course lists this equation for recall, in Physics and in Combined Science. The paper still gives you the sheet with it on. You can compare the full lists on the Edexcel equation list page and the equation sheet overview.
Rearranging F = k e
- k = F ÷ e
- e = F ÷ k
The force on a hanging spring is usually the weight of the load. Work that out first with W = m g, then use it here. The energy the stretched spring stores has its own equation: see elastic potential energy.
Worked examples
F
- Put the extension in metres: 5 cm = 0.05 m
- F = k e
- F = 40 × 0.05
- F = 2 N
2 N
k
- 12 cm = 0.12 m
- k = F ÷ e
- k = 3 ÷ 0.12
- k = 25 N/m
25 N/m
e
- e = F ÷ k
- e = 6 ÷ 200
- e = 0.03 m, which is 3 cm
0.03 m
Common mistakes
- Putting the total length in place of the extension. A spring that is 20 cm long unloaded and 26 cm long loaded has an extension of 6 cm.
- Leaving the extension in centimetres. With k in N/m, 5 cm entered as 5 gives a force 100 times too big.
- Using a mass in kilograms as the force. A 300 g load pulls with about 2.9 N, not 300 or 0.3.
- Using the equation beyond the limit of proportionality, where force and extension no longer rise together.
The Equation Rearranger gives you quick practice at making k or e the subject.
Questions people ask
What does a spring constant of 25 N/m mean?
Each newton of force stretches the spring by 1/25 of a metre, which is 4 cm. So a 25 N pull would stretch it by 1 m, if it could go that far.
Does F = kx work for squashing a spring?
Yes. Compression works the same way. The extension becomes the amount the spring gets shorter.