The spring constant equation

The spring constant equation comes from Hooke's law, F = ke: force = spring constant × extension. Rearranged, k = F ÷ e. The spring constant k is the force needed to stretch a spring by one metre, measured in newtons per metre (N/m). A stiff spring has a large k and a floppy one a small k. The equation only holds up to the limit of proportionality.

Hooke's law and the spring constant

Hooke's law says that the extension of a spring is directly proportional to the force stretching it, as long as the limit of proportionality is not passed. Double the force and the extension doubles. The spring constant is the number that links them:

The extension is how much longer the spring has become, not its total length. The same equation works for a spring being squashed, with e as the compression.

The Hooke's Law Calculator works out any one of these from the other two.

Spring constant equation: worked examples

Example 1: the basic calculation

A force of 6 N stretches a spring by 0.04 m. Find the spring constant.

Example 2: extension in centimetres

A 2 N force stretches a spring by 5 cm. Find k.

Convert first: 5 cm = 0.05 m. Then k = 2 ÷ 0.05 = 40 N/m.

Example 3: from the lengths

A spring is 12 cm long. When a 7 N force is applied it stretches to 15.5 cm. Find k.

Extension = 15.5 − 12 = 3.5 cm = 0.035 m, so k = 7 ÷ 0.035 = 200 N/m.

Example 4: finding the extension from a mass

A spring with k = 80 N/m holds a 0.4 kg mass. How far does it stretch? Use g = 9.8 N/kg.

The force is the weight: W = mg = 0.4 × 9.8 = 3.92 N.

The spring stretches by about 4.9 cm.

Example 5: finding the force

A spring with k = 250 N/m is stretched by 0.12 m. What force is acting? F = ke = 250 × 0.12 = 30 N.

Finding the spring constant from a force-extension graph

Plot force on the y-axis against extension on the x-axis. While the spring obeys Hooke's law, the graph is a straight line through the origin, and its gradient is k.

Here x is the extension in metres and y is the force in newtons. The straight part has a gradient of 5 N ÷ 0.1 m = 50 N/m, which is the spring constant. Beyond the limit of proportionality the line curves: the extension is no longer proportional to the force, and F = ke no longer applies. Beyond the elastic limit, the spring will not return to its original length when the force is removed.

In the GCSE required practical on force and extension, you hang masses on a spring, measure each extension and plot this graph. The Practical Data Lab plots your results and finds the gradient. Our guide to the elastic potential energy formula sets out the method step by step and shows how the area under the line gives the energy stored.

Here is a quick check.

Springs in series and in parallel

Joining springs changes how stiff the combination is. For two identical springs, each with spring constant k:

• In series, one below the other, the same force stretches both, so the total extension doubles. The combination has spring constant k ÷ 2.
• In parallel, side by side, the force is shared between them, so each stretches half as much. The combination has spring constant 2k.

Two 200 N/m springs in series act like one 100 N/m spring; in parallel, like one 400 N/m spring.

Common mistakes

• Using the total length instead of the extension. Subtract the original length first.
• Extension in cm or mm. Convert to metres before calculating k in N/m.
• Using the mass instead of the weight. A hanging mass in kg must be turned into a force with W = mg.
• Using F = ke past the limit of proportionality. Only the straight part of the graph obeys Hooke's law.
• Reading the graph the wrong way round. If extension is on the y-axis, the gradient is 1/k, not k.

Is the spring constant equation on the equation sheet?

Yes. AQA's GCSE Physics equations sheet, checked in November 2026, gives force = spring constant × extension, F = ke, next to the elastic potential energy equation, Ee = ½ke². AQA's A-level Physics data booklet gives Hooke's law as F = kΔL in its materials section. Other boards publish their own sheets, so check yours.

Practise forces in a short timed paper

GCSE Physics Foundation and GCSE Physics Higher each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, including forces and elasticity.

Read next

Physics · MechanicsHooke's law calculator: F = kx, spring constant, energyFind the force, spring constant or extension with F = kx, the elastic energy stored, and the combined spring constant of springs in series or parallel.