Hooke’s Law and Springs: Formula and Calculator
Hooke’s law explained with a calculator: F = kx, the spring constant, elastic limit, elastic energy, springs in series and parallel, and the classic experiment.
Key takeaways
- Hooke’s law: the force on a spring is proportional to its extension, F = k × x, up to the limit of proportionality.
- The spring constant k (N/m) measures stiffness; a stiffer spring has a bigger k.
- A stretched spring stores elastic potential energy: E = ½ × k × x².
On this page
Hooke’s law says that the force needed to stretch or compress a spring is proportional to how far it’s stretched or compressed: F = k × x. F is the force in newtons, x is the extension (or compression) in metres, and k is the spring constant in newtons per metre (N/m), which describes how stiff the spring is. The law holds only up to a point, the limit of proportionality. Beyond it, the spring stretches more than the law predicts, and past the elastic limit it won’t return to its original shape.
The formula
- F = k × x
- k = F ÷ x
- x = F ÷ k
- Elastic potential energy: E = ½ × k × x²
Extension means the change in length, not the total length: x = stretched length − original length.
Hooke’s law calculator
Enter any two of force, spring constant and extension. For a hanging mass, force = mass in kilograms × 9.81.
Hooke’s law calculator
Enter any two values.
Worked examples
- Find k: a 2 N force stretches a spring by 4 cm (0.04 m). k = 2 ÷ 0.04 = 50 N/m.
- Find extension: the same spring with 5 N: x = 5 ÷ 50 = 0.1 m (10 cm).
- Hanging mass: a 300 g mass hangs from a spring with k = 60 N/m. F = 0.3 × 9.81 ≈ 2.94 N, so x = 2.94 ÷ 60 ≈ 0.049 m (4.9 cm).
- Energy stored: stretching the 50 N/m spring by 0.1 m stores E = ½ × 50 × 0.1² = 0.25 J.
Limit of proportionality and elastic limit
| Region | What happens |
|---|---|
| Up to the limit of proportionality | Extension is proportional to force; the force–extension graph is a straight line through the origin |
| Between the limit of proportionality and the elastic limit | The graph curves, but the spring still returns to its original length |
| Beyond the elastic limit | The spring is permanently deformed (plastic deformation) |
On a force–extension graph, the gradient of the straight section is the spring constant k. On an extension–force graph, the gradient is 1 ÷ k, so check which way round your axes are.
The classic experiment
- Hang a spring from a clamp stand with a metre rule alongside, and note the pointer position for the unloaded length.
- Add masses one at a time (for example 100 g, weighing about 0.98 N each).
- Record the new length each time and calculate the extension.
- Plot force against extension and draw a line of best fit.
- Calculate k from the gradient of the straight section.
Safety: wear eye protection in case the spring snaps, and keep feet clear of falling masses. Accuracy: read the rule at eye level, wait for the spring to stop bouncing, and remove masses at the end to check the spring returns to its original length.
Springs in series and parallel
| Arrangement | Combined spring constant | Effect |
|---|---|---|
| Parallel (side by side) | k = k₁ + k₂ | Stiffer; each spring stretches less |
| Series (end to end) | 1/k = 1/k₁ + 1/k₂ | Softer; the total stretch is larger |
Two identical springs in parallel have twice the spring constant; in series, half. These formulas mirror the rules for resistors, but the other way round; compare with our guide to series and parallel circuits.
Where Hooke’s law is used
- Newton meters (spring balances), which measure force from a spring’s extension
- Car suspension, where springs absorb bumps
- Weighing scales, mechanical and some digital designs
- Mattresses, trampolines and pens
- Engineering design, where materials such as steel behave elastically under normal loads
Rubber bands and bungee cords don’t follow Hooke’s law closely; their force–extension graphs curve.
Engineers also use the idea in reverse: strain gauges bonded to beams and bridges measure tiny extensions, from which forces and stresses can be calculated. The same proportional behaviour underlies Young’s modulus, which describes the stiffness of a material rather than of one particular spring.
Energy in springs
The energy stored in a stretched or compressed spring is the area under the force–extension graph, which gives E = ½kx² for the straight-line region. When the spring is released, this energy can become kinetic energy, as in a toy launcher. See kinetic vs potential energy and work, energy and power. A spring bouncing with a mass on the end also shows how weight, set by gravity, balances the spring force at equilibrium.
Common mistakes
- Using total length instead of extension.
- Forgetting to convert centimetres to metres. 4 cm is 0.04 m.
- Using mass instead of force. Convert kilograms to newtons with W = m × g.
- Applying the law beyond the limit of proportionality.
- Reading the length while the spring is still bouncing. Wait until it settles.
Frequently asked questions
What does the spring constant tell you?
How stiff a spring is: the force needed to stretch it by one metre. A bigger k means a stiffer spring.
Does Hooke’s law apply to compression?
Yes. It works for compressing a spring as well as stretching it, within the limit of proportionality.
What is the unit of the spring constant?
Newtons per metre (N/m).
Sources
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