Hooke's Law states that the extension of a spring is directly proportional to the force applied to it, provided the elastic limit is not exceeded. Double the force and you double the extension — a beautifully predictable relationship that Robert Hooke first published in 1678 and that underpins everything from bathroom scales to car suspension.
What is Hooke's Law and how is it written?
Hooke's Law states:
Force (N) = Spring constant (N/m) × Extension (m)
In symbols: F = k × e
Where:
- F = force applied (in newtons, N)
- k = spring constant (in newtons per metre, N/m) — a measure of how stiff the spring is
- e = extension (in metres, m) — how much the spring has stretched beyond its natural (unstretched) length
The spring constant k is different for every spring. A stiff spring (large k) stretches only a little for a large force; a soft spring (small k) stretches a lot for the same force.
Extension is not the same as total length. If a spring has a natural length of 10 cm and stretches to 16 cm, the extension is 6 cm (0.06 m).
How do you calculate using Hooke's Law?
Worked example 1: A force of 6 N is applied to a spring with a spring constant of 300 N/m. What is the extension?
e = F ÷ k = 6 ÷ 300 = 0.02 m (2 cm)
Worked example 2: A spring extends by 5 cm (0.05 m) when a force of 10 N is applied. What is the spring constant?
k = F ÷ e = 10 ÷ 0.05 = 200 N/m
Worked example 3: A spring has k = 150 N/m. What force is needed to produce an extension of 8 cm?
F = k × e = 150 × 0.08 = 12 N
A key exam tip: always convert cm to m before substituting into the equation.
What is the elastic limit?
Hooke's Law only holds up to the elastic limit (also called the limit of proportionality at GCSE level). Beyond this point:
- Extension is no longer proportional to force.
- The spring does not return to its original length when the force is removed — it is permanently deformed (stretched beyond its elastic range).
This behaviour is called plastic deformation — the spring has been permanently stretched. Before the elastic limit, the deformation is elastic — the spring returns to its original shape when the force is removed.
The distinction between elastic and plastic deformation matters in engineering: a spring used in a mechanism must always operate below its elastic limit, otherwise it will not function correctly after the first overload.
How do you interpret a force-extension graph?
| Region of graph | What it shows |
|---|---|
| Straight line through the origin | Hooke's Law obeyed — extension is proportional to force |
| Curve beginning to deviate from straight | Elastic limit being approached |
| Steep curve with no return to origin | Plastic deformation — permanent stretching |
From a force-extension graph you can find the spring constant k by calculating the gradient of the straight-line section:
k = gradient = ΔF ÷ Δe (change in force divided by change in extension)
A steeper gradient means a stiffer spring (higher k).
You can also find the elastic potential energy stored in the spring from the area under the straight-line section of the graph:
Elastic potential energy (J) = ½ × k × e²
This is a GCSE extension point — at KS3 you focus on reading and drawing the graph rather than the area calculation.
How is Hooke's Law used in everyday devices?
| Device | How Hooke's Law applies |
|---|---|
| Newton meter (force meter) | Graduated scale calibrated using known weights; extension proportional to force lets you read the force directly |
| Bathroom scales (spring type) | Downward force compresses a spring; scale shows force or mass |
| Car suspension springs | Absorb shock from road bumps; designed so normal driving loads stay within the elastic limit |
| Diving board | Bends proportionally to load before springing back — elastic deformation |
| Mattress springs | Compress proportionally to a person's weight across many coil springs |
| Guitar strings | Tension in the string obeys a spring-like relationship that determines pitch |
What is the difference between elastic and plastic deformation?
- Elastic deformation: The material (spring, rubber band, bone) returns to its original shape after the force is removed. Energy is stored as elastic potential energy and released when the force is removed.
- Plastic deformation: The material does not return to its original shape — it has been permanently changed. Energy has gone into rearranging the internal structure of the material. This is irreversible.
Most engineering materials are designed to operate well within their elastic limit in normal use, with the plastic region acting as a safety margin before catastrophic failure.
Frequently asked questions
What does Hooke's Law state?
Hooke's Law states that the extension of a spring (or other elastic material) is directly proportional to the applied force, provided the elastic limit is not exceeded. This means if you double the force, the extension doubles; if you triple the force, the extension triples — a straight-line relationship through the origin on a force-extension graph.
What is the spring constant?
The spring constant (symbol k, unit N/m) is a measure of how stiff a spring is. A high spring constant means the spring is stiff and needs a large force for a small extension. A low spring constant means the spring is soft and extends a lot with a small force. It is the gradient of the straight-line section of a force-extension graph.
What happens when you go past the elastic limit?
Beyond the elastic limit, the spring no longer obeys Hooke's Law and the extension is no longer proportional to the force. More importantly, the spring is permanently deformed — it will not return to its original length even when all the force is removed. This is called plastic deformation and means the spring is damaged for its intended purpose.
How do you find the spring constant from a graph?
Draw a force-extension graph and identify the straight-line section that passes through the origin. Calculate the gradient of that straight line by choosing two points on the line and dividing the change in force by the change in extension: k = ΔF ÷ Δe. The steeper the line, the stiffer the spring and the higher the spring constant.
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