Stretch a spring and it pulls back; stretch it twice as far and it pulls back twice as hard. That simple, proportional relationship is Hooke’s law, and it holds for any elastic material — springs, wires, bungee ropes — right up until you stretch it too far. It gives you a clean equation, a straight-line graph, and one of the most reliable calculations in the whole forces topic.
📘 What you need to know
Hooke’s law: the extension of a material is directly proportional to the applied force, up to the limit of proportionality
As an equation: FH = −kx, where k is the spring constant and x the extension
The spring constantk (in N m−1) measures the stiffness — a bigger k means a stiffer material
On a force–extension graph, the Hooke’s law region is a straight line through the origin
The gradient of that straight-line region equals the spring constant k
The law and its equation
When you pull on a spring, it stretches. Hooke’s law tells you exactly how much:
Hooke’s law
The extension of a material is directly proportional to the applied force, up to the limit of proportionality
Written as an equation:
Hooke’s law equationFH = −kx
Here FH is the elastic restoring force in newtons (N), k is the spring constant in N m−1, and x is the extension in metres (m). The minus sign is there because the spring’s restoring force points opposite to the stretch — pull it down and it pulls back up. For most calculations you just need the magnitudes, so F = kx is what you’ll usually work with.
The spring constant k is a property of the material and measures its stiffness. A large k means a stiff spring that barely stretches; a small k means a floppy one that stretches easily under the same load.
Extension, not length
The x in the equation is the extension — how much longer the material has become, not its total length. You find it by subtracting:
Hooke’s law works for compression too: squash a spring and the extension is negative, but the same proportional relationship holds. Hang a load on a spring and it stretches; the heavier the load, the greater the extension — in direct proportion, as long as you don’t overstretch it.
Hanging a load stretches the spring. The extra length below the natural-length level is the extension, x — and it grows in proportion to the load, up to the limit of proportionality.
Force–extension graphs
Plot the applied force against the extension and you get a force–extension graph — the clearest way to see Hooke’s law at work. While the material obeys the law, the graph is a straight line through the origin: force and extension are directly proportional. Push past the limit of proportionality and the line starts to curve, because the material no longer stretches in proportion.
In the Hooke’s law region the graph is a straight line through the origin, and its gradient is the spring constant k. Beyond the limit of proportionality the line curves and Hooke’s law no longer holds.
Watch the axes on these graphs. If force is on the y-axis and extension on the x-axis, the gradient is k. But exam questions love to swap them — putting extension (or length) on the y-axis and load on the x-axis. Then the gradient is 1/k, not k. Always check which quantity is where before you read a gradient as the spring constant.
WE 1
A spring has a spring constant of 25 N m−1. It is stretched by an extension of 0.08 m. Calculate the force applied to the spring.
Step 1 — use Hooke’s law (magnitudes)
F = kx
Step 2 — substitute the valuesF = 25 × 0.08F = 2.0 NKeep the extension in metres, not centimetres — 0.08 m, not 8. That’s the most common slip in these questions.
WE 2
A spring is stretched with increasing load and the force–extension graph is a straight line through the origin. It passes through the point where an extension of 0.05 m corresponds to a force of 4.0 N. Determine the spring constant.
Step 1 — the spring constant is the gradient
k = ΔF ÷ Δx
Step 2 — use the origin and the given pointk = 4.0 ÷ 0.05k = 80 N m⁻¹Because the line passes through the origin, you can use any point on it with (0,0) to find the gradient.
🛠️ Reading k off a graph
Check the axes. Force on y and extension on x means the gradient is k.
Pick two clear points on the straight-line region, far apart for accuracy.
Find the gradient:k = ΔF ÷ Δx.
If the axes are swapped (extension on y), the gradient is 1/k — take the reciprocal.
💡 Top tips
Extension, not total length. Subtract the natural length first: x = stretched − natural.
Convert to metres. A spring constant in N m−1 needs the extension in metres — watch for cm and mm.
Check the axes before reading a gradient — swapped axes give 1/k, not k.
Hooke’s law has a limit. Past the limit of proportionality the line curves and F = kx no longer applies.
Quick recap: Hooke’s law says extension is proportional to applied force, F = kx, up to the limit of proportionality. The spring constant k measures stiffness. Extension is stretched length minus natural length. On a force–extension graph the Hooke region is a straight line through the origin, and its gradient is k — unless the axes are swapped, when it’s 1/k.
⚠ Common mistakes
Using the total length instead of the extension — always subtract the natural length
Leaving the extension in cm or mm when k is in N m−1
Reading the gradient as k when the axes are swapped — then it’s 1/k
Applying F = kxbeyond the limit of proportionality, where the graph curves
Forgetting the spring constant has units of N m−1
Hooke’s law is a quick detour into stretchy solids. Now we return to resistive forces in fluids. When a small sphere moves slowly through a liquid or gas, the drag it feels follows a neat equation of its own — that’s next: Stokes’ Law, and the viscous drag on a sphere.
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