Stretch a spring and it pulls back — the harder you pull, the more it stretches, in direct proportion. That’s Hooke’s law, and it holds up to a point called the limit of proportionality. The constant that links force to extension is the spring constant, k.
📘 What you need to know
Hooke’s law: the extension of a spring is directly proportional to the force applied, up to the limit of proportionality.
Equation: FH = −kx (the restoring force opposes the stretch).
k = spring constant (N m⁻¹), x = extension (m); the bigger k, the stiffer the spring.
Extension x = stretched length − natural (unstretched) length.
On a force–extension graph, the straight region through the origin obeys Hooke’s law, and its gradient is k.
What Hooke’s law says
When you apply a force to a spring (or any elastic material, like a wire or a bungee cord), it stretches. Provided you don’t pull too hard, the extension is proportional to the force:
Hooke’s lawFH = −kx
Here FH is the elastic restoring force the spring exerts. The minus sign means it always acts opposite to the stretch — pull the spring down and it pulls back up. For the size of the force you just use F = kx.
The extension is how much longer the spring has become:
Extensionx = stretched length − natural length
The same idea works for compression (squashing it shorter), and the spring constantk measures the stiffness: a large k means a stiff spring that barely stretches.
The load stretches the spring; the extra length beyond its natural length is the extension x.
Force–extension graphs
Plotting the force against the extension shows how a material behaves. While Hooke’s law holds, the graph is a straight line through the origin, and its gradient is the spring constantk. Beyond the limit of proportionality, the line starts to curve and Hooke’s law no longer applies.
The straight part through the origin obeys Hooke’s law; its gradient (ΔF/Δx) is the spring constant. Past the limit of proportionality the line curves.
Quick reference:F = kx • k = F/x = gradient of a force–extension graph • x = stretched − natural length • big k = stiff spring.
Worked examples
WE 1
Finding the spring constant
A spring stretches by 0.040 m when a 2.0 N load is hung from it. Calculate its spring constant.
Use F = kx, so k = F / xk = 2.0 ÷ 0.040k = 50 N m⁻¹
WE 2
When the axes are swapped
A spring is loaded and its length is plotted on the y-axis against load on the x-axis. The straight line rises by 0.045 m of extension for every 0.36 N of load. Find the spring constant.
Length on y, load on x → gradient = Δx / ΔFgradient = 0.045 ÷ 0.36 = 0.125 m N⁻¹Here the spring constant is 1 ÷ gradientk = 1 ÷ 0.125k = 8.0 N m⁻¹
💡 Top tips
Check the axes before reading a gradient. Force on y and extension on x → gradient is k. Swapped → gradient is 1/k.
Use the extension, not the length. Subtract the natural length first.
Stay in the straight part. Hooke’s law only applies up to the limit of proportionality.
Big gradient = stiff spring. A steeper force–extension line means a larger k.
The minus sign is about direction — the restoring force opposes the stretch; for magnitudes use F = kx.
⚠ Common mistakes
Putting the total length into x instead of the extension (stretched − natural).
Reading the gradient as k when the axes are swapped — then it’s 1/k.
Applying Hooke’s law beyond the limit of proportionality, where the line is no longer straight.
Mixing units — convert cm to m and keep force in N so k comes out in N m⁻¹.
WE 2 is the classic exam trap: the data look like a normal straight line, but with length on the vertical axis the gradient gives you 1/k, not k. Always read the axes first. Up next: Stokes’ Law, the drag force on a small sphere moving through a fluid.
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