Hooke's Law

Hooke's law — the extension of a spring is directly proportional to the force applied, provided the limit of proportionality is not exceeded.

F = kx

k, the force constant — the force needed per unit extension, measured in N m⁻¹. A large k means a stiff spring. Note that x is the extension, the increase in length, not the total length of the spring.

Limit of proportionality — the point beyond which F is no longer proportional to x, so the force–extension graph stops being a straight line. Beyond it Hooke's law simply does not apply, and using F = kx there gives wrong answers.

Example: a spring of force constant 25 N m⁻¹ extended by 0.12 m needs F = 25 × 0.12 = 3.0 N. Doubling the force to 6.0 N gives 0.24 m of extension — but only while the spring remains within its limit of proportionality.

Force–extension graphs — with force on the y-axis and extension on the x-axis, the gradient is the force constant k. Plotted the other way round the gradient is 1/k, so always read the meaning off the axes rather than from memory.

Elastic potential energy — the work done stretching the spring is stored and can be recovered. It is the area under the force–extension graph, which for a straight line is a triangle:

E = ½Fx = ½kx²

Why the one half is there — the force is not constant during the stretch. It starts at zero and rises to F, so the average force is ½F, and the work done is average force × distance.

Springs in combination — two identical springs side by side (in parallel) share the load, so each takes half the force and the combination is twice as stiff. The same two end to end (in series) each carry the full load, so each extends fully and the combination is half as stiff — it extends twice as far for the same force.