Hooke’s Law — Cambridge IGCSE Co-ordinated Sciences 0654 - Physics Revision Notes & Topical Questions

For a spring (or wire) stretched within its limit of proportionality, the extension is directly proportional to the force applied:

F = k x

F is the force in newtons (N), x is the extension in metres (m) and k is the spring constant in N/m. A larger k means a stiffer spring. The law applies equally to compression, where x is the decrease in length.

Hooke's law is a statement about a material, not a universal law: it describes the behaviour of springs and many metal wires only while the stretching is small.

With force on the y-axis and extension on the x-axis:

For many springs these two limits are very close together.

The spring constant depends on the material and construction of the spring: thicker wire, fewer coils and a smaller coil radius all make a spring stiffer.

In elastic deformation the object returns to its original shape when the force is removed. In plastic deformation some extension remains. A spring overloaded beyond its elastic limit becomes longer and weaker.

Metal wires obey Hooke's law at small loads and then show a yield point and plastic flow. Materials used for car crumple zones are chosen to deform plastically and absorb energy, while materials for trampoline springs are chosen to stay elastic. Whether an object is elastic or plastic depends on how far it is stretched.

Work is done when a spring is stretched, and this is stored as elastic potential energy. It equals the area under the force–extension graph. For a spring within the limit of proportionality the area is a triangle:

E = ½ F x = ½ k x²

E is in joules (J). If the extension is doubled, the stored energy increases by four times.

When a spring is stretched, the work done by the force is the average force times the extension.

A rubber band does not obey Hooke's law: its force–extension graph is an S-shaped curve, and it takes a different path on unloading (hysteresis). A copper wire behaves like a spring at small loads but shows plastic stretching earlier.

To find the spring constant from experimental data, choose two points far apart on the straight part of the graph and use k = ΔF/Δx, which also removes the effect of any zero error in the measurement of the original length. Quote the result to a sensible number of significant figures, and give the unit, N/m.

Two identical springs side by side share the load, so the system is twice as stiff. Two identical springs end to end each carry the full force, so the total extension doubles and the system is half as stiff. These ideas appear in suspensions, trampolines and weighing scales.