Hooke’s Law

When you hang weights on a spring, it gets longer. Hooke’s Law explains how the stretching (extension) links to the pulling force, as long as the spring is not stretched too much.

Key idea

Hooke’s Law: the extension of a spring is directly proportional to the force applied.

$$F = kx$$

• F is the force (load) in newtons (N)
• x is the extension in metres (m) or centimetres (cm)
• k is the spring constant (stiffness) in N/m or N/cm

A large k means a stiff spring (it hardly stretches). A small k means an easy-to-stretch spring.

Limit of proportionality

This is the point where the straight-line relationship ends. Before this point, doubling the force doubles the extension. After this point, the graph curves and $F$ is no longer proportional to $x$. You should be able to identify this point on a load–extension graph.

Load–extension graphs

Plot extension on the horizontal axis (x) and force on the vertical axis (F). For a good spring, the graph is a straight line through the origin up to the limit of proportionality.

• Straight line region: Hooke’s Law holds
• Gradient (slope) equals the spring constant $k$
• Curved region: not proportional

$$k = \frac{\Delta F}{\Delta x}$$ Choose two points on the straight line to find $k$.

Simple experiment (procedure)

1. Measure the spring’s original length $L_0$.
2. Add a known load (use weights in newtons). Measure new length $L$.
3. Calculate extension: $x = L - L_0$.
4. Repeat with equal extra loads. Plot F against x. Draw a best-fit straight line through the origin for the proportional region.