Hooke's Law

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

The equation

$F = kx$

Symbol	Quantity	Unit
$F$	Force applied	N
$k$	Spring constant	N/m
$x$	Extension	m

The spring constant $k$ tells you how stiff the spring is. A larger $k$ means a stiffer spring that extends less for the same force.

Limit of proportionality

Beyond this limit the spring no longer obeys Hooke’s law. The force-extension graph curves, and the spring may not return to its original length when unloaded (it has been permanently deformed).

On a force-extension graph:

• The straight-line region shows proportional behaviour (Hooke’s law applies).
• The point where the line begins to curve is the limit of proportionality.
• The gradient of the straight section equals the spring constant $k$.

Worked example

A spring has a spring constant of 40 N/m. Calculate the extension when a force of 6.0 N is applied.

$x = \frac{F}{k} = \frac{6.0}{40} = 0.15\text{ m}$

The extension is 0.15 m (15 cm).

Common errors and how to correct them

Using total length instead of extension. Extension is the change in length: $x = \text{stretched length} - \text{natural length}$. If the natural length is 20 cm and the stretched length is 35 cm, the extension is 15 cm, not 35 cm.

Forgetting to convert cm to m before substituting. The spring constant is in N/m, so the extension must be in metres for the calculation to work.

How to apply this in an exam

Read force-extension graphs carefully. If asked for the spring constant, calculate the gradient of the straight section. If the graph curves, state that Hooke’s law no longer applies beyond the limit of proportionality.