Stress, strain and the Young modulus

Hooke's law

Up to the limit of proportionality, extension is proportional to force: F = kΔx, where k is the stiffness (N/m).

Stress and strain

• Stress σ = F / A (Pa), where A is the cross-sectional area
• Strain ε = Δx / x (no units), extension divided by original length

Young modulus

E = stress / strain = (F x) / (A Δx), measured in Pa. On a stress-strain graph, E is the gradient of the straight-line section.

Worked example

A wire of area 1.0 x 10^-6 m² carries 50 N. Find the stress.

1. σ = F / A = 50 / (1.0 x 10^-6)

Answer: σ = 5.0 x 10⁷ Pa

Check yourself

1. A spring of stiffness 200 N/m stretches 0.05 m. Find the force.
   Show answer

   10 N. F = kΔx = 200 x 0.05 = 10 N.

2. A 2.0 m wire extends by 1.0 mm. Find the strain.
   Show answer

   0.0005. ε = 0.0010 / 2.0 = 5.0 x 10^-4.

3. Stress = 2.0 x 10⁸ Pa and strain = 1.0 x 10^-3. Find the Young modulus in GPa.
   Show answer

   200 GPa. E = 2.0 x 10⁸ / 1.0 x 10^-3 = 2.0 x 10¹¹ Pa = 200 GPa.

4. What is the unit of strain?
   • Pa
   • It has no unit
   • N/m
   • m
   Show answer

   It has no unit. Strain is a length divided by a length.

5. On a stress-strain graph, the Young modulus is
   • the area under the graph
   • the maximum stress
   • the strain at breaking
   • the gradient of the straight-line section
   Show answer

   the gradient of the straight-line section. E = stress / strain in the linear region.

Practise with instant feedback

Sources

• Pearson Information Manual 2026/2027: International A level

Last updated 7 November 2026. Spotted a mistake? Email alevel@lifereadysociety.com.