The Young modulusAQA A-Level Physics: Revision notes

A material under tension is described using two quantities that do not depend on the size of the sample.

Tensile stress is the force per unit cross-sectional area: $\sigma = \frac{F}{A}$, measured in pascals (Pa, or N m⁻²).

Tensile strain is the extension per unit original length: $\varepsilon = \frac{\Delta L}{L}$. It is a ratio, so it has no unit.

For a wire of diameter $d$, the area is $A = \frac{\pi d^2}{4}$. Convert mm to m before calculating.

Within the limit of proportionality, stress is proportional to strain. The constant of proportionality is the Young modulus:

$E = \frac{\text{tensile stress}}{\text{tensile strain}} = \frac{F L}{A \Delta L}$

It is measured in pascals (Pa). A material with a high Young modulus, such as steel, is stiff: it needs a large stress to produce a given strain. Because stress and strain remove the effect of size, the Young modulus is a property of the material and not of a particular sample.

A copper wire of length 2.0 m and diameter 0.50 mm carries a tension of 30 N. $E = 1.2 \times 10^{11}$ Pa.

Area: $A = \pi \times (0.25 \times 10^{-3})^2 = 1.96 \times 10^{-7}$ m²

Stress: $\sigma = 30 / 1.96 \times 10^{-7} = 1.53 \times 10^{8}$ Pa

Strain: $\varepsilon = 1.53 \times 10^{8} / 1.2 \times 10^{11} = 1.27 \times 10^{-3}$

Extension: $\Delta L = 1.27 \times 10^{-3} \times 2.0 = 2.5 \times 10^{-3}$ m

A graph of stress (vertical axis) against strain (horizontal axis) for a metal is a straight line through the origin at low stress. Its gradient is the Young modulus.

To find $E$ from a graph, draw a large triangle on the straight section and calculate change in stress ÷ change in strain. Do not use points beyond the straight line, because stress is no longer proportional to strain there. A steeper line means a larger Young modulus, so a stiffer material.

If the graph is plotted as force against extension, the gradient is $F / \Delta L$, so $E = \text{gradient} \times \frac{L}{A}$.

Method: clamp a long thin wire (about 2 m) at one end and pass it over a pulley, adding masses to a hanger at the other end. Use a fiducial marker on the wire and a metre rule to measure its extension.

1. Measure the unstretched length $L$ from the clamp to the marker with a metre rule
2. Measure the diameter with a micrometer at several points along the wire and in two perpendicular directions, then take the mean
3. Add masses in equal steps, recording the extension each time, and also record while unloading
4. Plot force against extension, find the gradient, and calculate $E = \text{gradient} \times L / A$

Errors: the diameter is the largest source of percentage uncertainty because it is small and is squared. Use a long wire to make the extension large, and keep the loads small enough to stay within the limit of proportionality. Wear safety goggles in case the wire snaps, and place a box of sand or cloth below to catch the masses.