The Young modulus

AQA·Physics·3.4.2.2

Young modulus

When a material is pulled, it extends. The Young modulus describes how much tensile stress is needed to produce a given tensile strain, provided the material is in its linear elastic region.

In this region, stress is proportional to strain, and the material returns to its original length when the force is removed.

Tensile stress and strain

Tensile stress is the pulling force per unit original cross-sectional area. Tensile strain is the extension per unit original length:

σ=FA,ε=ΔLL\sigma = \frac{F}{A}, \qquad \varepsilon = \frac{\Delta L}{L}

Here, $$F$$ is the tensile force, $$A$$ is the original cross-sectional area, $$L$$ is the original length and $$\Delta L$$ is the extension. Stress has the SI unit pascal ($$\mathrm{Pa}$$); strain has no unit because it is a ratio of two lengths.

Calculating Young modulus

The Young modulus $$E$$ is the ratio of tensile stress to tensile strain in the linear elastic region:

E=σε=FLAΔLE = \frac{\sigma}{\varepsilon} = \frac{FL}{A\Delta L}

Because strain has no unit, the SI unit of Young modulus is also the pascal ($$\mathrm{Pa}$$).

On a tensile stress–strain graph, with stress on the vertical axis and strain on the horizontal axis, the Young modulus is the gradient of the straight-line region. A steeper gradient means a larger Young modulus: a greater stress is needed to produce the same strain.

Measurement of the Young modulus

A wire stretches when a tensile force pulls on it. To measure its Young modulus, apply known forces and measure the extension produced by each one. The measurements are used to find stress and strain; the relationship between them gives the Young modulus.

Measuring the wire

Secure a long, straight wire and measure its original length, $$L$$. Measure its diameter, $$d$$, using a micrometer. Calculate its original cross-sectional area, $$A$$:

A=πd24A = \frac{\pi d^2}{4}

Measure $$L$$ and $$d$$ in metres so that $$A$$ is in square metres.

Measuring force and extension

Place a fixed reference scale beside the wire and record the position of a marker attached to the wire before adding the loads. Add known masses and record the marker position for each load. The change from the original reading is the extension, $$\Delta L$$.

For each load, calculate the tensile force from the total supported mass: $$F = mg$$, where $$m$$ is the supported mass and $$g$$ is the gravitational field strength. Keep the loads within the elastic region, so the wire returns to its original length when unloaded.

Calculating the Young modulus

For each load, use the original area and length to calculate stress and strain:

stress=FAstrain=ΔLL\text{stress} = \frac{F}{A} \qquad \text{strain} = \frac{\Delta L}{L}

Stress is measured in pascals ($$\mathrm{Pa}$$). Strain has no unit because it is a ratio of two lengths.

Plot stress on the vertical axis against strain on the horizontal axis. The gradient of the straight-line region is the Young modulus, $$E$$:

E=change in stresschange in strainE = \frac{\text{change in stress}}{\text{change in strain}}

Calculate the gradient using two points on the straight line. Since strain has no unit, the Young modulus has the same unit as stress: the pascal.