Young's modulus and the stretching of wires

Young's modulus

Within the proportional limit, stress over strain is a constant for each material. For stretching or squeezing along a line it is Young's modulus:

Strain has no unit, so $Y$ has the unit of stress, the pascal. Some values:

A large $Y$ means a stiff material: it needs a big stress for a small strain. In physics, steel is more elastic than rubber.

How far a wire stretches

Turn the formula round:

A wire stretches more if it is long, thin, or made of a soft material.

You should find that doubling the length doubles the stretch, but doubling the diameter cuts it to a quarter.

Watch out. $Y$ belongs to the material, not to the wire. A thick and a thin steel wire have the same $Y$; the thin one just stretches more.

Measuring Y

In Searle's apparatus, two long wires hang side by side from one support. One carries a fixed weight and a scale; the other carries the loads and a spirit level. As loads are added, a screw gauge measures the tiny stretch. Using a second wire cancels out any sag of the support and any change of temperature. A graph of load against stretch is a straight line of slope $\dfrac{AY}{L}$.

A worked example

An aluminium wire $2.2\ \text{m}$ long and $1.0\ \text{mm}$ across holds a $5.0\ \text{kg}$ load. How far does it stretch? ($Y = 7.0 \times 10^{10}\ \text{Pa}$.)

1. Area. $A = \pi(0.50 \times 10^{-3}\ \text{m})^2 = 7.85 \times 10^{-7}\ \text{m}^2$.
2. Stretch. $\Delta L = \dfrac{49\ \text{N}}{7.85 \times 10^{-7}\ \text{m}^2} \times \dfrac{2.2\ \text{m}}{7.0 \times 10^{10}\ \text{Pa}} = 1.96 \times 10^{-3}\ \text{m}$.
3. Answer. About $1.96\ \text{mm}$.

Think first. A steel wire of the same size would stretch only $\dfrac{70\ \text{GPa}}{200\ \text{GPa}}$ as much, about $0.69\ \text{mm}$.

Practise with new numbers

Try it in Python

The program works out the aluminium wire of the example.

Remember these

• $Y = \dfrac{FL}{A\,\Delta L}$, in pascal; a property of the material.
• $\Delta L = \dfrac{FL}{AY}$: more for long, thin, soft wires.
• A large $Y$ means stiff ('more elastic').

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A wire stretches $0.6\ \text{mm}$. A wire of the same metal, the same length, and half the diameter carries the same load. How far does it stretch?

2. Which stretches more under the same load: a steel or a brass wire of the same size? By how many times?

3. Why does Searle's apparatus use two wires?

4. Does a thicker steel wire have a larger Young's modulus than a thin one?