Wires in series, wires in parallel and thermal stress

Wires end to end

Hang one wire from another, and a load from the bottom. Each wire holds the whole load, so the force in each is the same, and the stretches add:

The thinner or softer wire carries a larger stress and is the one likely to break.

Wires side by side

Hang a rigid bar from two wires and put the load on the bar so that it stays level. Both wires now have the same stretch, and the forces add: $F = F_1 + F_2$. Each takes a share in proportion to its stiffness $\dfrac{YA}{L}$, so a steel wire carries more than a brass one of the same size.

Thermal stress

A rod heated by $\Delta T$ would grow by $\alpha L\,\Delta T$. If clamps stop it, they squeeze it back by the same strain, $\alpha\,\Delta T$. That needs a stress

and a force $Y\alpha A\,\Delta T$ on each clamp. Notice that the length does not matter.

You should find that aluminium, though softer, pushes nearly as hard as steel, because it tries to expand about twice as much.

Watch out. The stress does not depend on the length of the rod, but the force on the clamps does depend on its area.

A worked example

A steel wire $1.5\ \text{m}$ long hangs from a hook, and a copper wire $1.0\ \text{m}$ long hangs from it. Each has a cross-section of $1.0\ \text{mm}^2$. A $120\ \text{N}$ load hangs at the bottom. How far does it drop?

1. Steel. $\Delta L = \dfrac{120\ \text{N}}{1.0 \times 10^{-6}\ \text{m}^2} \times \dfrac{1.5\ \text{m}}{2.0 \times 10^{11}\ \text{Pa}} = 0.90\ \text{mm}$.
2. Copper. $\Delta L = \dfrac{120\ \text{N}}{1.0 \times 10^{-6}\ \text{m}^2} \times \dfrac{1.0\ \text{m}}{1.1 \times 10^{11}\ \text{Pa}} = 1.09\ \text{mm}$.
3. Total. $0.90\ \text{mm} + 1.09\ \text{mm} = 1.99\ \text{mm}$.

Think first. The shorter copper wire stretches more, because copper is softer.

Practise with new numbers

Try it in Python

The program works out the two wires of the example.

Remember these

• In series: same force, stretches add.
• In parallel: same stretch, forces add, shared in proportion to $\dfrac{YA}{L}$.
• Thermal stress $= Y\alpha\,\Delta T$, whatever the length.

Check yourself

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

1. Two identical wires hang side by side and share a $90\ \text{N}$ load. What force does each carry?

2. A thin and a thick wire of the same metal are joined end to end with a load below. Which is more likely to break?

3. A clamped copper rod ($Y = 110\ \text{GPa}$, $\alpha = 17 \times 10^{-6}\ \text{K}^{-1}$) is warmed by $25\ \text{K}$. What stress builds up?

4. Why are railway tracks laid with small gaps between the rails?