Determine the ratio of the lengths of an iron rod and an aluminium...

Let $L_1$ and $L_2$ be the lengths of the iron and aluminium rods respectively.

Let $\alpha_1$ and $\alpha_2$ be their coefficients of linear expansion.

The change in length for a temperature change $\Delta T$ is given by $\Delta L = L \alpha \Delta T$.

For the difference in lengths to remain independent of temperature, the change in length of both rods must be equal.

$\Delta L_1 = \Delta L_2$

$L_1 \alpha_1 \Delta T = L_2 \alpha_2 \Delta T$

$L_1 \alpha_1 = L_2 \alpha_2$

$\dfrac{L_1}{L_2} = \dfrac{\alpha_2}{\alpha_1}$

Substituting the given values, \alpha_1 = 12 \times 10^{-6} ^{}^{{}^{\circ}}\textC^{-1} and \alpha_2 = 23 \times 10^{-6} ^{}^{{}^{\circ}}\textC^{-1}:

$\dfrac{L_1}{L_2} = \dfrac{23 \times 10^{-6}}{12 \times 10^{-6}} = \dfrac{23}{12}$

The ratio of their lengths is $23 : 12$.