Question

Find the ratio of the lengths of an iron rod  and an aluminium rod for which the difference in the lengths is independent of temperature. Coefficients of linear expansion of iron and aluminium are 12 × 10^–6 °C^–1 and 23 × 10^–6 °C^–1 respectively.

Answer 1

Let the original length of iron rod be L_Fe and L^'​​​_Fe be its length when temperature is increased by ΔT.
Let the original length of aluminium rod be L_Al and L^'​​​_Al be its length when temperature is increased by ΔT. 

Coefficient of linear expansion of iron, 

\[\alpha_{Fe}\] = 12 × 10^–6 °C​

\[-\] 1

Coefficient of linear expansion of aluminium, α_Al = 23 × 10^–6 °C^​​

\[-\] 1

Since the difference in length is independent of temperature, the difference is always constant.

\[L '_{Fe} = L_{Fe} \left( 1 + \alpha_{Fe} \times ∆ T \right)\]

\[and\] \[  L '_{Al} = L_{Al} \left( 1 + \alpha_{Al} \times ∆ T \right)\]

\[ \Rightarrow \begin{array}\[L '_{Fe} - L '_{Al} = L_{Fe} - L_{Al} + L_{Fe} \times \alpha_{Fe} ∆ T - L_{Al} \times \alpha_{Al} \times ∆ T & - (1)\end{array}\]

\[Given: \]

\[L '_{Fe} - L '_{Al} = L_{Fe} - L_{Al} \]

\[Hence, L_{Fe} \alpha_{Fe} = L_{Al} \alpha_{Al} [using (1)]\]

\[ \Rightarrow \frac{L_{Fe}}{L_{Al}} = \frac{23}{12}\]

The ratio of the lengths of the iron to the aluminium rod is 23:12.