Question

A railway track (made of iron) is laid in winter when the average temperature is 18°C. The track consists of sections of 12.0 m placed one after the other. How much gap should be left between two such sections, so that there is no compression during summer when the maximum temperature rises to 48°C? Coefficient of linear expansion of iron = 11 × 10^–6 °C^–1.

Answer 1

Given:
Length of the iron sections when there's no effect of temperature on them, L_o = 12.0 m
​Temperature at which the iron track is laid in winter, t_​w​ = 18 ^oC
Maximum temperature during summers, t_s = 48 ^oC
Coefficient of linear expansion of iron, $\alpha$ = 11 × 10^–6 °C^–1
Let the new lengths attained by each section due to expansion of iron in winter and summer be L_w and L_s, respectively, which can be calculated as follows:
$$\begin{gathered} L_w=L_0\left(1+\alpha t_w\right) \\ \Rightarrow L_w=12\left(1+11\times {10}^{-6}\times 18\right) \\ \Rightarrow L_w=12.00237\mathrm{m} \\ {L}_s=L_0\left(1+\alpha t_s\right) \\ \Rightarrow L_s=12\left(1+11\times {10}^{-6}\times 48\right) \\ \Rightarrow L_s=12.006336\mathrm{m} \\ \therefore ∆L=L_s-L_w \\ \Rightarrow \Delta L=12.006336-12.002376 \\ \Rightarrow \Delta L=0.00396\mathrm{m} \\ \Rightarrow \Delta L\approx 0.4\mathrm{cm} \end{gathered}$$
Therefore, the gap ($\Delta$L) that should be left between two iron sections, so that there is no compression during summer, is 0.4 cm.