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:

\[L_w = L_0 \left( 1 + \alpha t_w \right)\]

\[ \Rightarrow L_w = 12 \left( 1 + 11 \times {10}^{- 6} \times 18 \right)\]

\[ \Rightarrow L_w = 12 . 00237 m\]

\[ L_s = L_0 \left( 1 + \alpha t_s \right)\]

\[ \Rightarrow L_s = 12 \left( 1 + 11 \times {10}^{- 6} \times 48 \right)\]

\[ \Rightarrow L_s = 12 . 006336 m\]

\[ \therefore ∆ L = L_s - L_w \]

\[ \Rightarrow ∆ L= 12 . 006336 - 12 . 002376\]

\[ \Rightarrow ∆ L= 0 . 00396 m\]

\[ \Rightarrow ∆ L \approx 0 . 4 cm\]

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.