Question

A railway track (made of iron) is laid in winter when the average temperature is $18\,{}^ \circ C$. The track consists of sections of $12.0\,m$ placed one after the other. How much gap should be left between the two such section so that there is no compression during summer when the maximum temperature goes to $48\,{}^ \circ C$? Coefficient of linear expansion of iron is $11 \times {10^{ - 6}}\,{}^ \circ C$.

Answer 1

Hint The length of the gap so that there is no compression during the season change is determined by the difference of the length of the rod in winter and the length of the rod in summer. The length of the rod is determined by using the thermal expansion formula.

Useful formula
The thermal expansion of the rod is given by,
${L_x} = L\left( {1 + \alpha {t_x}} \right)$
Where, ${L_x}$ is the length of the required season, $L$ is the original length, $\alpha $ is the coefficient of linear expansion and ${t_x}$ is the temperature of the season.

Complete step by step solution
Given that,
The temperature of the winter is, ${t_w} = 18\,{}^ \circ C$,
The original length is, $L = 12\,m$,
The temperature of the summer is, ${t_s} = 48\,{}^ \circ C$,
The coefficient of linear expansion of iron is $11 \times {10^{ - 6}}\,{}^ \circ C$.
Now, the length of the rod in the winter season is given by,
${L_w} = L\left( {1 + \alpha {t_w}} \right)\,....................\left( 1 \right)$
By substituting the original length of the rod, coefficient of the linear expansion and the temperature of the winter season in the above equation (1), then the above equation is written as,
${L_w} = 12\left( {1 + \left( {11 \times {{10}^{ - 6}} \times 18} \right)} \right)$
On multiplying the terms in the above equation, then
${L_w} = 12\left( {1 + 1.98 \times {{10}^{ - 4}}} \right)$
By adding the terms in the above equation, then
${L_w} = 12\left( {1.000198} \right)$
On multiplying the terms in the above equation, then
${L_w} = 12.002\,m$
Now, the length of the rod in the summer season is given by,
${L_s} = L\left( {1 + \alpha {t_s}} \right)\,....................\left( 2 \right)$
By substituting the original length of the rod, coefficient of the linear expansion and the temperature of the summer season in the above equation (1), then the above equation is written as,
${L_s} = 12\left( {1 + \left( {11 \times {{10}^{ - 6}} \times 48} \right)} \right)$
On multiplying the terms in the above equation, then
${L_s} = 12\left( {1 + 5.28 \times {{10}^{ - 4}}} \right)$
By adding the terms in the above equation, then
${L_s} = 12\left( {1.000528} \right)$
On multiplying the terms in the above equation, then
${L_s} = 12.006\,m$
The gap between the two section is, $\Delta L = {L_s} - {L_w}$, then
$\Delta L = 12.006 - 12.002$
On subtracting the above equation, then
$\Delta L = 0.004\,m \Rightarrow 0.4\,cm$
Thus, the above equation shows that the gap should be left between the two such sections

so that there is no compression during summer is $0.4\,cm$.

Note: During winter season the temperature of the atmosphere is reduced so that the material structure will get compressed, because that the internal molecules will get very close at low temperature. During the summer, the atmospheric temperature is high, when the temperature is high the material starts expanding, the bond between the molecules will expand.