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प्रश्न
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.
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उत्तर
Given:
Length of the iron sections when there's no effect of temperature on them, Lo = 12.0 m
Temperature at which the iron track is laid in winter, tw = 18 oC
Maximum temperature during summers, ts = 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 Lw and Ls, 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.
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