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 two such sections so that there is no compression during summer when the maximum temperature goes to $48^{\circ} C$ ? Coefficient of linear expansion of iron = $11 \times 10^{- 6}^{\circ} C^{- 1}$

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Solution

$L_{0} = 12 m$ $α = 11 \times 10^{- 6} /^{\circ} C$

$t_{w} = 18^{\circ} C$ $t_{8} = 48^{\circ}$

$L_{w} = L_{0} (1 + α Δ t_{w})$

$= 12 (1 + 11 \times 10^{- 6} \times 18)$

= 12.00237 m

$L_{a} = L_{0} (1 + α Δ t_{8})$

$= 12 (1 + 11 \times 10^{- 8} \times 48)$

= 12.006336 m

$∴ Δ L$ = 12.006336 - 12.002376

= 0.00396 m $\approx$ 0.4 cm

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L0=12m α=11×10−6/∘C tw=18∘C t8=48∘ Lw=L0(1+αΔtw) =12(1+11×10−6×18) = 12.00237 m La=L0(1+α Δt8) =12(1+11×10−8×48) = 12.006336 m ∴ ΔL = 12.006336 - 12.002376 = 0.00396 m ≈ 0.4 cm