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Difficulty: MediumThermal Expansion of Solids (Linear, Area, and Volume Expansivity)

A railway line is laid using steel rails, each of length 15 m15\text{ m}, at a temperature of 20∘C20^\circ\text{C}. What minimum gap, in millimetres (mm\text{mm}), must be left between consecutive rails so that they just touch without buckling when heated to 60∘C60^\circ\text{C}? [Linear expansivity of steel = 1.2×10−5 K−11.2 \times 10^{-5}\text{ K}^{-1}]

Answer: 7.2 mm

Answer

The minimum gap required between consecutive rails is 7.2 mm7.2\text{ mm}.
The expansion in length ΔL\Delta L is given by ΔL=L0αΔT\Delta L = L_0 \alpha \Delta T. Substituting the initial length L0=15 mL_0 = 15\text{ m}, linear expansivity α=1.2×10−5 K−1\alpha = 1.2 \times 10^{-5}\text{ K}^{-1}, and temperature change ΔT=40 K\Delta T = 40\text{ K} gives ΔL=7.2×10−3 m\Delta L = 7.2 \times 10^{-3}\text{ m}, which corresponds to 7.2 mm7.2\text{ mm}.

Step-by-Step Solution

1
Determine the change in temperature
\Delta T = 60^\circ\text{C} - 20^\circ\text{C} = 40\text{ K}
Thermal expansion is driven by the temperature difference between the final and initial states.
2
Set up the linear expansion formula
\Delta L = L_0 \alpha \Delta T
The linear expansion of a solid bar depends on its initial length, the material's linear expansivity, and the temperature change.
3
Calculate the expansion in metres and convert to millimetres
\Delta L = 15 \times (1.2 \times 10^{-5}) \times 40 = 7.2 \times 10^{-3}\text{ m} = 7.2\text{ mm}
Multiplying the value in metres by 10001000 yields the required measurement in millimetres.

Key Concept

Linear Expansivity and Thermal Expansion of Solids
Estimated Time:1m 30s
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Topics

Thermal Expansion of Solids (Linear, Area, and Volume Expansivity)

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