Worked exercise — The railway track in winter

Problem

A railway track $30$ m long is laid at $T_0=20\ ^{\circ}$C. In full summer sun, in August, the track’s temperature rises to $T=55\ ^{\circ}$C. For steel, $\lambda=1.2\cdot10^{-5}\ \mathrm{K}^{-1}$. By how much does it lengthen?

Solution

Setting up. Linear expansion is $\Delta\ell=\ell_0\,\lambda\,\Delta T$, with $\Delta T=T-T_0=55-20=35\ \mathrm{K}$. Numerical substitution: $\Delta\ell=30\cdot 1.2\cdot10^{-5}\cdot 35\approx 1.26\cdot10^{-2}\ \mathrm{m}=1.26\ \mathrm{cm}$ $\ev{\Delta\ell\approx 1.26\ \mathrm{cm}}$ Without expansion joints the track would buckle: the joints are the small gaps left between one rail and the next precisely to absorb this lengthening.

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Topics: Termologia Concepts: Dilatazione termica