Linear Thermal Expansion

Question

A metal railway track is 50 m long at $20^\circ	ext{C}$ and the coefficient of linear expansion of the metal is $1.2	imes10^{-5}\,/^\circ	ext{C}$. How much does the length of the track increase when its temperature rises to $60^\circ	ext{C}$?

RankGuru · Accepted Answer

Most solids expand when heated because rising temperature increases the average separation between vibrating atoms in the lattice. For a rod the change in length is $\Delta L = L_0\alpha\Delta T$, where $L_0$ is the original length, $\alpha$ the coefficient of linear expansion and $\Delta T$ the temperature rise. The temperature increases from $20^\circ	ext{C}$ to $60^\circ	ext{C}$, so $\Delta T = 40^\circ	ext{C}$. Substituting $L_0 = 50\,	ext{m}$, $\alpha = 1.2	imes10^{-5}\,/^\circ	ext{C}$ and $\Delta T = 40$ gives $\Delta L = 50	imes1.2	imes10^{-5}	imes40 = 2.4	imes10^{-2}\,	ext{m} = 24\,	ext{mm}$. The value 12 mm uses only half the temperature change. The value 36 mm overstates $\Delta T$ as 60. The value 48 mm doubles the correct expansion. This is the standard NCERT relation for linear thermal expansion and explains why expansion gaps are left between rail segments. As a check, a 24 mm expansion over 50 m is a tiny relative change of about $5	imes10^{-4}$, which is the small magnitude expected for solid expansion over a modest temperature rise.

Linear Thermal Expansion

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