Problem — Three bars at the same temperature

Problem

Three metal bars, each $\ell_0 = 2.0$ m long at $20\ ^{\circ}$C, are heated up to $120\ ^{\circ}$C. The linear expansion coefficients are:

Bar	Metal	$\lambda$ [K⁻¹]
A	brass	$19 \cdot 10^{-6}$
B	invar	$1.2 \cdot 10^{-6}$
C	zinc	$30 \cdot 10^{-6}$

Order the bars by absolute expansion $\Delta\ell$, from smallest to largest.

Solution

Law of linear expansion. The elongation is $\Delta\ell = \ell_0 \, \lambda \, \Delta T$, with $\Delta T = 120 - 20 = 100$ K and $\ell_0 = 2.0$ m, so $\Delta\ell = 200 \cdot \lambda$ (in metres). $\Delta\ell_A = 200 \cdot 19 \cdot 10^{-6} = 3.8 \cdot 10^{-3}\ \mathrm{m} = 3.8\ \mathrm{mm}$ $\Delta\ell_B = 200 \cdot 1.2 \cdot 10^{-6} = 2.4 \cdot 10^{-4}\ \mathrm{m} = 0.24\ \mathrm{mm}$ $\Delta\ell_C = 200 \cdot 30 \cdot 10^{-6} = 6.0 \cdot 10^{-3}\ \mathrm{m} = 6.0\ \mathrm{mm}$ Conclusion. For the same initial length and temperature rise, the expansion is proportional to $\lambda$. Invar, almost non-expanding, lengthens very little; zinc is the one that expands the most. $\ev{B \ll A < C}$

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Topics: Termologia Concepts: Dilatazione termica