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Question

Find the ratio of the length of an iron and an aluminium rod for which the difference in the lengths is independent of temperature. Coefficients of linear expansion of iron and aluminium are 12×106C1 and 12×106C1 respectively.

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Solution

Iron rod
LFe
αFe=12×108/oC

Aluminium rod
LAl
αAl=23×108/oC

Since the difference in length is independent of temp. Hence the different always remains constant.
LFe=LFe(1+αFe×ΔT) ...(1)

LAl=LAl(1+αAl×ΔT) ...(2)

LFeLAl=LFeLAl+LFe×αFe×ΔT=LAl×αAl×ΔT

LFeLAl=αAlαFe=2312=23:12

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