Question

An iron rod of length $50 \, cm$ is joined at an end to an aluminium rod of length $100 \, cm$ . All measurements refer to $20 \,{}^\circ C$ . The coefficients of linear expansion of iron and aluminium are $12\times 10^{- 6} \,{}^\circ C^{- 1}$ and $24\times 10^{- 6} \,{}^\circ C^{- 1}$ respectively. The average coefficient of expansion of the composite system is

• A. $36 \times 10^{-6}{ }^{\circ} C ^{-1}$
• B. $12 \times 10^{-6}{ }^{\circ} C ^{-1}$
• C. $20 \times 10^{-6}{ }^{\circ} C ^{-1}$
• D. $48 \times 10^{-6}{ }^{\circ} C ^{-1}$

Answer 1

Answer: (C)

Length of iron rod at 100^oC,
L₁ = 50[1 + 12 x 10^-6 x (100 - 20)] = 50.048 cm
Length of aluminum rod at 100^oC,
L₂ = 100[1 + 24 x 10^-6 x (100 - 20)] = 100.192 cm
The length of composite system at 20^oC
= 50 + 100 = 150 cm
and length of composite system at 100^oC
= 50.048 + 100.192 = 150.24 cm
$∴ \, \, \, A verage \alpha = \frac{\text{0.24 cm}}{1 5 0 cm \times \left(1 0 0 - 2 0\right)}$
= 20 x 10^-6/^oC