An iron rod of length 50 cm is joined at one end to an aluminium rod of length 100 cm . All measurem

Question

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

Professor PC Thomas Classes · Answer

The length of the iron rod at $100^{\circ} \mathrm{C}$ is
$$
\begin{aligned}
l_1 & =(50 \mathrm{~cm})\left[1+\left(12 	imes 10^{-6}{ }^{\circ} \mathrm{C}^{-1}ight)\left(100^{\circ} \mathrm{C}-20^{\circ} \mathrm{C}ight)ight] \
& =50 \cdot 048 \mathrm{~cm}
\end{aligned}
$$The length of the aluminium rod at $100^{\circ} \mathrm{C}$ is
$$
\begin{aligned}
l_2 & =(100 \mathrm{~cm})\left[1+\left(24 	imes 10^{-6}{ }^{\circ} \mathrm{C}^{-1}ight)\left(100^{\circ} \mathrm{C}-20^{\circ} \mathrm{C}ight)ight] \
& =100 \cdot 192 \mathrm{~cm} .
\end{aligned}
$$The length of the composite system at $100^{\circ} \mathrm{C}$ is
$$
50 \cdot 048 \mathrm{~cm}+100 \cdot 192 \mathrm{~cm}=150 \cdot 24 \mathrm{~cm} .
$$The length of the composite system at $20^{\circ} \mathrm{C}$ is 150 cm . So, the average coefficient of linear expansion of the composite rod is
$$
\begin{aligned}
\alpha & =\frac{0 \cdot 24}{150 \mathrm{~cm} 	imes\left(100^{\circ} \mathrm{C}-20^{\circ} \mathrm{C}ight)} \
& =20 	imes 10^{-6}{ }^{\circ} \mathrm{C}^{-1}
\end{aligned}
$$