Question

A composite bar of length $L=L_{1}+L_{2}$ is made up from a rod of material 1 and of length $L_{1}$ attached to a rod of material 2 and of length $L_{2}$ as shown. If $\alpha_{1}$ and $\alpha_{2}$ are their respective coefficients of linear expansion, then equivalent coefficient of linear expansion for the composite rod is:

• A. $\frac{\alpha_{1} L_{2}+\alpha_{2} L_{1}}{L}$
• B. $\frac{\alpha_{1} L_{2}+\alpha_{2} L_{2}}{L}$
• C. $\frac{\alpha_{1} L_{1}+\alpha_{2} L_{2}}{L}$
• D. $\frac{\alpha_{1} \alpha_{2}\left(L_{1}^{2}+L_{2}\right)}{\left(\alpha_{1} L_{1}+\alpha_{2} L_{2}\right)}$

Answer 1

Answer: (C)

When it has been given a temperature change $\Delta T$, the new length of the composite bar is the sum of the individual final lengths (Let ' $\alpha$ ' be required coefficient of linear expansion)
$L'=L_{1}'+L_{2}'$
$L(1+\alpha \Delta T)=L_{1}\left(1+\alpha_{1} \Delta T\right)+L_{2}\left(1+\alpha_{2} \Delta T\right)$
$\Rightarrow L+L \alpha \Delta T=L_{1}+L_{1} \alpha_{1} \Delta T+L_{2}+L_{2} \alpha_{2} \Delta T$
$\Rightarrow L+L \alpha \Delta T=L+L_{1} \alpha_{1} \Delta T+L_{2} \alpha_{2} \Delta T$
$\left(\because L_{1}+L_{2}=L\right)$
$\Rightarrow L \alpha \Delta T=L_{1} \alpha_{1} \Delta T+L_{2} \alpha_{2} \Delta T$
$\Rightarrow \alpha=\frac{L_{1} \alpha_{1}+L_{2} \alpha_{2}}{L}$