Question

The coefficient of linear expansion of an inhomogeneous rod changes linearly from $\alpha_{1}$ to $\alpha_{2}$ from one end to the other end of the rod. The effective coefficient of linear expansion of rod is :

• A. $\alpha_{1}+\alpha_{2}$
• B. $\frac{\alpha_{1}+\alpha_{2}}{2}$
• C. $\sqrt{\alpha_{1} \alpha_{2}}$
• D. $\alpha_{1}-\alpha_{2}$

Answer 1

Answer: (B)

We consider an element at a distance $x$ from one end of width $d x$ where coefficient of expansion will be
$\alpha=\alpha_{1}+\frac{\alpha_{2}-\alpha_{1}}{L} \cdot x$
Expansion in element is $d l=\alpha d x \Delta T$
Total expansion $\Delta L=\int \alpha \Delta T d x$
$\Delta L=\Delta T \int\limits_{0}^{L}\left(\alpha_{1}+\frac{\alpha_{2}-\alpha_{1}}{L} \cdot x\right) d x$
$=\Delta T\left[\alpha_{1} L+\frac{\alpha_{2}-\alpha_{1}}{L}\left(\frac{L^{2}}{2}\right)\right]$
$=\Delta T\left(\frac{\alpha_{1}-\alpha_{2}}{2}\right)$
$=\alpha_{e q} L \Delta T$
Where we get
$\alpha_{e q}=\frac{\alpha_{1}+\alpha_{2}}{2}$