Question

If two rods of lengths $L$ and $2L$, having coefficients of linear expansion $\alpha$ and $2\alpha$ respectively, are connected so that total length becomes $3L$. The average coefficient of linear expansion of the composite rod is equal to:

• A. $\frac{3}{2}\alpha$
• B. $\frac{5}{2}\alpha$
• C. $\frac{5}{3}\alpha$
• D. None of these

Answer 1

The correct option is C $\frac{5}{3}\alpha$


Total length $=L+2L=3L$
Increase in length in the first rod $=L\alpha \Delta T$.
Increase in length in second rod $=2L\left(2\alpha \right)\Delta T.=4L\alpha \Delta T$
So total increase in length $=L\alpha \Delta T+4L\alpha \Delta T$
$=5L\alpha \Delta T$
Coefficient of linear expansion $=\frac{\text{increases in length}}{\text{original length}\times \Delta T}=\frac{5L\alpha \Delta T}{3L\Delta T}$
$=\frac{5}{3}\alpha$

Hence, $\left(C\right)$ is the correct answer.