Question

If two rods of length $L$ 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 $\frac{x\alpha }{3}$. Find $x$.

Answer 1

$\Delta L=\alpha L\Delta T$
$\Delta \left(2L\right)=2\alpha \left(2L\right)\Delta T$
$\Delta \left(3L\right)={\alpha }_{composite}\left(3L\right)\Delta T$
$\therefore \alpha L\Delta T+2\alpha \left(2L\right)\Delta T={\alpha }_{composite}\left(3L\right)\Delta T$
$\Rightarrow 5\alpha =3{\alpha }_{composite}$
$\Rightarrow {\alpha }_{composite}=\frac{5}{3}\alpha$
Comparing with $\frac{5x}{3}$, $x=5$