Question

What should be the sum of lengths of an aluminium and steal rod at $0^{o}C$ is, so that at all temperatures their difference in length is $0.25m$ (Take coefficient linear expansion for aluminium and steel at $0^{o}C$ as $22\times {10}^{-6}{/}^{}C$ and $11\times {10}^{-6}{/}^{}C$ respectively.

Answer 1

Thermal expansion $-:$

in steel,

$△l_s={\alpha }_3{l}_s△T$ in aluminium

$△l_a={\alpha }_a{l}_a△T$

As at all temp.

$|l_a-l_s|=0.25m$ ----- $\left(i\right)$

$\Rightarrow △l_a=△l_s=0$

$\Rightarrow △l_a=△l_s$

$\Rightarrow {\alpha }_a\left(l_a\right)△T={\alpha }_s{l}_s△T$

$\Rightarrow 22\times {10}^{-6}\times l_a=11\times {10}^{-6}\times l_s$

$\Rightarrow 2l_a=l_s$ ------- $\left(ii\right)$

From $\left(i\right)$ and $\left(ii\right)$

$l_s-l_a=0.25$

$\Rightarrow 2l_a-l_a=0.25$

$\Rightarrow l_a=0.25m$

$l_s=2l_a=0.5m$

Sum of length $=l_s+l_a=0.5+0.25=0.75m$