An iron rod of length 50 cm is joined at an end to an Aluminium rod of length 100 cm. All measurements refer to ${20}^{0}C.$  Now the temperature is increased to ${100}^{0}C.$The coefficient of linear expansion of iron and Aluminium are $12\times {10}^{-6}/{}^{0}Cand24\times {10}^{-6}/{}^{0}C,$ respectively. The average coefficient of composition system is $x\times {10}^{-5}/{}^{0}C$ . The value of $x$ is

Initially $\left(at{20}^{0}C\right)$length of composite system  $L=50+100=150cm$

 Length of iron rod at ${100}^{0}C=50\left[1+12\times {10}^{-6}\times \left(100-20\right)\right]=50.048cm$

Length of aluminum rod at ${100}^{0}C=100\left[1+24\times {10}^{-6}\times \left(100-20\right)\right]=100.192cm$

Change in length of the composite system $L^{\text{'}}=50.048+100.192=150.24cm$

Change in length of the composite system $\Delta L=L^{\text{'}}-L=150.24-150=0.24cm$

 $\therefore$ Average coefficient of expansion at ${100}^{0}C$

$\alpha =\frac{\Delta L}{L\times \Delta T}=\frac{0.24}{150\times \left(100-20\right)}=2\times {10}^{-5}/{}^{0}C$