Question

Two rods, one of aluminium and the other made of steel, having initial length $L_1\text{and}{L}_2$ are connected together to form a single rod of length $L_1+L_2$. The coefficient of linear expansion for aluminium and steel are ${\alpha }_a\text{and}{\alpha }_s$ respectively. If the length of each rod increases by the same amount when their temperatures are raised by $\Delta T{}^{\circ }\text{C}$, then find the value of $\frac{L_1}{L_1+L_2}$.

• A. $\frac{{\alpha }_s}{{\alpha }_a}$
• B. $\frac{{\alpha }_a}{{\alpha }_s}$
• C. $\frac{{\alpha }_s}{({\alpha }_a+{\alpha }_s)}$
• D. $\frac{{\alpha }_a}{({\alpha }_a+{\alpha }_s)}$

Answer 1

The correct option is C $\frac{{\alpha }_s}{({\alpha }_a+{\alpha }_s)}$


With raise in temperature, $\Delta T,L_1\text{and}{L}_2$ increase by equal amount

$\Delta L_1=\Delta L_2$

$\Rightarrow L_1{\alpha }_a\Delta T=L_2{\alpha }_s\Delta T$

$\Rightarrow \frac{L_2}{L_1}=\frac{{\alpha }_a}{{\alpha }_s}$

$\Rightarrow (\frac{L_2}{L_1}+1)=(\frac{{\alpha }_a}{{\alpha }_s}+1)$

$\Rightarrow \frac{L_2+L_1}{L_1}=\frac{{\alpha }_a+{\alpha }_s}{{\alpha }_s}$

$\Rightarrow \frac{L_1}{L_1+L_2}=\frac{{\alpha }_s}{{\alpha }_a+{\alpha }_s}$

Hence, option $\left(\text{C}\right)$ is correct.