Question

Two rods, one of aluminium and the other made of steel, having initial lengths $l_{1}$ and $l_{2}$ are connected together to form a single rod of length $\left(l_{1}+l_{2}\right) .$ The coefficient of linear expansion of aluminium and steel are $\alpha_{a}$ and $\alpha_{s}$ respectively. If the length of each rod increases by the same amount, when their temperatures are raised by $T^{\circ} C$, then the ratio $\frac{l_{1}}{l_{1}+l_{2}}$ is

• 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

Answer: (C)

As $\Delta l_{1}=\Delta l_{2} $
$\therefore \alpha_{a} l_{1} T=\alpha_{s} l_{2} T$
or $\frac{l_{2}}{l_{1}}=\frac{\alpha_{a}}{\alpha_{s}}$ or
$\frac{l_{2}}{l_{1}}+1=\frac{\alpha_{a}}{\alpha_{s}}+1 $
or $\frac{l_{2}+l_{1}}{l_{1}}=\frac{\alpha_{a}+\alpha_{s}}{\alpha_{s}}$
$\therefore \frac{l_1}{l_1 + l_2} = \frac{\alpha_s}{\alpha_a + \alpha_s}$