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 $\frac{l_2+l_1}{l_1}=\frac{{\alpha }_a+{\alpha }_s}{{\alpha }_s}$
$\therefore \frac{l_1}{l_1+l_2}=\frac{{\alpha }_s}{{\alpha }_a+{\alpha }_s}$