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 $l_1+l_2$. The coefficients of linear expansion for 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 find the ratio $\frac{l_1}{l_1+l_2}$.

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

Answer 1

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

The length of each rod increases by the same amount
$\therefore △l_a=△l_s\Rightarrow l_1{\alpha }_{a}t=l_2{\alpha }_{s}t$
$\Rightarrow \frac{l_2}{l_1}=\frac{{\alpha }_a}{{\alpha }_s}\Rightarrow \frac{l_2}{l_1}+1=\frac{{\alpha }_a}{{\alpha }_s}+1$
$\Rightarrow \frac{l_2+l_1}{l_1}=\frac{{\alpha }_a+{\alpha }_s}{{\alpha }_s}\Rightarrow \frac{l_1}{l_1+l_2}=\frac{{\alpha }_s}{{\alpha }_a+{\alpha }_s}$.