Question

Two rods, one of aluminum and the other made of steel, having initial length $l_{1}$ and $l_{2}$ are connected together to form a single rod of length $l_{1} + l_{2}$.

The coefficients of linear expansion for aluminum and steel are $\alpha_{a}$ and $\alpha_{s}$ respectively. If the length of each rod increases by the same amount when their temperature are raised by $t^{o}C$, then find the ratio $\frac{l_{1}}{(l_{1} + l_{2})}$

• 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

Answer: (C)

$\frac{\alpha_{s}}{(\alpha_{a} + \alpha_{s})}$

Answer 2

Given $\Delta l_{1} = \Delta l_{2}$ or $l_{1}\alpha_{a}t = l_{2}\alpha_{s}t$

$\therefore$ $\frac{l_{1}}{l_{2}} = \frac{\alpha_{s}}{\alpha_{a}}$ or $\frac{l_{1}}{l_{1} + l_{2}} = \frac{\alpha_{s}}{\alpha_{a} + \alpha_{s}}$.