Question

An aluminium rod (length $l_1$ and coefficient of linear expansion ${\alpha }_A$) and a steel rod (length $l_2$ and coefficient of linear expansion ${\alpha }_B$) are joined together. If the length of each rod increased by the same amount when their temperature are raised by $t^{\circ }C$, then $l_1/l_2+1$ is:

• A. ${\alpha }_A/{\alpha }_B$
• B. ${\alpha }_B/{\alpha }_A$
• C. ${\alpha }_B/{\alpha }_A+1$
• D. ${\alpha }_A/{\alpha }_B+1$

Answer 1

Answer: (C)

${\alpha }_B/{\alpha }_A+1$

Increase in the length of aluminum be $\Delta L_A=l_1{\alpha }_A\Delta T$

Increase in the length of steel be $\Delta L_s=l_2{\alpha }_B\Delta T$

Given that $\Delta L_1=\Delta L_2$

So $l_1{\alpha }_A=l_2{\alpha }_B$

So $\frac{l_1}{l_2}=\frac{{\alpha }_B}{{\alpha }_A}$

So $\frac{l_2}{l_1}=\frac{{\alpha }_A}{{\alpha }_B}$

Adding 1 on both sides we get $\frac{l_2+l_1}{l_1}=\frac{{\alpha }_A+{\alpha }_B}{{\alpha }_B}$

$\frac{l_1}{l_1+l_2}=\frac{{\alpha }_B}{{\alpha }_B+{\alpha }_A}$