Question

Co-efficient of linear expansion of brass and steel rods are ${\alpha }_1$ and ${\alpha }_2$ respectively. Lengths of brass and steel rods are $L_1$ and $L_2$ respectively. If $(L_2-L_1)$ is maintained same at all the temperatures, which one of the following relations holds good?

• A. ${\alpha }_1{L}_1={\alpha }_2{L}_2$
• B. ${\alpha }_1{L}_2={\alpha }_2{L}_1$
• C. ${\alpha }_1{L}_2^2={\alpha }_2{L}_1^2$
• D. ${\alpha }_1^2{L}_2={\alpha }_2^2{L}_1$

Answer 1

The correct option is A ${\alpha }_1{L}_1={\alpha }_2{L}_2$

As per the question, $L_2-L_1=$ constant
Let $L_1^{\prime }$ and $L_2^{\prime }$ be the lengths of the rods when temperatures are changed.
$\Rightarrow L_2^{\prime }-L_1^{\prime }=L_2-L_1$
$\Rightarrow L_2(1+{\alpha }_2\Delta T)-L_1(1+{\alpha }_1\Delta T)=L_2-L_1$
$[$ temperature change will be same for both the rods $]$
$\Rightarrow L_2{\alpha }_2=L_1{\alpha }_1$