Question

Two metal rods A and B are having their initial lengths in the ratio 2:3 and coefficients of linear expansion in the ratio 3:4. When they are heated through the same temperature difference, the ratio of their linear expansion is :

• A. 1:2
• B. 2:3
• C. 3:4
• D. 4:3

Answer 1

The correct option is A 1:2

We know that the change in the length of the rod due to the thermal expansion is given by,

$\Delta l=l\times \alpha \times \Delta T$

where $l$ = original length,

$\alpha$ = coefficient of linear expansion and

$\Delta T$ = change in temperature.

For the same temperature difference $\Delta T$ we have

$\Delta l\propto l\times \alpha$
$\Rightarrow \frac{\Delta l_1}{\Delta l_2}=\left(\frac{l_1}{l_2}\right)\times \left(\frac{{\alpha }_1}{{\alpha }_2}\right)$

$\Rightarrow \frac{\Delta l_1}{\Delta l_2}=\frac{2}{3}\times \frac{3}{4}=\frac{1}{2}$.