Question

The ratio of lengths of two rods is 1: 2 and the ratio of coefficient of expansions is 2 : 3. The first rod is heated through 60C. Find the temperature through which the second rod is to be heated so that its expansion is twice that of first is

• A. ${60}^{0}C$
• B. ${40}^{0}C$
• C. ${30}^{0}C$
• D. ${10}^{0}C$

Answer 1

The correct option is B ${40}^{0}C$

Given,

$\frac{l_1}{l_2}=\frac{1}{2}$

$\frac{{\alpha }_1}{{\alpha }_2}=\frac{2}{3}$ ($\alpha =$ Coefficient of expansion)

$\Delta t_1={60}^{0}C$ (Assuming initial temperature $=0^{0}C$)

$L$ (at temperature $t$ ) $=L_0(1+\alpha \Delta t)=L_0+L_0\alpha \Delta t$

Therefore,

Expansion $=L$ (at temperature $t$) $-L_0=L_0\alpha \Delta t$

$\frac{Expansio{n}_1}{Expansio{n}_2}=\frac{L_1{\alpha }_1\Delta t_1}{L_2{\alpha }_2\Delta t_2}=\frac{1}{2}$

$⟹\frac{1}{2}\times \frac{2}{3}\times \frac{60}{\Delta t_2}=\frac{1}{2}$

$⟹\Delta t_2={40}^{0}C$