Question

Two rods of different materials, having coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$ and Young's moduli $Y_1$ and $Y_2$ respectively, are fixed between two rigid walls. The rods are heated to the same temperature. There is no bending of the rods. If ${\alpha }_1:{\alpha }_2=2:3$, the thermal stress developed in the two rods will be equal provided $Y_1:Y_2$ is equal to

• A. 2:3
• B. 1:1
• C. 3:2
• D. 4:9

Answer 1

The correct option is C

3:2

Young's modulus $Y=\frac{Stress}{Strain}=\frac{\sigma }{ϵ}$, where $ϵ=\frac{l}{L}$.
Now, if the temperature of a rod is increased by $\theta$, the increase in its length due to thermal expansion is $l=\alpha L\theta$
$\therefore$ Strain, $ϵ=\frac{l}{L}=\alpha \theta$.
and stress, $\sigma =Yϵ=Y\alpha \theta$
For the two rods, the stress is given by ${\sigma }_1=Y_1{\alpha }_1\theta$ and ${\sigma }_2=Y_2{\alpha }_2\theta$
But given that, ${\sigma }_1={\sigma }_2$
$\Rightarrow Y_1{\alpha }_1\theta =Y_2{\alpha }_2\theta$
$\Rightarrow \frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}=\frac{3}{2}$.
Hence, the correct choice is (c).