Question

Two rods of different materials having coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$. Young modulus ${\gamma }_1$ and ${\gamma }_2$ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of rods.
If ${\alpha }_1:{\alpha }_2=2:3$, the thermal stress developed in the two rods are equal provided ${\gamma }_1:{\gamma }_2$ is equal to

• A. $1:1$
• B. $2:3$
• C. $3:2$
• D. $4:9$

Answer 1

Answer: (C)

$3:2$

We have $\gamma =\frac{\sigma }{\epsilon }$
hence $\epsilon =\frac{\Delta L}{L}=\frac{\sigma }{\gamma }$
But $\frac{\Delta L}{L}=\alpha \Delta \theta$ where $\Delta \theta$ is the change in temperature. Hence $\sigma =Y\alpha \Delta \theta$ in this case stress is same
Hence, $Y_1{\alpha }_1\Delta \theta =Y_2{\alpha }_2\Delta \theta$
$\therefore \frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}=\frac{3}{2}$