Question

Two rods of different materials having coefficients of thermal expansion ${\alpha }_1$ and ${\alpha }_2$ and Young's modulli $Y_1$ and $Y_2$ are fixed between two rigid and massive walls. The rods are heated to the same temperature. If there is no bending in the rods, the thermal stress developed in them are equal if

• A. $\frac{Y_1}{Y_2}=\sqrt{\frac{{\alpha }_1}{{\alpha }_2}}$
• B. $\frac{Y_1}{Y_2}=\sqrt{\frac{{\alpha }_2}{{\alpha }_1}}$
• C. $\frac{Y_1}{Y_2}=\frac{{\alpha }_1}{{\alpha }_2}$
• D. $\frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}$

Answer 1

The correct option is D $\frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}$

Stress=$Y\times Strain$

$=Y\times \frac{\Delta l}{l}$

$=Y\times \frac{\alpha \Delta T}{l}\propto Y\alpha$

Thus $Y_1{\alpha }_1=Y_2{\alpha }_2$

$⟹\frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}$