Question

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

• 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

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

If a rod of length L and coefficient fo thermal expansion $\alpha$ is heated to a temperature t, the increase in the length is given by $l=\alpha Lt$
$\therefore$ Strain $=\frac{l}{L}=\alpha t$
and stress $=Y\times strain=Y\alpha t$
Since the value of t is the same for the two rods, the stresses in them will be equal if $Y_1{\alpha }_{1}t=Y_2{\alpha }_{2}t$, i.e., $\frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}$
Hence, the correct choice is (d).