Question

Two rods of different materials having coefficients of thermal expansion and Young's moduli $Y_1,Y_2$, 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 the rods. If ${\alpha }_1:{\alpha }_2=2:3$, the thermal stresses developed in the two rods are 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 $3:2$

We know, $l_2=l_1(1+\alpha \Delta t)$ where $\Delta t$ = change in temperature, if $dl$ = change in length,
we have $\frac{\Delta l}{l}=\alpha \Delta t$ = strain
The stress developed is same for both the rods.
So $Y_1{\alpha }_1\Delta t=Y_2{\alpha }_2\Delta t$
or
$\frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}=\frac{3}{2}$