Question

Two rods of different materials having coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$ and Young’s modulus
$Y_1$ and $Y_2$ 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 rods. If
${\alpha }_1$ : ${\alpha }_2$ = 2:3, the thermal stress developed in the two rods are equal provided $Y_1:Y_2$ equal to

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

Answer 1

The correct option is D 3 : 2

From the formula for thermal stress
$(F/A{)}_1={\alpha }_1{y}_1△T$
$(F/A{)}_2={\alpha }_2{y}_2△T$
Since,temperature $△T$ is same
$\frac{(\frac{F}{A}{)}_1}{(\frac{F}{A}{)}_2}=\frac{{\alpha }_1{y}_1}{{\alpha }_2{y}_2}$
For thermal stress to be equal
${\alpha }_1{y}_1={\alpha }_2{y}_2$
$or\frac{y_1}{y_2}=\frac{{\alpha }_2}{{\alpha }_1}=\frac{3}{2}$ since $\frac{{\alpha }_1}{{\alpha }_2}=2:3)$