Question

Two rods of different materials having coefficients of linear expansion ${\alpha }_1:{\alpha }_2$ and Young's modulii $Y_1:Y_2$ 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 stress developed in the 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 $3:2$

$\frac{Stress}{Strain}=Y$
$Stress=YStrain$
$=Y\frac{△l}{l}$
$=Y\alpha △T$
$Stres{s}_1=Y_1{\alpha }_1△T$
$Stres{s}_2=Y_2{\alpha }_2△T$

By problem
$Stres{s}_1=Stres{s}_2$
$Y_1{\alpha }_1△T=Y_2{\alpha }_2△T$

$\frac{Y_1}{Y_2}=\frac{{\alpha }_2}{{\alpha }_1}=\frac{3}{2}(\because \frac{{\alpha }_1}{{\alpha }_2}=\frac{2}{3})$