Question

Two rods of different materials having coefficients of thermal expansion ${\alpha }_1$ and ${\alpha }_2$ and Youngs
moduli $Y_1$ 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 the rods. If ${\alpha }_1$ and ${\alpha }_2$ are
in the ratio 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 that $\gamma =\frac{stress}{strain}\gamma =$Young's modulus

Stress$=\gamma .$ strain$\to 1$

Also, Strain$=\frac{△L}{L}=\frac{\alpha △T}{L}$

using equation $1$

${\gamma }_1.$strain $1={\gamma }_2.$strain $2$

${\gamma }_1.\frac{{\gamma }_1△T}{L}={\gamma }_2.\frac{{\gamma }_2△T}{L}$

$\frac{{\gamma }_1}{{\gamma }_2}=\frac{{\alpha }_2}{{\alpha }_1}=\frac{3}{2}$

Hence ratio of $Y_1:Y_2$ is $3:2$