Question

Two rods of different materials having coefficients of thermal expansion ${\alpha }_1,{\alpha }_2$ and Young's modulus $Y_1,Y_2$ 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 $3:2$

As we know ,

$\Delta l=l\alpha \Delta T$

ie, $\frac{\Delta l}{l}=\alpha \Delta T$

Stress =Y x stain

stress=$Y\alpha \Delta T$

For first rod stress =$Y_1{\alpha }_1\Delta T$

For second rod stress= $Y_2{\alpha }_2\Delta T$

Since, stresses are equal

$Y_1{\alpha }_1\Delta T$=$Y_2{\alpha }_2\Delta T$

or,

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