Question

Two rods of different materials having coefficients of thermal expansions $ α_1 $ and $ α_2 $ and Young’s moduli $ Y_1 $ and $ Y_2 $ respectively are fixed between two rigid walls. The rods are heated, such that they undergo the same increase in temperature. There is no bending of rods. If $ α_1/α_2 = 2/3 $ and stresses developed in the two rods are equal, then $ \frac{Y_{1}}{Y_{2}} $ is :

• A. $ 3/2 $
• B. $ 1 $
• C. $ 2/3 $
• D. $ 1/2 $

Answer 1

Answer: (A)

Thermal stress is a measure of the internal distribution of force per unit area within a body that is applied to the body, in the form of heat.
Thermal stress $=Y \,\alpha \,\Delta \,T$
where $Y$ is Young's modulus, $\alpha$ the coefficient of linear expansion and $\Delta T$ the change in temperature.
Both the rods are heated,
$\therefore \, Y_{1} \,\alpha_{1} \,\Delta T_{1}=Y_{2} \,\alpha_{2} \,\Delta T_{2}$
Since, $\Delta \,T_{1}=\Delta \,T_{2}$
$\Rightarrow \, \frac{Y_{1}}{Y_{2}}=\frac{\alpha_{2}}{\alpha_{1}}=\frac{3}{2}$