Question

Two rods of different materials having coefficients of thermal expansion ${\alpha }_1,{\alpha }_2$ and Young's modulii $Y_1,Y_2$ 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 ${\alpha }_1:{\alpha }_2=2:3$ and thermal stresses developed in the two rods are equal, then $Y_1:Y_2$ is equal to

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

Answer 1

The correct option is C $3:2$

Data given:
For rod $1$,
Coefficient of thermal expansion $={\alpha }_1$
Young's modulus $=Y_1$
For rod $2$,
Coefficient of thermal expansion $={\alpha }_2$
Young's modulus $=Y_2$
Ratio of coefficients of thermal expansion, $\frac{{\alpha }_1}{{\alpha }_2}=\frac{2}{3}$
To find:
Ratio of Young modulii, $\frac{Y_1}{Y_2}=?$

Let us suppose, $L$ is the initial length of the rods and $\Delta T$ is the equal change in temperature.
By definition of linear expansion,
For rod $1$,
$\Delta L_1={\alpha }_{1}L\Delta T$ ...........(1)
For rod $2$,
$\Delta L_2={\alpha }_{2}L\Delta T$ ...........(2)
From (1) and (2)
$\frac{\Delta L_1}{\Delta L_2}=\frac{{\alpha }_1}{{\alpha }_2}=\frac{2}{3}$ [given]
$\Rightarrow \frac{\Delta L_1/L}{\Delta L_2/L}=\frac{2}{3}$
$\Rightarrow \frac{{ϵ}_1}{{ϵ}_2}=\frac{2}{3}$ ............(3)
$[ϵ\to$ Thermal Strain $]$

Now, by definition of Young's modulus,
$\frac{Y_1}{Y_2}=\frac{\frac{{\sigma }_1}{{ϵ}_1}}{\frac{{\sigma }_2}{{ϵ}_2}}=\frac{{\sigma }_1{ϵ}_2}{{\sigma }_2{ϵ}_1}$
$[\sigma \to$ Thermal Stress $]$
$\Rightarrow \frac{Y_1}{Y_2}=\frac{{ϵ}_2}{{ϵ}_1}$
[ given thermal stresses are equal ]
$\Rightarrow \frac{Y_1}{Y_2}=\frac{3}{2}=3:2$
[from (3)]