Question

Two metal rods of the same lengths and area of cross-sections are fixed end to end between rigid supports. The materials of the rods have Young's moduli $Y_1$ and $Y_2$ and coefficients of linear expansion are ${\alpha }_1\text{ and }{\alpha }_2$ . The junction between the rods does not shift if the rods are cooled. Then

• A. $Y_1{\alpha }_1=Y_2{\alpha }_2$
• B. $Y_1{\alpha }_2=Y_2{\alpha }_1$
• C. $Y_1{\alpha }_1^2=Y_2{\alpha }_2^2$
• D. $Y_1^2{\alpha }_1=Y_2^2{\alpha }_2$

Answer 1

Answer: (A)

Thermal strain in a linear rod is given, $\Delta l=l\alpha t$
where $l$ is the length of the rod and $t$ is the temperature change.
Here $Y_1$ and $Y_2$ are Young's moduli of the material of the rods and ${\alpha }_1$ and ${\alpha }_2$ are the coefficients of thermal expansion.
Using Young's modulus, $Y=\frac{\text{stress}}{\text{strain}}=\frac{T/A}{\alpha t}$
$\Rightarrow T=YA\alpha t$ , where $T$ is the tension in rod.
Tension must be the same in both the rods for their junction to be in equilibrium.
$\therefore Y_{1}A{\alpha }_{1}t=Y_{2}A{\alpha }_{2}t\Rightarrow Y_1{\alpha }_1=Y_2{\alpha }_2$