Question

A rod, of length $L$ at room temperature and uniform area of cross-section $A$, is made of a metal having coefficient of linear expansion $\alpha {}^{\circ }{\text{C}}^{-1}$. It is observed that an external compressive force $F$, is applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by $\Delta T\text{K}$. Young's modulus, $Y$, for this metal is:

• A. $\frac{F}{A\alpha \Delta T}$
• B. $\frac{2F}{A\alpha \Delta T}$
• C. $\frac{F}{2A\alpha \Delta T}$
• D. $\frac{F}{A\alpha (\Delta T-273)}$

Answer 1

The correct option is A $\frac{F}{A\alpha \Delta T}$

$\text{Young's modulus},Y=\frac{\text{stress}}{\text{strain}}$

$Y=\frac{\left(\frac{F}{A}\right)}{\left(\frac{\Delta l}{l}\right)}$

Using, coefficient of linear expansion,

$\alpha =\frac{\Delta l}{l\Delta T}\Rightarrow \frac{\Delta l}{l}=\alpha \Delta T$

$\therefore Y=\frac{F}{A\alpha \Delta T}$

Hence, option $\left(\text{A}\right)$ is correct.