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 $\alpha {/}^{\circ }C$. 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 $\Delta TK$. 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 $\frac{F}{A\alpha \Delta T}$

We know that:

$Y=\frac{\text{stress}}{\text{strain}},\frac{\Delta l}{l}=\alpha \Delta T$

Expansion due to heat caused is balanced by compression cause by compressive force $F$.

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

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

$=\frac{F}{A\left(\alpha \Delta T\right)}$

Final Answer; (c)