Question

A compressive force, $F$ is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta T$. The net change in its length is zero. Let $l$ be the length of the rod, $A$ its area of cross-section, $Y$ its Young’s modulus, and $\alpha$ its coefficient of linear expansion. Then, $F$ is equal to :

• A. $l^2 Y \alpha \, \Delta T $
• B. $l A \, Y \alpha \, \Delta T $
• C. $A \, Y \alpha \, \Delta T $
• D. $\frac{A Y}{\alpha \Delta T } $

Answer 1

Answer: (C)

Net change in length = 0
Thermal Exp. = $l \propto \Delta t$
$y = \frac{F/A}{\Delta l / l}$
$\frac{\Delta l}{l} = \frac{F}{Ay}$
$\Delta l = \frac{Fl}{Ay}$
$\frac{Fl}{Ay} = l \propto \Delta t$
$F = Ay \propto \Delta t$