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. $lAY\alpha \Delta T$
• C. $AY\alpha \Delta T$
• D. $\frac{AY}{\alpha \Delta T}$

Answer 1

Answer: (C)

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