Question

A metal rod of length ${}^{\prime }{L}^{\prime }$ and cross-sectional area ${}^{\prime }{A}^{\prime }$ is heated through ${}^{\prime }{T}^{\prime \circ }C$. What is the force required to prevent the expansion of the rod lengthwise?
$[Y=$ Young's modulus of the material of rod, $\alpha -$ coefficient of linear expansion]

• A. $\frac{YA\alpha T}{(1-\alpha T)}$
• B. $\frac{YA\alpha T}{(1+\alpha T)}$
• C. $\frac{(1-\alpha T)}{YA\alpha T}$
• D. $\frac{(1+\alpha T)}{YA\alpha T}$

Answer 1

The correct option is B $\frac{YA\alpha T}{(1+\alpha T)}$

change in length $\delta L=L\alpha T$

final length $L_f=L+\delta L$

now in order to to constrain the expansion ,there should be force that should compress the rod by $\delta L$

$strain=\frac{\delta L}{L_f}$

$Y=\frac{F/A}{strain}$

$F=\frac{YA\alpha T}{1+\alpha T}$