Question

A uniform cylindrical rod of length $L$ and radius $r$, is made from a material whose Young's modulus of Elasticity equals $Y$. When this rod is heated by temperature $T$ and simultaneously subjected to a net longitudinal compressional force $F$, its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equals to :

• A. $F /\left(3 \pi r ^{2} YT \right)$
• B. $3 F /\left(\pi r ^{2} YT \right)$
• C. $6 F /\left(\pi r ^{2} YT \right)$
• D. $9 F /\left(\pi r ^{2} YT \right)$

Answer 1

Answer: (B)

$3 F /\left(\pi r ^{2} YT \right)$

Answer 2

$\therefore$ Length of cylinder remains unchanged
so $\bigg(\frac{F}{A}\bigg)_{Compressive} \, \, = \bigg(\frac{F}{A}\bigg)_{Thermal}$
$\frac{F}{\pi r^2} \, Y \alpha T$
($\alpha$ is linear coefficient of expansion)
$\therefore \, \, \alpha \frac{F}{YT \pi r^2}$
$\therefore$ The coefficient of volume expansion $\gamma$ = $3\alpha$
$\therefore \, \, \gamma \, \, = \, \, 3 \frac{F}{YT \pi r^2}$