Q. 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 :
Solution:
$\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 $\gamma$ = $3\alpha$
$\therefore \, \, \gamma \, \, = \, \, 3 \frac{F}{YT \pi r^2}$
