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

Answer 1

Answer: (B)

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