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) equal to :

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

Answer 1

The correct option is C $\frac{3F}{\left(\pi r^{2}YT\right)}$

The thermal stress produced in the rod is,
${\sigma }_{th}=Y\alpha \Delta T=Y\alpha T$

Also, $\sigma =\frac{F}{A}=\frac{F}{\pi r^2}$

$\Rightarrow \frac{F}{\pi r^2}=Y\alpha T$

$\Rightarrow \alpha =\frac{F}{\pi r^{2}YT}$

The coefficient of volume expansion is given by,
$\gamma =3\alpha =\frac{3F}{\pi r^{2}YT}$

Hence, $\left(C\right)$ is the correct answer.