Question

A uniform metal rod fixed at its ends of $2m{m}^2$ cross-section is cooled from ${40}^{\circ }C$ to ${20}^{\circ }C$. The coefficient of the linear expansion of the rod is $12\times {10}^{-6}$per degree celsius and its Young's modulus of elasticity is ${10}^{11}N/m^2$. The energy stored per unit volume of the rod is

• A. $2880J/m^3$
• B. $1500J/m^3$
• C. $5760J/m^3$
• D. $1440J/m^3$

Answer 1

The correct option is A $2880J/m^3$

$\Delta l=l\alpha \Delta \Theta$

Strain$=\frac{\Delta l}{l}=\left(\alpha \Delta \Theta \right)$

Stress$=Y\times Strain=Y\alpha \Delta \Theta$

Energy stored per unit volume
$=\frac{1}{2}\times Stress\times Strain$
$=\frac{1}{2}\times Y\times (\alpha \Delta \Theta {)}^2$
$=\frac{1}{2}\times {10}^{11}\times (12\times {10}^{-6}\times 20{)}^2$
$=2880J/m^2$