Q. A uniform metal rod of $2\ mm^{2}$ cross-section is heated from $0^{0}C\ to\ 20^{0}C$. The coefficient of the linear expansion of the rod is $12 \times 10^{- 6}/{^\circ}C.\ $ Its Young’s modulus of elasticity is $10^{11}\ {Nm}^{- 2}.\ $ The energy stored per unit volume of the rod is
A) $1440\ {Jm}^{- 3}$
B) $15750\ {Jm}^{- 3}$
C) $1500\ {Jm}^{- 3}$
D) $2880\ {Jm}^{- 3}$
Explanation
(d) Energy stored per unit volume is given by
$W = \frac{Y \times (strain)^{2}}{2}$
$= \frac{10^{11}}{2} \times \left( \frac{change\ in\ length}{original\ length} \right)^{2}$
where Y is Young’s modulus
$= \frac{10^{11}}{2}\left( \frac{\propto L\mathrm{\Delta}\theta}{L} \right)^{2}$
$= \frac{10^{11}}{2}(12 \times 10^{- 6} \times 20)^{2} = 2880\ {Jm}^{- 3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.