Question

A metallic rod of length L, area of cross-section A, Young's modulus Y has coefficient of linear expansion a. If the rod is heated through a temperature T, the energy stored per unit volume will be:

• A. $\frac 12 Y \alpha T$
• B. $\frac 12 Y\alpha ^{2} T^{2}$
• C. $\frac 12 Y \alpha T^2$
• D. $\frac 12 Y ^{2}\alpha T^{2}$

Answer 1

Answer: (B)

$\frac 12 Y\alpha ^{2} T^{2}$

Answer 2

We know that the energy stored per unit volume $=\frac{1}{2}$ (stress)(strain) = $\frac{1}{2}( Y )(\text { strain })^{2}$ Now strain = fractional change in length $=\alpha T$ (using thermal expansion formula) So energy stored per unit volume $=\frac{1}{2} Y \alpha^{2} T ^{2}$