Question

A metallic rod of length 'L' and cross-section 'A' has Young's modulus 'Y' and coefficient of linear expansion $\alpha$. If the rod is heated to a temperature 'T', then the energy stored per unit volume is:

• A. $\frac{1}{2}Y{\alpha }^2{T}^2$
• B. $\frac{1}{2}YA{\alpha }^2{T}^2$
• C. $\frac{1}{2}YA\alpha T$
• D. $\frac{1}{2}Y{A}^2{\alpha }^2{T}^2$

Answer 1

Answer: (A)

$\frac{1}{2}Y{\alpha }^2{T}^2$

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