Question

The elastic energy stored per unit volume in terms of longitudinal strain \( e \) and Young’s modulus \( Y \) is:

• A. \( \frac{Y e^2}{2} \)
• B. \( \frac{1}{2} Y e \)
• C. \( 2 Y e^2 \)
• D. \( 2 Y e \)

Hint:

Elastic potential energy per unit volume is given by \( U = \frac{1}{2} \sigma e \). Using Hooke’s Law \( \sigma = Y e \), we derive \( U = \frac{Y e^2}{2} \).

Answer 1

The Correct Option is A

Step 1: Elastic Potential Energy Per Unit Volume
The elastic potential energy stored per unit volume (also called strain energy density) is given by: \[ U = \frac{1}{2} \sigma e. \] where: - \( \sigma \) is the stress,
- \( e \) is the strain.
Step 2: Expressing Stress in Terms of Young's Modulus
From Hooke’s Law, \[ \sigma = Y e. \] Substituting this into the strain energy equation: \[ U = \frac{1}{2} Y e \cdot e. \] \[ U = \frac{1}{2} Y e^2. \] Step 3: Conclusion
Thus, the elastic energy stored per unit volume is: \[ \frac{Y e^2}{2}. \]