Question

The elastic potential energy stored per unit volume (energy density) in a stretched string under a longitudinal tension stress $\sigma$ and material Young's modulus $Y$ is expressed as:

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

Hint:

Memorize the three symmetric expressions for elastic energy density to handle any variable pairing given in a question: \[ u = \frac{1}{2}(\text{Stress})(\text{Strain}) = \frac{1}{2}Y(\text{Strain})^2 = \frac{\text{Stress}^2}{2Y} \] This is completely analogous to electrostatics, where the energy density of an electric field is given by \(u = \frac{1}{2}\varepsilon_0 E^2 = \frac{D^2}{2}\varepsilon_0\)!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the mathematical expression for the elastic potential energy stored per unit volume (energy density) in a stretched string, in terms of stress \(\sigma\) and Young's modulus \(Y\).
Step 2: Key Formula or Approach:
The elastic potential energy density \(u\) is defined as:
\[ u = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2} \cdot \sigma \cdot \varepsilon \]
Young's modulus \(Y\) maps stress directly to strain according to Hooke's Law:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{\sigma}{\varepsilon} \]
Step 3: Detailed Explanation:
1. We express the strain \(\varepsilon\) in terms of stress \(\sigma\) and Young's modulus \(Y\):
\[ \varepsilon = \frac{\sigma}{Y} \]
2. Substitute this expression for strain into the energy density formula:
\[ u = \frac{1}{2} \cdot \sigma \cdot \left( \frac{\sigma}{Y} \right) \]
3. Combining the terms in the numerator and denominator:
\[ u = \frac{\sigma^2}{2Y} \]
4. This derived expression matches Option (A).
Step 4: Final Answer:
The elastic potential energy stored per unit volume is \( \frac{\sigma^2}{2Y} \), which corresponds to Option (A).