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

Concept: The elastic energy density (\(u\)), which represents the structural potential energy stored per unit volume inside a deformed elastic medium, is fundamentally defined as: \[ u = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2} \cdot \sigma \cdot \varepsilon \] From Hooke's Law, Young's Modulus (\(Y\)) maps stress directly to strain via the linear equation: \[ \sigma = Y \cdot \varepsilon \quad \implies \quad \varepsilon = \frac{\sigma}{Y} \]

Step 1: Substituting the strain variable to match stress form.
Because the problem requests the solution expressed explicitly in terms of stress (\(\sigma\)) and Young's modulus (\(Y\)), we replace the strain parameter (\(\varepsilon\)) using our Hooke's Law identity: \[ u = \frac{1}{2} \cdot \sigma \cdot \left( \frac{\sigma}{Y} \right) \]

Step 2: Simplifying the expression layout.
Combining the numerator products: \[ u = \frac{\sigma^2}{2Y} \] This perfectly derives the identity corresponding to Option (A).