Question

When a wire made of material with Young's modulus Y is subjected to a stress S, the elastic potential energy per unit volume stored in the wire is

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

Hint:

- Stress (\(S\)) = Force / Area - Strain (\(\epsilon\)) = Change in length / Original length (\( \Delta L / L \)) - Young's Modulus (\(Y\)) = Stress / Strain = \( S/\epsilon \) - Elastic Potential Energy per unit volume (Energy Density) \( U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \). Substitute relations between \(S, \epsilon, Y\) to get different forms: \( U = \frac{1}{2} S \epsilon = \frac{S^2}{2Y} = \frac{1}{2} Y \epsilon^2 \).

Answer 1

The Correct Option is C

Young's modulus \( Y = \frac{\text{Stress}}{\text{Strain}} \).
Let Stress be \(S\) and Strain be \( \epsilon \).
So, \( Y = \frac{S}{\epsilon} \implies \epsilon = \frac{S}{Y} \).
Elastic potential energy per unit volume (Energy Density, U) stored in a stretched wire is given by: \[ U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \] Substitute the expressions for Stress and Strain: \[ U = \frac{1}{2} \times S \times \epsilon \] Substitute \( \epsilon = \frac{S}{Y} \): \[ U = \frac{1}{2} \times S \times \frac{S}{Y} = \frac{S^2}{2Y} \] This matches option (3).
Other forms of energy density: Using \( S = Y\epsilon \): \( U = \frac{1}{2} (Y\epsilon) \epsilon = \frac{1}{2}Y\epsilon^2 \).