Question

A rod has volume \(V\) and Young's modulus \(Y\) and is subjected to stress \(\tau\). Find elastic energy stored in the rod.

• A. \( \dfrac{1}{2}\dfrac{\tau^2 V}{Y} \)
• B. \( \dfrac{1}{2}\dfrac{\tau V}{Y} \)
• C. \( \dfrac{1}{2}\dfrac{\tau V}{Y^2} \)
• D. \( \dfrac{1}{2}\dfrac{\tau V^2}{Y} \)

Hint:

Elastic energy density in a stretched body: \[ u = \frac{1}{2}\times \text{stress} \times \text{strain} \] Total energy = energy density \(\times\) volume.

Answer 1

The Correct Option is A

Concept: Elastic potential energy stored in a stretched material is \[ U = \frac{1}{2}\times \text{Stress} \times \text{Strain} \times \text{Volume} \] Also, \[ Y = \frac{\text{Stress}}{\text{Strain}} \]
Step 1: Find strain. \[ \text{Strain} = \frac{\text{Stress}}{Y} \] \[ \text{Strain} = \frac{\tau}{Y} \]
Step 2: Substitute in energy expression. \[ U = \frac{1}{2} \times \tau \times \frac{\tau}{Y} \times V \] \[ U = \frac{1}{2}\frac{\tau^2 V}{Y} \] \[ \boxed{U = \frac{1}{2}\frac{\tau^2 V}{Y}} \]