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

Step 1: Understanding the Concept:
When a material is subjected to stress, it deforms and elastic potential energy is stored in it.
The stored strain energy per unit volume (energy density) can be expressed in terms of stress ($\tau$) and strain.
Step 2: Key Formula or Approach:
Energy stored $U = (\text{Energy Density}) \times \text{Volume}$.
$\text{Energy Density} = \frac{1}{2} \times \text{Stress} \times \text{Strain}$.
Young's Modulus $Y = \frac{\text{Stress}}{\text{Strain}} \implies \text{Strain} = \frac{\text{Stress}}{Y}$.
Step 3: Detailed Explanation:
Substitute the strain into the energy density formula:
\[ \text{Energy Density} = \frac{1}{2} \times \tau \times \frac{\tau}{Y} = \frac{\tau^2}{2Y} \]
The total elastic potential energy $U$ stored in the rod is Energy Density multiplied by the total volume $V$:
\[ U = \frac{\tau^2}{2Y} \times V = \frac{1}{2} \frac{\tau^2 V}{Y} \]
Step 4: Final Answer:
The stored elastic energy is $\frac{1}{2}\frac{\tau^2\text{V}}{\text{Y}}$.