If x longitudinal strain is produced in a wire of Young's modulus y, then energy stored in the material of the wire per unit volume is

Options

A. `yx^(2)`
B. `2yx^(2)`
C. `(1)/(2)y^(2)x`
D. `(1)/(2)yx^(2)`

Answer

To find the energy stored in the material of a wire per unit volume when a longitudinal strain \( x \) is produced in a wire of Young's modulus \( y \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Longitudinal strain \( x \) is defined as the change in length per unit original length of the wire.
- Young's modulus \( y \) is defined as the ratio of stress to strain.

2. **Relate Stress and Strain**:
- Stress \( \sigma \) is defined as the force \( F \) applied per unit area \( A \):
\[ \sigma = \frac{F}{A} \]
- According to Young's modulus:
\[ y = \frac{\sigma}{x} \]
- Rearranging gives us:
\[ \sigma = y \cdot x \]

3. **Energy Stored Per Unit Volume**:
- The energy stored per unit volume \( U \) in a material under stress can be expressed as:
\[ U = \frac{1}{2} \cdot \text{stress} \cdot \text{strain} \]
- Substituting the expressions for stress and strain:
\[ U = \frac{1}{2} \cdot (y \cdot x) \cdot x \]

4. **Final Expression**:
- Simplifying the equation gives:
\[ U = \frac{1}{2} y x^2 \]

Thus, the energy stored in the material of the wire per unit volume is:
\[ U = \frac{1}{2} y x^2 \]