A wire of length L and cross sectional area A is made of a material of Young's modulus Y. If the wire is streched by an amount x, the work done is………………

Explanation

To find the work done when a wire of length \( L \) and cross-sectional area \( A \) is stretched by an amount \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's Modulus \( Y \) is defined as the ratio of stress to strain. Stress is the force applied per unit area, and strain is the relative change in length. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
where \( F \) is the force applied, \( A \) is the cross-sectional area, \( \Delta L \) is the change in length (which is \( x \)), and \( L \) is the original length.

2. **Calculating Strain**:
The strain \( \epsilon \) when the wire is stretched by \( x \) is given by:
\[ \text{Strain} = \frac{\Delta L}{L} = \frac{x}{L} \]

3. **Calculating Stress**:
The stress \( \sigma \) in the wire can be calculated using the formula:
\[ \text{Stress} = \frac{F}{A} \]
Rearranging the Young's Modulus equation gives us:
\[ F = Y \cdot \frac{A \cdot x}{L} \]

4. **Calculating Work Done**:
The work done \( W \) in stretching the wire can be calculated using the formula for work done under variable force:
\[ W = \frac{1}{2} \times \text{Force} \times \text{Displacement} \]
Here, the average force is \( \frac{F}{2} \) since the force increases linearly from 0 to \( F \) as the wire is stretched. Thus:
\[ W = \frac{1}{2} \cdot F \cdot x \]

5. **Substituting the Expression for Force**:
Substituting the expression for \( F \):
\[ W = \frac{1}{2} \cdot \left(Y \cdot \frac{A \cdot x}{L}\right) \cdot x \]

6. **Simplifying the Equation**:
Simplifying the equation gives:
\[ W = \frac{1}{2} \cdot Y \cdot \frac{A \cdot x^2}{L} \]

### Final Answer:
Thus, the work done in stretching the wire by an amount \( x \) is:
\[ W = \frac{Y \cdot A \cdot x^2}{2L} \]