A wire fixed at the upper end stretches by length l by applying a force F. The work done in stretching is

Options

A. `(F)/(2 1)`
B. F 1
C. 2 F 1
D. `(F1)/(2)`

Explanation

To find the work done in stretching a wire that is fixed at one end and stretches by a length \( l \) when a force \( F \) is applied, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Work Done**:
The work done \( W \) in stretching a wire can be expressed in terms of stress and strain. The formula for work done in stretching a wire is:
\[ W = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times \text{Volume} \]

2. **Define Stress and Strain**:
- **Stress** is defined as the force applied per unit area. Mathematically, it is given by:
\[ \text{Stress} = \frac{F}{A} \]
where \( F \) is the force applied and \( A \) is the cross-sectional area of the wire.
- **Strain** is defined as the change in length divided by the original length. It is given by:
\[ \text{Strain} = \frac{\Delta L}{L} \]
where \( \Delta L \) is the change in length (which is \( l \) in this case) and \( L \) is the original length of the wire.

3. **Calculate the Volume of the Wire**:
The volume \( V \) of the wire can be expressed as:
\[ V = A \times L \]

4. **Substitute Stress, Strain, and Volume into the Work Done Formula**:
Now, substituting the expressions for stress, strain, and volume into the work done formula:
\[ W = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{l}{L}\right) \times (A \times L) \]

5. **Simplify the Expression**:
In the equation, the area \( A \) cancels out:
\[ W = \frac{1}{2} \times F \times \frac{l}{L} \times L \]
This simplifies to:
\[ W = \frac{1}{2} \times F \times l \]

6. **Final Result**:
Thus, the work done in stretching the wire is:
\[ W = \frac{F \cdot l}{2} \]

### Conclusion:
The work done in stretching the wire by a length \( l \) when a force \( F \) is applied is given by:
\[ W = \frac{F \cdot l}{2} \]