A wire of cross section `A` is stretched horizontally between two clamps located `2lm` apart. A weight `W kg` is suspended from the mid-point of the wire. If the mid-point sags vertically through a distance `x lt 1` the strain produced is

Options

A. `(2x^(2))/(l^(2))`
B. `(x^(2))/(l^(2))`
C. `(x^(2))/(2l^(2))`
D. none of these

Explanation

To solve the problem, we need to determine the strain produced in the wire when a weight is suspended from its midpoint, causing it to sag a certain distance.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a wire of cross-sectional area \( A \) stretched horizontally between two clamps that are \( 2L \) meters apart.
- A weight \( W \) kg is suspended from the midpoint of the wire, causing it to sag vertically by a distance \( x \).

2. **Identifying the Lengths**:
- The original length of the wire between the clamps is \( 2L \).
- When the weight is applied, the wire sags downwards, creating two segments of the wire from the clamps to the weight. Each segment can be considered as a hypotenuse of a right triangle where:
- The horizontal leg is \( L \) (half of the total length).
- The vertical leg is \( x \) (the sag).

3. **Calculating the New Length of the Wire**:
- The length of each segment of the wire can be calculated using the Pythagorean theorem:
\[ \text{Length of each segment} = \sqrt{L^2 + x^2} \]
- Since there are two segments, the total length of the wire when the weight is applied is:
\[ \text{Total length} = 2 \times \sqrt{L^2 + x^2} \]

4. **Calculating the Change in Length**:
- The change in length \( \Delta L \) of the wire due to the sag is given by:
\[ \Delta L = \text{Total length with weight} - \text{Original length} \]
- Substituting the values:
\[ \Delta L = 2\sqrt{L^2 + x^2} - 2L \]

5. **Calculating the Strain**:
- Strain is defined as the change in length divided by the original length:
\[ \text{Strain} = \frac{\Delta L}{\text{Original length}} = \frac{2\sqrt{L^2 + x^2} - 2L}{2L} \]
- Simplifying this gives:
\[ \text{Strain} = \frac{\sqrt{L^2 + x^2} - L}{L} \]

6. **Final Expression for Strain**:
- Therefore, the strain produced in the wire is:
\[ \text{Strain} = \frac{\sqrt{L^2 + x^2} - L}{L} \]