A sonometer wire is divided in many segments using bridges. If fundamental natural frequencies of the segments are `n_(1), n_(2), n_(3)…..` then the fundamental natural frequency of entire sonometer wire will be (if the divisons were not made):

Options

A. `n = n_(1) + n_(2) + n_(3) + …..`
B. `n + sqrt(n_(1) xx n_(2) xx n_(3) xx….)`
C. `(1)/(n) = (1)/(n_(1)) + (1)/(n_(2)) + (1)/(n_(3)) + …..`
D. none of the above

Explanation

To find the fundamental natural frequency of the entire sonometer wire when it is divided into segments with fundamental natural frequencies \( n_1, n_2, n_3, \ldots \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Sonometer Wire**:
- A sonometer wire is a stretched wire that can vibrate to produce sound. It is typically fixed at one end and has weights applied to the other end to create tension.

2. **Identify the Lengths of Segments**:
- Let the total length of the wire be \( L \). When divided into segments, let the lengths of these segments be \( L_1, L_2, L_3, \ldots \). Therefore, we have:
\[ L = L_1 + L_2 + L_3 + \ldots \]

3. **Frequency Formula**:
- The fundamental frequency \( n \) for a segment of wire is given by the formula:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{M}} \]
- Here, \( T \) is the tension in the wire, and \( M \) is the mass of the wire segment.

4. **Relate Frequencies of Segments**:
- For each segment, the frequency can be expressed as:
\[ n_1 = \frac{1}{2L_1} \sqrt{\frac{T}{M_1}}, \quad n_2 = \frac{1}{2L_2} \sqrt{\frac{T}{M_2}}, \quad n_3 = \frac{1}{2L_3} \sqrt{\frac{T}{M_3}}, \ldots \]

5. **Velocity of Wave**:
- The speed of the wave \( v \) on the wire can be expressed as:
\[ v = \sqrt{\frac{T}{M}} \]
- Since the tension \( T \) is constant for all segments, we can relate the lengths and frequencies.

6. **Combine Frequencies**:
- From the frequency formula, we can express the lengths in terms of frequencies:
\[ L_1 = \frac{v}{2n_1}, \quad L_2 = \frac{v}{2n_2}, \quad L_3 = \frac{v}{2n_3}, \ldots \]
- Thus, the total length \( L \) can be expressed as:
\[ L = \frac{v}{2n_1} + \frac{v}{2n_2} + \frac{v}{2n_3} + \ldots \]

7. **Factor Out Common Terms**:
- Factoring out \( \frac{v}{2} \):
\[ L = \frac{v}{2} \left( \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} + \ldots \right) \]

8. **Finding the Overall Frequency**:
- The overall frequency \( n \) of the entire wire can be expressed as:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{M}} \]
- Substituting the expression for \( L \):
\[ n = \frac{1}{\frac{v}{2} \left( \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} + \ldots \right)} = \frac{v}{2} \cdot \frac{1}{\left( \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} + \ldots \right)} \]

9. **Final Expression**:
- Thus, the fundamental natural frequency of the entire sonometer wire is given by:
\[ \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} + \ldots \]

### Conclusion:
The fundamental natural frequency of the entire sonometer wire when divided into segments is given by the reciprocal of the sum of the reciprocals of the individual segment frequencies.