A string is cut into three parts, having fundamental frequencies `n_(1),n_(2)` and `n_(3)` respectively. Then original fundamental frequency 'n' related by the expression as (other quantities are identical) :-

Options

A. `(1)/(n)=(1)/(n_(1))+(1)/(n_(2))+(1)/(n_(3))`
B. `n=n_(1) xxn_(2) xxn_(3)`
C. `n=n_(1)+n_(2)+n_(3)`
D. `n=(n_(1)+n_(2)+n_(3))/(3)`

Explanation

To solve the problem, we need to relate the fundamental frequencies of the original string and the three parts after it has been cut.

### Step-by-Step Solution:

1. **Understanding the Fundamental Frequency**:
The fundamental frequency \( n \) of a string is given by the formula:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( L \) is the length of the string, \( T \) is the tension, and \( \mu \) is the mass per unit length.

2. **Cutting the String into Parts**:
Let the original string of length \( L \) be cut into three parts with lengths \( L_1, L_2, \) and \( L_3 \). The fundamental frequencies of these parts are \( n_1, n_2, \) and \( n_3 \) respectively.

3. **Relating Lengths and Frequencies**:
The fundamental frequency of each part can be expressed as:
\[ n_1 = \frac{1}{2L_1} \sqrt{\frac{T}{\mu}}, \quad n_2 = \frac{1}{2L_2} \sqrt{\frac{T}{\mu}}, \quad n_3 = \frac{1}{2L_3} \sqrt{\frac{T}{\mu}} \]
From these equations, we can express the lengths in terms of the frequencies:
\[ L_1 = \frac{T}{2\mu n_1^2}, \quad L_2 = \frac{T}{2\mu n_2^2}, \quad L_3 = \frac{T}{2\mu n_3^2} \]

4. **Total Length of the String**:
The total length of the original string is:
\[ L = L_1 + L_2 + L_3 \]
Substituting the expressions for \( L_1, L_2, \) and \( L_3 \):
\[ L = \frac{T}{2\mu n_1^2} + \frac{T}{2\mu n_2^2} + \frac{T}{2\mu n_3^2} \]

5. **Factoring Out Common Terms**:
We can factor out \( \frac{T}{2\mu} \):
\[ L = \frac{T}{2\mu} \left( \frac{1}{n_1^2} + \frac{1}{n_2^2} + \frac{1}{n_3^2} \right) \]

6. **Relating Original Frequency to Parts**:
The original frequency \( n \) can also be expressed in terms of the total length \( L \):
\[ n = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Substituting \( L \) from the previous step:
\[ n = \frac{1}{2 \left( \frac{T}{2\mu} \left( \frac{1}{n_1^2} + \frac{1}{n_2^2} + \frac{1}{n_3^2} \right) \right)} \sqrt{\frac{T}{\mu}} \]

7. **Simplifying the Expression**:
After simplification, we find:
\[ \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \]

### Conclusion:
Thus, the relationship between the original fundamental frequency \( n \) and the frequencies \( n_1, n_2, n_3 \) of the cut parts is given by:
\[ \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \]