If `n_(1), n_(2 ) "and" n_(3)` are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency n of the string is given by

Options

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

Answer

To find the original fundamental frequency \( n \) of a string that is divided into three segments with fundamental frequencies \( n_1, n_2, \) and \( n_3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between frequency and length**:
The fundamental frequency of a vibrating string is inversely proportional to its length. This means that if you have a string of length \( L \), the frequency \( n \) can be expressed as:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the string and \( \mu \) is the mass per unit length.

2. **Express the frequencies of the segments**:
Let the lengths of the three segments be \( L_1, L_2, \) and \( L_3 \). The fundamental frequencies for these segments can be expressed as:
\[ n_1 = \frac{1}{2L_1} \sqrt{\frac{T}{\mu}}, \quad n_2 = \frac{1}{2L_2} \sqrt{\frac{T}{\mu}}, \quad n_3 = \frac{1}{2L_3} \sqrt{\frac{T}{\mu}} \]

3. **Write the relationship for the original string**:
The total length of the string is:
\[ L = L_1 + L_2 + L_3 \]
Therefore, the original fundamental frequency \( n \) of the entire string can be expressed as:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]

4. **Relate the original frequency to the segment frequencies**:
Since frequency is inversely proportional to length, we can write:
\[ \frac{1}{n} = \frac{2L}{\sqrt{T/\mu}} = \frac{2L_1}{\sqrt{T/\mu}} + \frac{2L_2}{\sqrt{T/\mu}} + \frac{2L_3}{\sqrt{T/\mu}} \]
This simplifies to:
\[ \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \]

5. **Final expression for the original frequency**:
Rearranging the equation gives us the expression for the original fundamental frequency \( n \):
\[ \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \]
Therefore, the original frequency \( n \) can be calculated using:
\[ n = \frac{1}{\frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}} \]