A string of length l is divided into four segments of fundamental frequencies `f_(1), f_(2), f_(3), f_(4)` respectively. The original fundamental frequency f of the string is given by

Options

A. `f^(2) = f_(1)^(2) + f_(2)^(2) + f_(3)^(2) + f_(4)^(4)`
B. `(1)/(sqrt(f)) = (1)/(sqrt(f_(1))) + (1)/(sqrt(f_(2))) + (1)/(sqrt(f_(3))) + (1)/(sqrt(f_(3)))`
C. `(1)/(f) = (1)/(f_(1)) + (1)/(f_(2)) + (1)/(f_(3)) + (1)/(f_(4))`
D. `f = f_(1) + f_(2) + f_(3) + f_(4)`

Explanation

To find the original fundamental frequency \( f \) of a string that is divided into four segments with fundamental frequencies \( f_1, f_2, f_3, \) and \( f_4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Length and Frequency**:
The fundamental frequency \( f \) of a string is related to its length \( L \), tension \( T \), and linear mass density \( \nu \) by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\nu}} \]

2. **Divide the String into Segments**:
Let the lengths of the four segments be \( L_1, L_2, L_3, \) and \( L_4 \). The total length of the string can be expressed as:
\[ L = L_1 + L_2 + L_3 + L_4 \]

3. **Express Each Segment's Length in Terms of Its Frequency**:
For each segment, we can express the lengths in terms of their respective frequencies:
\[ L_1 = \frac{1}{2f_1} \sqrt{\frac{T}{\nu}}, \quad L_2 = \frac{1}{2f_2} \sqrt{\frac{T}{\nu}}, \quad L_3 = \frac{1}{2f_3} \sqrt{\frac{T}{\nu}}, \quad L_4 = \frac{1}{2f_4} \sqrt{\frac{T}{\nu}} \]

4. **Substitute into the Total Length Equation**:
Substitute the expressions for \( L_1, L_2, L_3, \) and \( L_4 \) into the total length equation:
\[ L = \frac{1}{2f_1} \sqrt{\frac{T}{\nu}} + \frac{1}{2f_2} \sqrt{\frac{T}{\nu}} + \frac{1}{2f_3} \sqrt{\frac{T}{\nu}} + \frac{1}{2f_4} \sqrt{\frac{T}{\nu}} \]

5. **Factor Out Common Terms**:
Notice that \( \sqrt{\frac{T}{\nu}} \) and \( 2 \) are common factors:
\[ L = \sqrt{\frac{T}{\nu}} \left( \frac{1}{2f_1} + \frac{1}{2f_2} + \frac{1}{2f_3} + \frac{1}{2f_4} \right) \]

6. **Solve for the Original Frequency \( f \)**:
Rearranging the equation gives:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \frac{1}{f_4} \]

7. **Final Result**:
Thus, the original fundamental frequency \( f \) of the string is given by:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \frac{1}{f_4} \]