When a string is divided into three segments of length `l_1,l_2` and `l_3` the fundamental frequencies of these three segments are `f_1,f_2` and `f_3` respectively. The original fundamental frequency `f` of the string is

Options

A. (a) `f = f_(1) + f_(2) + f_(3)`
B. (b) `1/f = 1/f_(1)` +`1/f_(2)` + `1/f_(3)`
C. (c) `1/sqrtf = 1/sqrtf_(1) + 1/sqrtf_(2) + 1/sqrtf_(3)`
D. (d) None of these

Explanation

To find the original fundamental frequency \( f \) of a string that has been divided into three segments of lengths \( l_1, l_2, \) and \( l_3 \) with fundamental frequencies \( f_1, f_2, \) and \( f_3 \) respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between frequency and length**:
The fundamental frequency \( f \) of a string is inversely proportional to its length \( L \). The relationship can be expressed as:
\[ f = \frac{V}{2L} \]
where \( V \) is the speed of the wave on the string.

2. **Express lengths in terms of frequencies**:
For each segment of the string, we can express the lengths \( l_1, l_2, \) and \( l_3 \) in terms of their respective frequencies:
\[ l_1 = \frac{V}{2f_1}, \quad l_2 = \frac{V}{2f_2}, \quad l_3 = \frac{V}{2f_3} \]

3. **Combine the lengths**:
The total length \( L \) of the original string can be expressed as:
\[ L = l_1 + l_2 + l_3 = \frac{V}{2f_1} + \frac{V}{2f_2} + \frac{V}{2f_3} \]

4. **Factor out common terms**:
We can factor out \( \frac{V}{2} \) from the right-hand side:
\[ L = \frac{V}{2} \left( \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \right) \]

5. **Substitute back into the frequency formula**:
Now, substituting this expression for \( L \) back into the frequency formula:
\[ f = \frac{V}{2L} = \frac{V}{2 \left( \frac{V}{2} \left( \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \right) \right)} \]

6. **Simplify the expression**:
This simplifies to:
\[ f = \frac{1}{\left( \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \right)} \]

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

### Final Answer:
The original fundamental frequency \( f \) of the string is given by:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \]