If `f_(1) and f_(2)` be the fundamental frequencies of the two segments into which a stretched string is divided by means of a bridge , then find the original fundamental frequency `f` of the complete string.

Options

A. `(f_1f_2)/(f_1+f_2) = f`
B. `2f=f_1 +f_2`
C. `sqrt(f)=sqrt(f_1)+sqrt(f_2)`
D. `sqrt(f_1+f_2)=2f`

Answer

To find the original fundamental frequency \( f \) of the complete string given the fundamental frequencies \( f_1 \) and \( f_2 \) of the two segments, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between frequency and length**:
The frequency of a vibrating string is inversely proportional to its length. This means that if a string has a longer length, it will have a lower frequency.

\[ f \propto \frac{1}{L} \]

2. **Express frequencies in terms of lengths**:
For the two segments of the string, we can express the frequencies \( f_1 \) and \( f_2 \) in terms of their respective lengths \( L_1 \) and \( L_2 \):

\[ f_1 = \frac{k}{L_1} \quad \text{and} \quad f_2 = \frac{k}{L_2} \]

Here, \( k \) is a constant.

3. **Express the total length**:
The total length \( L \) of the string is the sum of the lengths of the two segments:

\[ L = L_1 + L_2 \]

4. **Express the frequency of the complete string**:
The frequency \( f \) of the complete string can be expressed as:

\[ f = \frac{k}{L} \]

5. **Substitute lengths in terms of frequencies**:
From the expressions for \( f_1 \) and \( f_2 \), we can express \( L_1 \) and \( L_2 \) in terms of \( f_1 \) and \( f_2 \):

\[ L_1 = \frac{k}{f_1} \quad \text{and} \quad L_2 = \frac{k}{f_2} \]

6. **Substitute into the total length equation**:
Substitute \( L_1 \) and \( L_2 \) into the equation for \( L \):

\[ L = \frac{k}{f_1} + \frac{k}{f_2} \]

7. **Factor out \( k \)**:
This gives us:

\[ L = k \left( \frac{1}{f_1} + \frac{1}{f_2} \right) \]

8. **Substitute back into the frequency equation**:
Now substitute this expression for \( L \) back into the equation for \( f \):

\[ f = \frac{k}{L} = \frac{k}{k \left( \frac{1}{f_1} + \frac{1}{f_2} \right)} = \frac{1}{\left( \frac{1}{f_1} + \frac{1}{f_2} \right)} \]

9. **Simplify the expression**:
This can be rewritten using the formula for the sum of fractions:

\[ f = \frac{f_1 f_2}{f_1 + f_2} \]

Thus, the original fundamental frequency \( f \) of the complete string is given by:

\[ f = \frac{f_1 f_2}{f_1 + f_2} \]