To solve the problem of how the original fundamental frequency \( n \) relates to the fundamental frequencies \( n_1, n_2, \) and \( n_3 \) of the three parts of a string, we can follow these steps:
### Step 1: Write the formula for the fundamental frequency of a string
The fundamental frequency \( n \) of a vibrating 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 in the string,
- \( \mu \) is the mass per unit length of the string.
### Step 2: Express the length of the string in terms of the frequencies
If the string is cut into three parts with lengths \( L_1, L_2, \) and \( L_3 \), then the total length \( L \) can be expressed as:
\[
L = L_1 + L_2 + L_3
\]
### Step 3: Relate the lengths to the frequencies
Using the frequency formula, we can express the lengths of each part in terms of their respective frequencies:
\[
L_1 = \frac{K}{n_1}, \quad L_2 = \frac{K}{n_2}, \quad L_3 = \frac{K}{n_3}
\]
where \( K = \frac{1}{\sqrt{\frac{T}{\mu}}} \).
### Step 4: Substitute the lengths into the total length equation
Substituting the expressions for \( L_1, L_2, \) and \( L_3 \) into the total length equation gives:
\[
L = \frac{K}{n_1} + \frac{K}{n_2} + \frac{K}{n_3}
\]
### Step 5: Factor out \( K \)
Factoring out \( K \) from the right-hand side, we have:
\[
L = K \left( \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \right)
\]
### Step 6: Relate the original frequency to the total length
Now, we can express the original frequency \( n \) in terms of the total length \( L \):
\[
n = \frac{1}{2L} \sqrt{\frac{T}{\mu}}
\]
### Step 7: Substitute \( L \) into the frequency equation
Substituting the expression for \( L \) into the frequency equation gives:
\[
n = \frac{1}{2 \left( K \left( \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \right) \right)} \sqrt{\frac{T}{\mu}}
\]
### Step 8: Simplify the equation
Since \( K \) cancels out, we can simplify this to:
\[
\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}
\]
### Final Result
Thus, the relationship between the original fundamental frequency \( n \) and the fundamental frequencies \( n_1, n_2, n_3 \) of the three parts is given by:
\[
\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}
\]
