If the frequencies of the sound second and fifth harmonics of a string differ by 54 Hz. What is the fundamental frequency of the string ?

Explanation

To solve the problem of finding the fundamental frequency of a string given that the frequencies of the second and fifth harmonics differ by 54 Hz, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Harmonics**:
- The fundamental frequency of a string is denoted as \( f \).
- The frequency of the second harmonic (2nd harmonic) is given by \( f_2 = 2f \).
- The frequency of the fifth harmonic (5th harmonic) is given by \( f_5 = 5f \).

2. **Set Up the Equation**:
- According to the problem, the difference between the frequencies of the fifth and second harmonics is 54 Hz. Therefore, we can write the equation:
\[ f_5 - f_2 = 54 \text{ Hz} \]

3. **Substitute the Harmonic Frequencies**:
- Substitute the expressions for \( f_2 \) and \( f_5 \) into the equation:
\[ 5f - 2f = 54 \]

4. **Simplify the Equation**:
- Combine like terms:
\[ 3f = 54 \]

5. **Solve for the Fundamental Frequency**:
- Divide both sides by 3 to find \( f \):
\[ f = \frac{54}{3} = 18 \text{ Hz} \]

6. **Conclusion**:
- The fundamental frequency of the string is \( 18 \text{ Hz} \).

### Final Answer:
The fundamental frequency of the string is \( 18 \text{ Hz} \).

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