A string when stretched with a force of 18 kg it produces a note of frequency 652 Hz. What is the force required to produce an octave of the note?

Answer

To solve the problem, we need to find the force required to produce an octave of the note given that the original force produces a frequency of 652 Hz.

### Step-by-Step Solution:

1. **Understand the Concept of Octave**:
An octave means that the frequency of the new note is double that of the original note. Therefore, if the original frequency \( f \) is 652 Hz, the frequency of the octave \( f' \) will be:
\[ f' = 2 \times f = 2 \times 652 \, \text{Hz} = 1304 \, \text{Hz} \]

2. **Relationship Between Frequency and Tension**:
The frequency of a vibrating string is given by the formula:
\[ f \propto \sqrt{T} \]
where \( T \) is the tension (or force) in the string. This means that the frequency is directly proportional to the square root of the tension.

3. **Set Up the Proportionality**:
From the relationship above, we can express the ratio of the frequencies and the tensions:
\[ \frac{f'}{f} = \sqrt{\frac{T'}{T}} \]
where \( T' \) is the new tension required to produce the octave.

4. **Substituting Values**:
We know:
- \( f = 652 \, \text{Hz} \)
- \( f' = 1304 \, \text{Hz} \)
- \( T = 18 \, \text{kg} \) (the original force)

Substituting these values into the equation gives:
\[ \frac{1304}{652} = \sqrt{\frac{T'}{18}} \]

5. **Calculate the Ratio**:
Simplifying the left side:
\[ \frac{1304}{652} = 2 \]
Thus, we have:
\[ 2 = \sqrt{\frac{T'}{18}} \]

6. **Square Both Sides**:
Squaring both sides to eliminate the square root:
\[ 4 = \frac{T'}{18} \]

7. **Solve for \( T' \)**:
Multiplying both sides by 18 gives:
\[ T' = 4 \times 18 = 72 \, \text{kg} \]

### Final Answer:
The force required to produce an octave of the note is **72 kg**.

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