The fundamental frequency of a stretched string with a weight of `9kg` is `289Hz`. The weight required to produce its octave is

Options

A. `9kg wt`
B. `16 kg wt`
C. `25 kg wt`
D. `36kg wt`

Explanation

To solve the problem of finding the weight required to produce the octave of a stretched string, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between frequency and tension**: The frequency of a stretched string is given by the formula:
\[ f \propto \sqrt{T} \]
where \( f \) is the frequency and \( T \) is the tension in the string.

2. **Identify the fundamental frequency**: The fundamental frequency (first harmonic) of the string is given as \( f_1 = 289 \, \text{Hz} \).

3. **Determine the frequency for the octave**: The frequency of the octave is double the fundamental frequency:
\[ f_2 = 2 \times f_1 = 2 \times 289 \, \text{Hz} = 578 \, \text{Hz} \]

4. **Set up the ratio of frequencies**: Since the frequency is proportional to the square root of the tension, we can write:
\[ \frac{f_2}{f_1} = \sqrt{\frac{T_2}{T_1}} \]
Substituting the known frequencies:
\[ \frac{578}{289} = \sqrt{\frac{T_2}{T_1}} \]

5. **Calculate the ratio of tensions**: The ratio of the frequencies is:
\[ \frac{578}{289} = 2 \]
Squaring both sides gives:
\[ 4 = \frac{T_2}{T_1} \]
This implies:
\[ T_2 = 4 T_1 \]

6. **Relate tension to weight**: The tension in the string is equal to the weight hanging from it. Given that the original weight \( T_1 \) is \( 9 \, \text{kg} \), we can find \( T_2 \):
\[ T_2 = 4 \times T_1 = 4 \times 9 \, \text{kg} = 36 \, \text{kg} \]

7. **Conclusion**: The weight required to produce the octave is \( 36 \, \text{kg} \).

### Final Answer:
The weight required to produce its octave is **36 kg**.

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