The fundamental frequency of a string stretched with a weight of 4 kg is 256 Hz . The weight required to produce its octave is

Options

A. 16 kg - wt
B. 12 kg - wt
C. 24 kg - wt
D. 8 kg - wt

Answer

To solve the problem, we need to find the weight required to produce the octave of a string that has a fundamental frequency of 256 Hz when stretched with a weight of 4 kg.

### Step-by-Step Solution:

1. **Understand the Concept of Octave**:
- An octave means doubling the frequency. Therefore, if the fundamental frequency (n) is 256 Hz, the frequency for the octave (n') will be:
\[ n' = 2 \times n = 2 \times 256 \text{ Hz} = 512 \text{ Hz} \]

2. **Frequency and Tension Relationship**:
- The frequency of a vibrating string is directly proportional to the square root of the tension (T) in the string:
\[ n \propto \sqrt{T} \]
- This can be expressed as:
\[ \frac{n'}{n} = \sqrt{\frac{T'}{T}} \]
- Where \( T' \) is the tension corresponding to the new frequency \( n' \), and \( T \) is the tension corresponding to the original frequency \( n \).

3. **Substituting Values**:
- We know:
\[ n' = 512 \text{ Hz}, \quad n = 256 \text{ Hz} \]
- Therefore:
\[ \frac{512}{256} = \sqrt{\frac{T'}{T}} \]
- Simplifying the left side:
\[ 2 = \sqrt{\frac{T'}{T}} \]

4. **Squaring Both Sides**:
- To eliminate the square root, we square both sides:
\[ 2^2 = \frac{T'}{T} \]
- This gives us:
\[ 4 = \frac{T'}{T} \]
- Therefore, we can express \( T' \) in terms of \( T \):
\[ T' = 4T \]

5. **Calculating the Original Tension**:
- The original tension \( T \) is produced by a weight of 4 kg. The tension due to a weight is given by:
\[ T = mg \]
- Where \( m = 4 \text{ kg} \) and \( g \) (acceleration due to gravity) is approximately \( 9.8 \text{ m/s}^2 \). Thus:
\[ T = 4 \text{ kg} \times 9.8 \text{ m/s}^2 = 39.2 \text{ N} \]

6. **Finding the New Tension**:
- Now substituting back to find \( T' \):
\[ T' = 4T = 4 \times 39.2 \text{ N} = 156.8 \text{ N} \]

7. **Calculating the Required Weight**:
- To find the weight corresponding to the new tension \( T' \):
\[ T' = m'g \quad \Rightarrow \quad m' = \frac{T'}{g} \]
- Substituting the values:
\[ m' = \frac{156.8 \text{ N}}{9.8 \text{ m/s}^2} \approx 16 \text{ kg} \]

### Conclusion:
The weight required to produce the octave is **16 kg**.