If a string is stretched with a weight 4 kg then the fundamental frequency is equal to 256 Hz. What weight is needed to produce its octave?

Options

A. 4 kg-wt
B. 12 kg-wt
C. 16 kg-wt
D. 24 kg-wt

Explanation

To solve the problem step by step, we will use the relationship between the fundamental frequency of a stretched string and the tension in the string.

### Step 1: Understand the relationship between frequency and tension
The fundamental frequency (F) of a string is given by the formula:
\[ F \propto \sqrt{\frac{T}{\mu}} \]
where:
- \( T \) is the tension in the string,
- \( \mu \) is the linear mass density of the string (which remains constant in this case).

### Step 2: Establish the relationship for the two frequencies
Given that the fundamental frequency with a weight of 4 kg is 256 Hz, we want to find the weight needed to produce its octave. The octave means that the second frequency (F2) is double the first frequency (F1):
\[ F_2 = 2F_1 \]

### Step 3: Set up the ratio of frequencies
From the relationship of frequencies and tension, we can write:
\[ \frac{F_1}{F_2} = \sqrt{\frac{T_1}{T_2}} \]
Substituting \( F_2 = 2F_1 \):
\[ \frac{F_1}{2F_1} = \sqrt{\frac{T_1}{T_2}} \]
This simplifies to:
\[ \frac{1}{2} = \sqrt{\frac{T_1}{T_2}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{1}{2}\right)^2 = \frac{T_1}{T_2} \]
\[ \frac{1}{4} = \frac{T_1}{T_2} \]

### Step 5: Express tensions in terms of weights
The tension in the string is equal to the weight hanging from it. Therefore:
- \( T_1 = 4 \, \text{kg} \times g \) (where \( g \) is the acceleration due to gravity)
- \( T_2 = W \, \text{kg} \times g \) (where \( W \) is the new weight we want to find)

Substituting these into the equation gives:
\[ \frac{1}{4} = \frac{4g}{Wg} \]
The \( g \) cancels out:
\[ \frac{1}{4} = \frac{4}{W} \]

### Step 6: Solve for W
Cross-multiplying gives:
\[ W = 4 \times 4 = 16 \, \text{kg} \]

### Conclusion
To produce the octave of the fundamental frequency of 256 Hz, a weight of **16 kg** is needed.