The length of a sonometer wire tuned to a frequency of 256 Hz is 0.6 m. Calculate the frequency of the tuning fork with which the vibrating wire will be in tune when the length is made 0.4 m

Options

A. 78Hz
B. 512Hz
C. 384Hz
D. 126Hz

Answer

To solve the problem, we will use the relationship between the frequency of a vibrating wire and its length. The fundamental frequency (f) of a wire is inversely proportional to its length (L), which can be expressed as:

\[ f \propto \frac{1}{L} \]

This means that if the length of the wire decreases, the frequency increases, and vice versa.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Initial frequency \( F_1 = 256 \, \text{Hz} \)
- Initial length \( L_1 = 0.6 \, \text{m} \)
- New length \( L_2 = 0.4 \, \text{m} \)

2. **Use the Relationship Between Frequencies and Lengths**:
According to the relationship, we can write:
\[ F_1 \cdot L_1 = F_2 \cdot L_2 \]
where \( F_2 \) is the frequency we need to find.

3. **Rearranging the Equation to Solve for \( F_2 \)**:
\[ F_2 = \frac{F_1 \cdot L_1}{L_2} \]

4. **Substituting the Known Values**:
\[ F_2 = \frac{256 \, \text{Hz} \cdot 0.6 \, \text{m}}{0.4 \, \text{m}} \]

5. **Calculating the Value**:
- First, calculate the numerator:
\[ 256 \cdot 0.6 = 153.6 \]
- Now divide by the new length:
\[ F_2 = \frac{153.6}{0.4} = 384 \, \text{Hz} \]

6. **Conclusion**:
The frequency of the tuning fork with which the vibrating wire will be in tune when the length is made 0.4 m is \( F_2 = 384 \, \text{Hz} \).

### Final Answer:
**The frequency of the tuning fork is 384 Hz.**