A metal bar clamped at its centre resonates in its fundamental mode to produce longitudinal waves of frequency `4 kHz`. Now the clamp is moved to one end . If `f_(1)` and `f_(2)` be the frequencies of first overtone and second overtone respectively then ,

Options

A. (a)`3 f_(2) = 5 f_(1)`
B. (b)`3 f_(1) = 5 f_(2)`
C. (c)`f_(2) = 2f_(1)`
D. (d)`2f_(2) = f_(1)`

Explanation

To solve the problem step by step, we need to analyze the situation of a metal bar clamped at one end and how it resonates in its fundamental mode, first overtone, and second overtone.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- The metal bar is clamped at its center and resonates at a fundamental frequency \( f_0 = 4 \, \text{kHz} \).
- The relationship between frequency, wavelength, and length for a bar clamped at the center is given by:
\[ f_0 = \frac{v}{2L} \]
where \( v \) is the speed of sound in the metal bar and \( L \) is the length of the bar.

2. **Finding the Speed of Sound**:
- Rearranging the equation gives:
\[ v = 2L f_0 \]
- Substituting \( f_0 = 4 \times 10^3 \, \text{Hz} \):
\[ v = 2L \times 4 \times 10^3 \]

3. **Changing the Clamp Position**:
- When the clamp is moved to one end, the bar behaves like a pipe closed at one end. The wavelengths of the harmonics are given by:
- Fundamental frequency: \( \lambda_1 = 4L \)
- First overtone (third harmonic): \( \lambda_2 = \frac{4L}{3} \)
- Second overtone (fifth harmonic): \( \lambda_3 = \frac{4L}{5} \)

4. **Finding Frequencies of Overtones**:
- The frequency for the nth overtone can be expressed as:
\[ f_n = \frac{n v}{4L} \]
- For the first overtone \( (n=2) \):
\[ f_1 = \frac{3v}{4L} \]
- For the second overtone \( (n=3) \):
\[ f_2 = \frac{5v}{4L} \]

5. **Substituting the Speed of Sound**:
- Substitute \( v = 8L \times 10^3 \) into the overtone frequencies:
- For \( f_1 \):
\[ f_1 = \frac{3 \times 8L \times 10^3}{4L} = 6 \times 10^3 \, \text{Hz} = 6 \, \text{kHz} \]
- For \( f_2 \):
\[ f_2 = \frac{5 \times 8L \times 10^3}{4L} = 10 \times 10^3 \, \text{Hz} = 10 \, \text{kHz} \]

6. **Finding the Relationship Between Frequencies**:
- We can express the relationship between \( f_1 \) and \( f_2 \):
\[ 3f_2 = 5f_1 \]
- Substituting \( f_1 \) and \( f_2 \):
\[ 3 \times 10 \, \text{kHz} = 5 \times 6 \, \text{kHz} \]
- This confirms the relationship holds true.

7. **Conclusion**:
- The frequencies of the first overtone and second overtone are:
\[ f_1 = 6 \, \text{kHz}, \quad f_2 = 10 \, \text{kHz} \]