To solve the problem step-by-step, we will analyze the situation of the metal bar being clamped at its center and then at one end, and derive the frequencies of the first and second overtones.
### Step 1: Understand the fundamental frequency when clamped at the center
When the metal bar is clamped at its center, it resonates in its fundamental mode. In this mode, there is a node at the center and antinodes at the ends. The length of the bar (L) can be related to the wavelength (λ) as follows:
\[
L = \frac{\lambda}{2}
\]
From the problem, we know the fundamental frequency \( f_0 = 4 \, \text{kHz} = 4 \times 10^3 \, \text{Hz} \).
### Step 2: Relate speed of sound to frequency and wavelength
The fundamental frequency is given by the formula:
\[
f_0 = \frac{V}{\lambda}
\]
Substituting \( \lambda = 2L \):
\[
f_0 = \frac{V}{2L}
\]
### Step 3: Calculate the speed of sound (V)
From the fundamental frequency, we can express the speed of sound \( V \):
\[
V = 2L \cdot f_0 = 2L \cdot (4 \times 10^3)
\]
### Step 4: Analyze the situation when the clamp is moved to one end
When the clamp is moved to one end, the bar now has one node at the clamped end and one antinode at the free end. The length of the bar (L) corresponds to:
\[
L = \frac{\lambda}{4}
\]
### Step 5: Determine the wavelengths for the new configuration
For this new configuration, the wavelengths can be described as:
\[
L = \frac{(2N - 1)\lambda}{4}
\]
Where \( N \) is the harmonic number (1 for the fundamental, 2 for the first overtone, etc.).
### Step 6: Calculate the frequencies for the first and second overtones
1. **First Overtone (N = 2)**:
\[
f_1 = \frac{(2 \cdot 2 - 1)V}{4L} = \frac{3V}{4L}
\]
2. **Second Overtone (N = 3)**:
\[
f_2 = \frac{(2 \cdot 3 - 1)V}{4L} = \frac{5V}{4L}
\]
### Step 7: Substitute V in terms of f0
From the earlier step, we know:
\[
V = 8L \times 10^3
\]
Substituting this into the equations for \( f_1 \) and \( f_2 \):
1. **First Overtone**:
\[
f_1 = \frac{3(8L \times 10^3)}{4L} = 6 \, \text{kHz}
\]
2. **Second Overtone**:
\[
f_2 = \frac{5(8L \times 10^3)}{4L} = 10 \, \text{kHz}
\]
### Step 8: Establish the relationship between \( f_1 \) and \( f_2 \)
Now, we have:
- \( f_1 = 6 \, \text{kHz} \)
- \( f_2 = 10 \, \text{kHz} \)
To find the relationship:
\[
5f_1 = 3f_2
\]
### Final Answer
Thus, the relationship between the first overtone and the second overtone is:
\[
5f_1 = 3f_2
\]
