A metal rod `40 cm` long is dropped on to a wooden floor and rebounds into air . Compressional waves of many frequencies are thereby set up in the rod . If the speed of compressional waves in the rod in `5500 m//s`, what is the lowest frequency of compressional waves to which the rod resonates as it rebounds?

Options

A. `675 Hz`
B. `6875 Hz`
C. `16875 Hz`
D. `0 Hz`

Explanation

To solve the problem of finding the lowest frequency of compressional waves in a metal rod that resonates as it rebounds, we can follow these steps:

### Step 1: Understand the relationship between speed, frequency, and wavelength
The fundamental relationship for waves is given by the equation:
\[ v = f \lambda \]
where:
- \( v \) is the speed of the wave,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.

### Step 2: Identify the parameters given in the problem
From the problem, we know:
- The length of the rod, \( L = 40 \, \text{cm} = 0.4 \, \text{m} \)
- The speed of compressional waves in the rod, \( v = 5500 \, \text{m/s} \)

### Step 3: Determine the conditions for resonance
For the rod to resonate, we need to consider the boundary conditions. Since both ends of the rod are fixed (acting as nodes), the fundamental mode of vibration will have one antinode in the middle.

### Step 4: Relate the wavelength to the length of the rod
In the case of a rod with both ends fixed, the relationship between the length of the rod and the wavelength for the fundamental frequency is given by:
\[ L = \frac{\lambda}{2} \]
This means that the wavelength \( \lambda \) can be expressed as:
\[ \lambda = 2L \]

### Step 5: Calculate the wavelength
Substituting the length of the rod into the equation:
\[ \lambda = 2 \times 0.4 \, \text{m} = 0.8 \, \text{m} \]

### Step 6: Calculate the lowest frequency
Now, we can use the wave speed formula to find the frequency:
\[ f = \frac{v}{\lambda} \]
Substituting the values we have:
\[ f = \frac{5500 \, \text{m/s}}{0.8 \, \text{m}} \]
\[ f = 6875 \, \text{Hz} \]

### Conclusion
The lowest frequency of compressional waves to which the rod resonates as it rebounds is approximately:
\[ \boxed{6875 \, \text{Hz}} \]

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