A `40 cm` long brass rod is dropped one end first onto a hard floor but is caught before it topples over . With an `3 kHz` tone . The speed of sound in brass is

Options

A. `600 m//s`
B. `1200 m//s`
C. `2400 m//s`
D. `4800 m//s`

Explanation

To find the speed of sound in the brass rod, we can follow these steps:

### Step 1: Understand the Problem
We have a brass rod that is 40 cm long, and it produces a sound with a frequency of 3 kHz when dropped. We need to find the speed of sound in brass.

### Step 2: Convert Units
First, we convert the length of the rod from centimeters to meters:
- Length of the rod, \( L = 40 \, \text{cm} = 0.4 \, \text{m} \)

### Step 3: Identify the Wave Properties
Since the rod is fixed at both ends when it is dropped, it can be modeled as a standing wave. The fundamental frequency (first harmonic) of a fixed rod has nodes at both ends.

### Step 4: Relate Wavelength and Length
For a rod fixed at both ends, the length of the rod is equal to half the wavelength (\( \lambda \)) of the fundamental frequency:
\[ L = \frac{\lambda}{2} \]
Thus, we can express the wavelength 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: Use the Speed of Sound Formula
The speed of sound (\( v \)) is given by the formula:
\[ v = f \cdot \lambda \]
where \( f \) is the frequency. We need to convert the frequency from kHz to Hz:
\[ f = 3 \, \text{kHz} = 3000 \, \text{Hz} \]

### Step 7: Calculate the Speed of Sound
Now we can substitute the values of \( f \) and \( \lambda \) into the speed of sound formula:
\[ v = 3000 \, \text{Hz} \times 0.8 \, \text{m} = 2400 \, \text{m/s} \]

### Step 8: Conclusion
The speed of sound in brass is \( 2400 \, \text{m/s} \).