When a sound wave of frequency 300 Hz passes through a medium the maximum displacement of a particle of the medium is 0.1 cm . The maximum velocity of the particle is equal to

Options

A. `60 pi cm //sec`
B. `30 pi cm //sec`
C. `30 cm //sec`
D. `60 cm //sec`

Answer

To find the maximum velocity of a particle in a sound wave, we can use the formula:

\[ V_{\text{max}} = A \cdot \omega \]

where:
- \( V_{\text{max}} \) is the maximum velocity,
- \( A \) is the amplitude of the wave,
- \( \omega \) is the angular frequency.

### Step 1: Identify the given values
From the problem, we have:
- Frequency \( f = 300 \, \text{Hz} \)
- Maximum displacement (amplitude) \( A = 0.1 \, \text{cm} = 0.001 \, \text{m} \) (converting to meters for standard SI units)

### Step 2: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is calculated using the formula:

\[ \omega = 2 \pi f \]

Substituting the given frequency:

\[ \omega = 2 \pi \times 300 \]

### Step 3: Calculate \( \omega \)
Now, calculating \( \omega \):

\[ \omega = 600 \pi \, \text{rad/s} \]

### Step 4: Substitute values into the maximum velocity formula
Now substitute \( A \) and \( \omega \) into the maximum velocity formula:

\[ V_{\text{max}} = A \cdot \omega = 0.001 \, \text{m} \cdot 600 \pi \]

### Step 5: Calculate \( V_{\text{max}} \)
Calculating \( V_{\text{max}} \):

\[ V_{\text{max}} = 0.001 \cdot 600 \pi \]
\[ V_{\text{max}} = 0.6 \pi \, \text{m/s} \]

To convert this back to centimeters per second:

\[ V_{\text{max}} = 60 \pi \, \text{cm/s} \]

### Final Answer:
The maximum velocity of the particle is:

\[ V_{\text{max}} = 60 \pi \, \text{cm/s} \]