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. (a)`60picms^(-1)`
B. (b)`30picms^(-1)`
C. (c)`30cms^(-1)`
D. (d)`60cms^(-1)`

Answer

To find the maximum velocity of a particle in a medium when a sound wave passes through it, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Frequency (f) = 300 Hz
- Maximum displacement (Amplitude, A) = 0.1 cm = 0.001 m (convert to meters for standard SI units)

2. **Understand the Relationship**:
The maximum velocity (V_max) of a particle in a medium due to a sound wave is given by the formula:
\[ V_{\text{max}} = A \cdot \omega \]
where \( \omega \) is the angular frequency.

3. **Calculate Angular Frequency**:
The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = 2 \pi f \]
Substituting the given frequency:
\[ \omega = 2 \pi \cdot 300 \, \text{Hz} \]

4. **Calculate V_max**:
Now substitute \( A \) and \( \omega \) into the maximum velocity formula:
\[ V_{\text{max}} = A \cdot \omega = 0.001 \, \text{m} \cdot (2 \pi \cdot 300) \]

5. **Perform the Calculation**:
First, calculate \( 2 \pi \cdot 300 \):
\[ 2 \pi \cdot 300 \approx 600 \pi \, \text{rad/s} \]
Now, substitute this back into the equation for \( V_{\text{max}} \):
\[ V_{\text{max}} = 0.001 \cdot 600 \pi \]
\[ V_{\text{max}} = 0.6 \pi \, \text{m/s} \]

6. **Convert to Centimeters per Second**:
Since \( 1 \, \text{m} = 100 \, \text{cm} \):
\[ V_{\text{max}} = 0.6 \pi \cdot 100 \, \text{cm/s} = 60 \pi \, \text{cm/s} \]

7. **Final Result**:
The maximum velocity of the particle is:
\[ V_{\text{max}} \approx 60 \pi \, \text{cm/s} \approx 188.5 \, \text{cm/s} \, (\text{using } \pi \approx 3.14) \]

### Conclusion:
The maximum velocity of the particle is \( 60 \pi \, \text{cm/s} \).