The angular frequency of a particle in a progressive wave in an elastic medium is `100 pi rads^(-1)` and it is moving with a velocity of `200ms^(-1)` The phase difference between two particles seperated by a distance of 20m is

Options

A. `31.4` rad
B. `pi` rad
C. `(3pi)/(4)` rad
D. 36 rad

Answer

To solve the problem, we need to find the phase difference between two particles separated by a distance of 20 meters in a progressive wave. We are given the angular frequency (ω) and the wave velocity (v).

### Step-by-Step Solution:

1. **Identify Given Values:**
- Angular frequency (ω) = \(100 \pi \, \text{radians/s}\)
- Wave velocity (v) = \(200 \, \text{m/s}\)
- Distance of separation (Δx) = \(20 \, \text{m}\)

2. **Calculate the Wave Number (k):**
The wave number (k) is related to the angular frequency (ω) and the wave velocity (v) by the formula:
\[ v = \frac{\omega}{k} \implies k = \frac{\omega}{v} \]
Substituting the given values:
\[ k = \frac{100 \pi}{200} = \frac{\pi}{2} \, \text{radians/m} \]

3. **Calculate the Phase Difference (Δφ):**
The phase difference (Δφ) between two points separated by a distance Δx in a wave can be calculated using the formula:
\[ \Delta \phi = k \cdot \Delta x \]
Substituting the values of k and Δx:
\[ \Delta \phi = \left(\frac{\pi}{2}\right) \cdot 20 = 10\pi \, \text{radians} \]

4. **Convert to Decimal:**
To express the phase difference in decimal form, we can use the approximation \( \pi \approx 3.14 \):
\[ \Delta \phi \approx 10 \cdot 3.14 = 31.4 \, \text{radians} \]

5. **Final Answer:**
The phase difference between the two particles separated by 20 meters is approximately \(31.4 \, \text{radians}\).

### Conclusion:
The correct option is \(31.4 \, \text{radians}\).

---