A simple harmonic progressive wave is representive by the equation `y=8 sin 2pi (0.1x -2t)` where x and y are in centimetres and t is in seconds. At any instant the phase difference between two particle separted by 2.0 cm along the x-direction is

Options

A. `18^(@)`
B. `36^(@)`
C. `54^(@)`
D. `72^(@)`

Explanation

To find the phase difference between two particles separated by 2.0 cm along the x-direction in the given wave equation \( y = 8 \sin(2\pi(0.1x - 2t)) \), we can follow these steps:

### Step 1: Identify the wave number (k)
The wave equation is in the form \( y = A \sin(kx - \omega t) \). Here, we can identify:
- \( k = 2\pi(0.1) = 0.2\pi \, \text{cm}^{-1} \)

### Step 2: Calculate the phase difference
The phase difference \( \Delta \phi \) between two points separated by a distance \( \Delta x \) is given by:
\[ \Delta \phi = k \Delta x \]
Substituting the values:
- \( \Delta x = 2 \, \text{cm} \)
- \( k = 0.2\pi \)

So,
\[ \Delta \phi = (0.2\pi)(2) = 0.4\pi \, \text{radians} \]

### Step 3: Convert radians to degrees
To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \Delta \phi \text{ (in degrees)} = 0.4\pi \times \frac{180}{\pi} = 0.4 \times 180 = 72^\circ \]

### Final Answer
The phase difference between the two particles separated by 2.0 cm along the x-direction is \( 72^\circ \).

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