Two points A and B are at a distance of 30 cm on a stretched string. A travelling wave of frequency 300 Hz and velocity 100 m/s is set up in the string. What will be the phase difference between points A and B ?

Answer

To find the phase difference between points A and B on a stretched string, we can follow these steps:

### Step 1: Identify the given values
- Distance between points A and B, \( \Delta x = 30 \, \text{cm} = 0.3 \, \text{m} \)
- Frequency of the wave, \( f = 300 \, \text{Hz} \)
- Velocity of the wave, \( v = 100 \, \text{m/s} \)

### Step 2: Calculate the wavelength (\( \lambda \))
Using the wave equation:
\[ v = f \cdot \lambda \]
We can rearrange this to find the wavelength:
\[ \lambda = \frac{v}{f} = \frac{100 \, \text{m/s}}{300 \, \text{Hz}} = \frac{1}{3} \, \text{m} \approx 0.333 \, \text{m} \]

### Step 3: Calculate the wave number (\( k \))
The wave number \( k \) is given by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{\frac{1}{3}} = 6\pi \, \text{rad/m} \]

### Step 4: Calculate the phase difference (\( \Delta \phi \))
The phase difference between two points separated by a distance \( \Delta x \) is given by:
\[ \Delta \phi = k \cdot \Delta x \]
Substituting the values of \( k \) and \( \Delta x \):
\[ \Delta \phi = 6\pi \cdot 0.3 = 1.8\pi \, \text{radians} \]

### Step 5: Convert to decimal form (if necessary)
Calculating \( 1.8\pi \):
\[ 1.8\pi \approx 5.654 \, \text{radians} \]

### Final Answer
The phase difference between points A and B is approximately \( 5.654 \, \text{radians} \).

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