In a string the speed of wave is 10 m/s and its frequency is 100 Hz. The value of the phase difference at a distance 2.5 cm will be:

Options

A. `pi//2`
B. `pi//8`
C. `3pi//2`
D. `2pi`

Explanation

To find the phase difference at a distance of 2.5 cm in a string where the speed of the wave is 10 m/s and the frequency is 100 Hz, we can follow these steps:

### Step 1: Convert the distance from cm to meters.
Given distance \( \Delta r = 2.5 \, \text{cm} \).

To convert centimeters to meters:
\[ \Delta r = \frac{2.5}{100} = 0.025 \, \text{m} \]

**Hint:** Always convert units to the standard SI units before performing calculations.

### Step 2: Calculate the wavelength (\( \lambda \)).
The wavelength can be calculated using the formula:
\[ \lambda = \frac{v}{f} \]
where \( v \) is the speed of the wave and \( f \) is the frequency.

Given:
- Speed \( v = 10 \, \text{m/s} \)
- Frequency \( f = 100 \, \text{Hz} \)

Substituting the values:
\[ \lambda = \frac{10 \, \text{m/s}}{100 \, \text{Hz}} = 0.1 \, \text{m} \]

**Hint:** Remember that frequency is in Hz (cycles per second), and speed is in meters per second.

### Step 3: Use the formula for phase difference (\( \Delta \phi \)).
The phase difference is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta r \]

Substituting the values we have:
- \( \lambda = 0.1 \, \text{m} \)
- \( \Delta r = 0.025 \, \text{m} \)

Now substituting into the formula:
\[ \Delta \phi = \frac{2\pi}{0.1} \times 0.025 \]

### Step 4: Calculate \( \Delta \phi \).
Calculating \( \Delta \phi \):
\[ \Delta \phi = \frac{2\pi \times 0.025}{0.1} = \frac{2\pi \times 0.025}{0.1} = 0.5\pi \]

### Final Answer:
Thus, the value of the phase difference at a distance of 2.5 cm is:
\[ \Delta \phi = 0.5\pi \, \text{radians} \]

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