The distance between two points differing in phase by `60^(@)` on a wave, velocity 360 m./s and frequency 500 Hz is

Options

A. 24 cm
B. 12 cm
C. 6 cm
D. 1 cm

Explanation

To solve the problem of finding the distance between two points differing in phase by \(60^\circ\) on a wave with a velocity of \(360 \, \text{m/s}\) and a frequency of \(500 \, \text{Hz}\), we can follow these steps:

### Step 1: Convert the phase difference from degrees to radians.
The phase difference given is \(60^\circ\). To convert degrees to radians, we use the formula:
\[ \text{radians} = \frac{\text{degrees} \times \pi}{180} \]
Thus,
\[ \Delta \phi = \frac{60 \times \pi}{180} = \frac{\pi}{3} \, \text{radians} \]

**Hint:** Remember that \(180^\circ\) is equivalent to \(\pi\) radians.

### Step 2: Calculate the wavelength (\(\lambda\)).
The wavelength can be calculated using the formula:
\[ \lambda = \frac{v}{f} \]
where \(v\) is the velocity of the wave and \(f\) is the frequency. Substituting the given values:
\[ \lambda = \frac{360 \, \text{m/s}}{500 \, \text{Hz}} = 0.72 \, \text{m} \]

**Hint:** The wavelength is the distance over which the wave's shape repeats.

### Step 3: Use the formula to find the distance (\(\Delta x\)) between two points differing in phase by \(\Delta \phi\).
The formula to find the distance between two points differing in phase is:
\[ \Delta x = \frac{\lambda}{2\pi} \times \Delta \phi \]
Substituting the values we found:
\[ \Delta x = \frac{0.72 \, \text{m}}{2\pi} \times \frac{\pi}{3} \]

**Hint:** The factor of \(\frac{1}{2\pi}\) relates the wavelength to the phase difference.

### Step 4: Simplify the expression.
\[ \Delta x = \frac{0.72 \, \text{m}}{2 \times 3.14} \times \frac{3.14}{3} = \frac{0.72}{6} = 0.12 \, \text{m} \]

**Hint:** When simplifying, remember that \(\frac{\pi}{\pi}\) cancels out.

### Step 5: Convert the distance to centimeters.
Since \(1 \, \text{m} = 100 \, \text{cm}\):
\[ \Delta x = 0.12 \, \text{m} = 12 \, \text{cm} \]

**Hint:** Always check the units to ensure they are in the required format.

### Final Answer:
The distance between the two points differing in phase by \(60^\circ\) is \(12 \, \text{cm}\).