If a sound wave of frequency 500 Hz and velocity 350 m/s. Then the distance between the two particles of a phase difference of `60^(@)` will be nearly

Options

A. (a)70 cm
B. (b)0.7 cm
C. (c)12.0 cm
D. (d)120.0 cm

Answer

To solve the problem step-by-step, we need to find the distance between two particles of a sound wave that have a phase difference of \(60^\circ\).

### Step 1: Calculate the Wavelength
We start by using the relationship between the velocity of the wave, frequency, and wavelength. The formula is:

\[ v = \lambda \nu \]

Where:
- \(v\) = velocity of the wave (350 m/s)
- \(\nu\) = frequency of the wave (500 Hz)
- \(\lambda\) = wavelength

Rearranging the formula to find the wavelength:

\[ \lambda = \frac{v}{\nu} \]

Substituting the values:

\[ \lambda = \frac{350 \, \text{m/s}}{500 \, \text{Hz}} = 0.7 \, \text{m} \]

### Step 2: Convert Wavelength to Centimeters
Since we need the answer in centimeters, we convert the wavelength:

\[ \lambda = 0.7 \, \text{m} = 70 \, \text{cm} \]

### Step 3: Calculate the Distance Corresponding to the Phase Difference
The distance between two particles corresponding to a phase difference (\(\Delta \phi\)) is given by:

\[ d = \frac{\lambda \Delta \phi}{2\pi} \]

Where:
- \(\Delta \phi\) = phase difference in radians

First, we convert \(60^\circ\) to radians:

\[ \Delta \phi = 60^\circ = \frac{\pi}{3} \, \text{radians} \]

Now, substituting the values into the distance formula:

\[ d = \frac{70 \, \text{cm} \cdot \frac{\pi}{3}}{2\pi} \]

### Step 4: Simplify the Expression
The \(\pi\) in the numerator and denominator cancels out:

\[ d = \frac{70 \, \text{cm}}{6} = \frac{70}{6} \, \text{cm} \]

Calculating this gives:

\[ d \approx 11.67 \, \text{cm} \]

### Step 5: Round to the Nearest Whole Number
Rounding \(11.67 \, \text{cm}\) gives us approximately \(12 \, \text{cm}\).

### Conclusion
Thus, the distance between the two particles with a phase difference of \(60^\circ\) is nearly \(12 \, \text{cm}\).

### Final Answer
The correct option is \(C\) which corresponds to \(12 \, \text{cm}\).

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