Question

Consider two points lying at a distance of \(10 \mathrm{~m}\) and $15 \mathrm{~m}$ from an oscillating source. If the periodic time of oscillation is \(0.05 \mathrm{~s}\) and the velocity of wave produced is $300 \mathrm{~m} / \mathrm{s}$, then what will be the phase difference the two points? (A) \(\pi\) (B) \(\pi / 6\) (C) \(\pi / 3\) (D) \(2 \pi / 3\)

Short answer

The phase difference between the two points is \(\frac{2\pi}{3}\), so the correct answer is (D) \(\frac{2\pi}{3}\).

Step-by-step solution

Determine the frequency of the wave

To determine the phase difference, we first need to find the frequency of the wave produced by the oscillating source. Given the periodic time of oscillation \(T\), the frequency \(f\) can be calculated using the formula: \[ f = \frac{1}{T} \]

Calculate the wavelength of the wave

Now we need to find the wavelength of the wave. We can use the wave speed formula, which relates frequency, wavelength, and wave velocity: \[ v = f\lambda \] Where \(v\) is the wave velocity, \(f\) is the frequency, and \(\lambda\) is the wavelength. We can solve this equation for the wavelength: \[ \lambda = \frac{v}{f} \]

Determine the path difference between the two points

The path difference between two points is simply the difference in distances from the oscillating source. In this case: \[ \Delta x = 15 \, m - 10 \, m = 5 \, m \]

Calculate the phase difference

Now we can determine the phase difference between the two points using the path difference and wavelength. The phase difference \(\Delta \phi\) can be calculated using the formula: \[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta x \] Substitute the values obtained in Steps 1, 2, and 3 into the formula: 1. Calculate the frequency \[ f = \frac{1}{0.05 \, s} = 20 \, Hz \] 2. Calculate the wavelength \[ \lambda = \frac{300 \, m / s}{20 \, Hz} = 15 \, m \] 3. Calculate the phase difference \[ \Delta \phi = \frac{2\pi}{15 \, m} \cdot 5 \, m = \frac{2\pi}{3} \] The phase difference between the two points is \(\frac{2\pi}{3}\), so the correct answer is (D) \(\frac{2\pi}{3}\).