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Chapter 9: Problem 79

The phase difference between two points separated by \(0.8 \mathrm{~m}\) in a wave of frequency \(120 \mathrm{~Hz}\) is \(0.5 \pi\). The wave velocity is (A) \(144 \mathrm{~m} / \mathrm{s}\) (B) \(256 \mathrm{~m} / \mathrm{s}\) (C) \(384 \mathrm{~m} / \mathrm{s}\) (D) \(720 \mathrm{~m} / \mathrm{s}\)

Short Answer

Expert verified
To find the wave velocity, first calculate the wavelength using the given phase difference and distance between two points: Wavelength = \(\dfrac{2\pi \times 0.8}{0.5\pi} = 3.2 \)m. Then, use the formula for wave velocity: Velocity = Frequency 脳 Wavelength, and substitute the given values: Velocity = \(120 \mathrm{Hz} \times 3.2 \mathrm{m} = 384 \mathrm{m/s}\). The correct answer is (C) \(384 \mathrm{~m} / \mathrm{s}\).

Step by step solution

01

1. Remember the formula for wavelength and phase difference

Using the formula for phase difference, we have: Phase Difference = \(\dfrac{2\pi\times Distance}{Wavelength}\). We can rearrange this equation to find the wavelength: Wavelength = \(\dfrac{2\pi\times Distance}{Phase Difference}\).
02

2. Substitute the given values into the equation

We have the phase difference (0.5\(\pi\)) and the distance (0.8 m). Substituting these values into the equation, we get: Wavelength = \(\dfrac{2\pi \times 0.8}{0.5\pi}\).
03

3. Calculate the wavelength

Now, we need to calculate the wavelength. After solving the equation, we get: Wavelength = \(\dfrac{2\pi \times 0.8}{0.5\pi} = \dfrac{1.6\pi}{0.5\pi} = 3.2 \)m.
04

4. Use the formula for wave velocity

Now that we have the wavelength and frequency, we can use the formula for wave velocity: Velocity = Frequency 脳 Wavelength.
05

5. Calculate the wave velocity

Substituting the given frequency (120 Hz) and the calculated wavelength (3.2 m) into the formula, we get: Velocity = \(120 \mathrm{Hz} \times 3.2 \mathrm{m} = 384 \mathrm{m/s}\). So the correct answer is (C) \(384 \mathrm{~m} / \mathrm{s}\).

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