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Chapter 10: Problem 1500

A stationary wave of frequency \(200 \mathrm{~Hz}\) are formed in air. If the velocity of the wave is \(360 \mathrm{~m} / \mathrm{s}\), the shortest distance between two antinodes is \(\ldots \ldots \ldots \ldots \mathrm{m}\) (A) \(1.8\) (B) \(3.6\) (C) \(0.9\) (D) \(0.45\)

Short Answer

Expert verified
The shortest distance between two antinodes is \(0.9\) m.

Step by step solution

01

Recall and Write Down the Wave Formula

We know that the speed (velocity) of a wave is given by the product of its frequency (f) and wavelength (λ). The formula is v = λ × f where v is the velocity of the wave, λ is the wavelength, and f is the frequency. We are given v = 360 m/s and f = 200 Hz, and we need to find the wavelength (λ).
02

Solve for the Wavelength (λ)

Using the formula v = λ × f, we can rearrange it to solve for λ: λ = v / f Substitute the given values of v and f to find the wavelength: λ = (360 m/s) / (200 Hz) λ = 1.8 m
03

Determine the Relationship between Wavelength and Antinodes Distance

Now we have the wavelength of the wave, but we need to find the shortest distance between two antinodes. A stationary wave has one antinode at every half-wavelength. Therefore, the distance between two antinodes is equal to half of the wavelength.
04

Calculate the Shortest Distance between Two Antinodes

We know the distance between antinodes is equal to half of the wavelength. So, divide the wavelength by 2 to get the shortest distance between two antinodes: Shortest distance = λ / 2 Shortest distance = (1.8 m) / 2 Shortest distance = 0.9 m
05

Select the Correct Answer

Comparing our answer with the given options, the correct answer is (C) \(0.9\) m.

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