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Chapter 10: Problem 1430

If the maximum frequency of a sound wave at room temperature is $20,000 \mathrm{~Hz}\( then its minimum wavelength will be approximately \)\ldots \ldots\left(\mathrm{v}=340 \mathrm{~ms}^{-1}\right)$ (A) \(0.2 \AA\) (B) \(5 \AA\) (C) \(5 \mathrm{~cm}\) to \(2 \mathrm{~m}\) (D) \(20 \mathrm{~mm}\)

Short Answer

Expert verified
The minimum wavelength can be found using the formula \(λ = \frac{v}{f}\). Given the maximum frequency (f) as 20,000 Hz and the wave speed (v) as 340 m/s, the minimum wavelength is found to be 0.017 m or 17 mm. The correct answer is (D) \(20 \mathrm{~mm}\).

Step by step solution

01

Write down the wave speed formula

The wave speed formula relates wave speed (v), frequency (f), and wavelength (λ) as follows: \(v = fλ\).
02

Rearrange the formula to find the wavelength

We are trying to find the minimum wavelength (λ), so we need to rearrange the formula to solve for λ. Divide both sides of the equation by the frequency (f) to get: \(λ = \frac{v}{f}\).
03

Substitute the given values into the formula

We are given the maximum frequency (f) as 20,000 Hz and the wave speed (v) as 340 m/s. Substitute these values into the formula: \(λ = \frac{340 \mathrm{~m/s}}{20,000 \mathrm{~Hz}}\).
04

Calculate the wavelength

Now, we can perform the division to find the minimum wavelength: \(λ = \frac{340 \mathrm{~m/s}}{20,000 \mathrm{~Hz}} = 0.017 \mathrm{~m}\).
05

Convert the wavelength to the appropriate unit

Since the answer choices are given in different units, we will convert the wavelength we found (0.017 m) into millimeters: \(0.017 \mathrm{~m} = 17 \mathrm{~mm}\).
06

Choose the correct answer choice

Based on our calculation, the minimum wavelength is approximately 17 mm. This value is closest to option (D) 20 mm. So, the correct answer is (D) \(20 \mathrm{~mm}\).

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