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Chapter 10: Problem 1453

An listener is moving towards a stationary source of sound with a speed (1/4) times the speed of sound. What will be the percentage increase in the frequency of sound heard by the listener? (A) \(20 \%\) (B) \(25 \%\) (C) \(2.5 \%\) (D) \(5 \%\)

Short Answer

Expert verified
The percentage increase in the frequency of sound heard by the listener is \(25\% (B)\).

Step by step solution

01

Write the Doppler effect formula

The Doppler effect formula for a moving observer and a stationary source is given by: \(f_{observed} = f_{actual} \frac{v + v_o}{v}\) Here, \(f_{observed}\) = Observed frequency by the listener \(f_{actual}\) = Actual frequency of the sound source \(v\) = Speed of sound \(v_o\) = Speed of the listener
02

Identify the given information

From the problem statement, we have: \(v_o = \frac{1}{4}v\) We need to find the percentage increase in the frequency of sound heard by the listener.
03

Substitute the given values in the Doppler effect formula

Substitute the given values in the Doppler effect formula: \(f_{observed} = f_{actual} \frac{v + \frac{1}{4}v}{v}\)
04

Simplify the expression

Simplify the expression to find the relationship between the observed frequency and actual frequency: \(f_{observed} = f_{actual} \frac{\frac{5}{4}v}{v}\) Divide both the numerator and the denominator by v: \(f_{observed} = f_{actual} \frac{5}{4}\)
05

Find the percentage increase in the frequency

To find the percentage increase in the frequency, use the following formula: Percentage Increase = \(\frac{f_{observed} - f_{actual}}{f_{actual}} * 100\%\) Substitute the expression for the observed frequency from Step 4: Percentage Increase = \(\frac{\frac{5}{4}*f_{actual} - f_{actual}}{f_{actual}} * 100\%\) Simplify the expression: Percentage Increase = \(\frac{\frac{1}{4}*f_{actual}}{f_{actual}} * 100\%\) Cancel out the actual frequency from the numerator and the denominator: Percentage Increase = \(\frac{1}{4} * 100\%\) Calculate the percentage increase: Percentage Increase = \(25\%\) Therefore, the percentage increase in the frequency of sound heard by the listener is \(25\% (B)\).

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Most popular questions from this chapter

The displacement for a particle performing S.H.M. is given by \(\mathrm{x}=\mathrm{A} \cos (\omega \mathrm{t}+\theta)\). If the initial position of the particle is \(1 \mathrm{~cm}\) and its initial velocity is $\pi \mathrm{cms}^{-1}$, then what will be its initial phase ? The angular frequency of the particle is \(\pi \mathrm{s}^{-1}\). (A) \((2 \pi / 4)\) (B) \((7 \pi / 4)\) (C) \((5 \pi / 4)\) (D) \((3 \pi / 4)\)

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