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Chapter 16: Problem 70

A police siren of frequency \(f_{siren}\) is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude \(A_p\) and frequency \(f_p\). (a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.

Short Answer

Expert verified
Maximum frequency is heard when moving upward at max speed; minimum when moving downward at max speed.

Step by step solution

01

Understanding the Basics of Frequency

The frequency of a sound we hear can change if the source is in motion. This phenomenon is known as the Doppler effect. If the source of sound moves toward you, the frequency increases, and if it moves away, the frequency decreases. In this problem, the siren, which is the source of sound, oscillates with a platform.
02

Identify Parameters Involved in the Problem

The problem involves a siren oscillating up and down at a frequency \( f_p \) and having a maximum amplitude \( A_p \). The original frequency of the siren is given as \( f_{siren} \). The motion of the platform causes the siren's frequency to change as perceived by an observer above.
03

Finding the Velocity of the Platform

As the platform moves in simple harmonic motion, its velocity \( v(t) \) at any point \( t \) can be given by \( v(t) = A_p \cdot 2\pi f_p \cdot \cos(2\pi f_p t) \), where \( \cos \) is used because maximum speed occurs at equilibrium position.
04

Determine Maximum and Minimum Frequencies Heard

Use the Doppler effect formula where perceived frequency \( f' = \frac{f_{siren}}{1 - \frac{v}{c}} \) (source approaching) and \( f' = \frac{f_{siren}}{1 + \frac{v}{c}} \) (source receding). The maximum velocity \( v = A_p \cdot 2\pi f_p \) gives maximum \( f' = \frac{f_{siren}}{1 - \frac{A_p \cdot 2\pi f_p}{c}} \) and minimum \( f' = \frac{f_{siren}}{1 + \frac{A_p \cdot 2\pi f_p}{c}} \), where \( c \) is the speed of sound.
05

Identifying Points of Maximum and Minimum Frequency

Maximum frequency is heard when the platform is moving upward at maximum speed, meaning it's passing through its equilibrium from lower to higher position. The minimum frequency is heard when the platform is moving downward at maximum speed, crossing equilibrium from higher to lower position.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Harmonic Motion
Simple harmonic motion (SHM) describes a type of periodic motion where an object oscillates about a central point, called equilibrium. This motion is very predictable and can be described by mathematical functions like sine and cosine.
SHM is characterized by four key quantities: amplitude, period, frequency, and phase.
  • Amplitude is the maximum displacement from the equilibrium position. In our exercise, it's denoted by \( A_p \).
  • Frequency is how many oscillations occur per second. It is denoted by \( f_p \) for the platform.
  • Period is the time taken for one complete oscillation, given by \( T = \frac{1}{f} \).
  • Phase describes the angle or position of the oscillating object at a particular time.
In the context of the siren and platform, the SHM affects the perceived sound frequency due to the Doppler effect, which we will explore below.
Sound Frequency
The frequency of sound refers to the number of waves that pass a point in one second. It is measured in Hertz (Hz). A higher frequency means a higher pitch sound.
In the given problem, we work with a siren having an original frequency \( f_{siren} \). However, due to the motion of the platform, this frequency changes for an observer.
  • When the siren moves towards the observer, the sound waves compress, increasing the frequency perceived.
  • When the siren moves away, the waves spread out, decreasing the frequency.
This change in frequency due to motion is the Doppler Effect, which plays a crucial role in determining the maximum and minimum frequencies heard above the siren.
Speed of Sound
Speed of sound is the speed at which sound waves travel through a medium. It varies with factors such as temperature and medium, but the standard speed of sound in air at room temperature is approximately 343 meters per second.
The speed of sound \( c \) is a critical factor in calculating perceived frequency using the Doppler Effect formula:
  • For the source moving towards the observer, the perceived frequency \( f' \) increases: \[ f' = \frac{f_{siren}}{1 - \frac{v}{c}} \]
  • For the source moving away, \( f' \) decreases: \[ f' = \frac{f_{siren}}{1 + \frac{v}{c}} \]
Understanding this speed helps explain why we hear changes in frequency based on relative motion between the sound source and the observer.
Oscillation
Oscillation refers to the repeated back-and-forth movement around an equilibrium point. This motion is the heart of simple harmonic motion.
In the context of this exercise, the oscillation comes from the platform's motion, which keeps moving up and down. This movement causes the frequency of the sound heard by an observer to change.
  • The upward motion results in compressing sound waves, this is when the maximum frequency is heard as the platform crosses the equilibrium upwards.
  • As it moves downward, the waves are stretched leading to the minimum frequency being heard as it crosses the equilibrium downwards at maximum speed.
Understanding oscillation helps us comprehend how periodic motion impacts sound frequency in the given exercise.

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Most popular questions from this chapter

The motors that drive airplane propellers are, in some cases, tuned by using beats. The whirring motor produces a sound wave having the same frequency as the propeller. (a) If one single-bladed propeller is turning at 575 rpm and you hear 2.0-Hz beats when you run the second propeller, what are the two possible frequencies (in rpm) of the second propeller? (b) Suppose you increase the speed of the second propeller slightly and find that the beat frequency changes to 2.1 Hz. In part (a), which of the two answers was the correct one for the frequency of the second single-bladed propeller? How do you know?

While sitting in your car by the side of a country road, you are approached by your friend, who happens to be in an identical car. You blow your car's horn, which has a frequency of 260 Hz. Your friend blows his car's horn, which is identical to yours, and you hear a beat frequency of 6.0 Hz. How fast is your friend approaching you?

The sound from a trumpet radiates uniformly in all directions in 20\(^\circ\)C air. At a distance of 5.00 m from the trumpet the sound intensity level is 52.0 dB. The frequency is 587 Hz. (a) What is the pressure amplitude at this distance? (b) What is the displacement amplitude? (c) At what distance is the sound intensity level 30.0 dB?

Horseshoe bats (genus \(Rhinolophus\)) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey's speed. (The "horseshoe" that gives the bat its name is a depression around the nostrils that acts like a focusing mirror, so that the bat emits sound in a narrow beam like a flashlight.) A \(Rhinolophus\) flying at speed \(v_{bat}\) emits sound of frequency \(f_{bat}\); the sound it hears reflected from an insect flying toward it has a higher frequency \(f_{refl}\). (a) Show that the speed of the insect is $$vinsect = v\Bigg[\frac{f_{refl}(v - v_{bat}) - f_{bat}(v + v_{bat})}{f_{refl}(v - v_{bat}) + f_{bat}(v + v_{bat})}\Bigg] $$ where \(v\) is the speed of sound. (b) If \(f_{bat} =\) 80.7 kHz, \(f_{refl} =\) 83.5 kHz, and \(v_{bat} =\) 3.9 m/s, calculate the speed of the insect.

At point \(A, 3.0 \mathrm{~m}\) from a small source of sound that is emitting uniformly in all directions, the sound intensity level is \(53 \mathrm{~dB}\). (a) What is the intensity of the sound at \(A ?\) (b) How far from the source must you go so that the intensity is one-fourth of what it was at \(A\) ? (c) How far must you go so that the sound intensity level is one-fourth of what it was at \(A ?\) (d) Does intensity obey the inverse-square law? What about sound intensity level?
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