Problem 76 - University Physics with Modern Physics

Question

A police siren of frequency firm 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.

Step-by-Step Solution

Answer

The maximum frequency is heard when the siren moves upward at maximum speed, and the minimum frequency is heard when it moves downward at maximum speed.

Step 1: Understanding the Doppler Effect

The Doppler effect explains the change in frequency of a wave in relation to an observer moving relative to the wave source. When the source moves towards the observer, the frequency increases; when it moves away, it decreases.

Step 2: Identify Relevant Equations

The observed frequency \( f' \) can be determined using the equation: \[ f' = \frac{v + v_o}{v - v_s} \cdot f \] where \( v \) is the speed of sound in air, \( v_o \) is the speed of the observer (0 in this case as the observer is stationary), and \( v_s \) is the speed of the source (siren).

Step 3: Calculate Maximum Frequency

Maximum frequency occurs when the siren is moving upwards towards the observer. The speed of the siren at maximum amplitude is the maximum speed: \( v_s = 2\pi f_p A_p \). Substitute this into the Doppler effect equation to find the maximum frequency \( f_{max} \).

Step 4: Calculate Minimum Frequency

Minimum frequency occurs when the siren is moving downwards, away from the observer. The minimum speed is also \( v_s = 2\pi f_p A_p \) but with opposite direction. Substitute this into the Doppler effect formula to find the minimum frequency \( f_{min} \).

Step 5: Identify Points for Maximum and Minimum Frequency

The maximum frequency is heard at the topmost point when the siren moves upwards, and the minimum frequency is heard at the bottommost point as it moves downwards. This is because these points represent the points where the speed towards and away from the observer is maximum.

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion, often abbreviated as SHM, is a type of periodic motion where an object moves back and forth around an equilibrium point. This type of motion is predictable and can be described mathematically by sine or cosine functions. When you think of SHM, imagine something like a swinging pendulum or a mass on a spring.
In the problem with the police siren, the siren and the platform it's on oscillate in simple harmonic motion. This means they move up and down continuously and repeatedly over time. The motion is defined by two key factors: amplitude and frequency.

• Amplitude \(A_p\) describes the maximum distance the motion reaches from the central equilibrium point. It shows how far the siren can move up or down from its resting position.
• Frequency \(f_p\) indicates how many oscillations occur in one second. It is measured in hertz (Hz).

Understanding these concepts helps us discern how the siren moves and how these movements affect sound perception.

Sound Frequency

Sound frequency is the number of vibrations or cycles that occur per second in a sound wave, usually measured in hertz (Hz). The frequency determines the pitch of the sound we hear. For instance, a higher frequency results in a higher-pitched sound, while a lower frequency corresponds to a lower-pitched sound.
In the given scenario, the sound frequency of the police siren changes due to its movement. These changes are perceptible when the siren experiences shifts in frequency, as it oscillates up and down.
A fixed frequency siren might sound consistent to a stationary listener, but because the siren vibrates, the frequency changes periodically. This is directly related to the Doppler effect, where the motion of the siren causes variations in the sound frequency as perceived by an observer. By understanding this principle, you can anticipate how the sound frequency changes in different positions of SHM.

Wave Source Movement

The movement of a wave source, such as our siren, directly affects the characteristics of the sound wave it emits. In the exercise, the siren moves as a part of SHM on a vibrating platform. As it moves closer and farther from a stationary observer, the frequency of the waves observed changes due to the relative motion.
This is a classic example of the Doppler effect. When the source moves towards the observer, the waves appear compressed, increasing the frequency, leading to a higher pitch. Conversely, when the source moves away, the waves stretch out, lowering the frequency and pitch.
The speed at which the siren moves contributes to the intensity of these frequency shifts. In this scenario, the maximum speed of the siren is given by \([v_s = 2\pi f_p A_p]\). This allows us to understand how and when the frequency changes during the motion.

Physics Problem Solving

Physics problem solving involves breaking down complex problems into manageable parts using laws and equations of physics. In this problem, the focus is on using the concept of the Doppler effect to solve the frequency changes.
The given solution steps guide us through how maximum and minimum sound frequencies are calculated when the siren travels in simple harmonic motion. We use the Doppler effect equation: \([f' = \frac{v + v_o}{v - v_s} \cdot f]\), \ where \(v_s\) represents the velocity of the siren. By substituting appropriate velocities when the siren moves towards and away from the observer, one calculates the changes in frequency.
This systematic approach simplifies the complexity of the physic problem into accessible steps:

• Understanding the principles involved (Doppler effect).
• Identifying the required equations and variables.
• Performing calculations step-by-step.

Following such methods helps to not only solve individual problems but also to develop a deep understanding of physics concepts.