Problem 80 - Physics

Question

A microphone is attached to a spring that is suspended from the ceiling, as the drawing indicates. Directly below on the floor is a stationary 440 -Hz source of sound. The microphone vibrates up and down in simple harmonic motion with a period of \(2.0 \mathrm{~s}\). The difference between the maximum and minimum sound frequencies detected by the microphone is \(2.1 \mathrm{~Hz}\). Ignoring any reflections of sound in the room and using \(343 \mathrm{~m} / \mathrm{s}\) for the speed of sound, determine the amplitude of the simple harmonic motion.

Step-by-Step Solution

Answer

The amplitude of the simple harmonic motion is approximately 26.1 cm.

Step 1: Understand the Problem

We have a microphone that moves in simple harmonic motion due to a spring, creating a variation in frequency detected by the microphone as it moves up and down relative to a stationary 440 Hz sound source. We need to find the amplitude of this motion.

Step 2: Identify Known Quantities

The frequency of the sound source is 440 Hz. The difference between the maximum and minimum detected frequencies is 2.1 Hz. The period of the microphone's oscillation is 2.0 s. The speed of sound is 343 m/s.

Step 3: Relate Frequency Variation to Velocity

The frequency variation detected by the microphone is due to the Doppler effect. The frequency difference of 2.1 Hz is related to the change in frequency due to the maximum velocity of the microphone.

Step 4: Doppler Effect Formula

For a moving observer and stationary source, the observed frequency is given by:\[ f' = f \left(1 + \frac{v_m}{v_s}\right)\]where \(f\) is the source frequency, \(v_m\) is the maximum velocity of the microphone, and \(v_s\) is the speed of sound.

Step 5: Solve for Maximum Velocity

Since the frequency difference is 2.1 Hz, and the microphone moves back and forth, this corresponds to a maximum frequency displacement of approximately \(\pm 1.05\) Hz. Using the Doppler effect:\[ f' - f = f \cdot \frac{v_{max}}{v_s} = 1.05\]Isolating \(v_{max}\):\[ v_{max} = \frac{1.05}{440} \cdot 343 = 0.819 \text{ m/s}\]

Step 6: Relate Maximum Velocity to Amplitude

The relationship between maximum velocity \(v_{max}\) and amplitude \(A\) in simple harmonic motion is given by:\[ v_{max} = A \cdot \omega\]where \(\omega = \frac{2\pi}{T}\) is the angular frequency, and \(T = 2\) s is the period of motion.

Step 7: Calculate Angular Frequency

Calculate angular frequency \(\omega\):\[ \omega = \frac{2\pi}{2} = \pi \text{ rad/s}\]

Step 8: Solve for Amplitude

Using the maximum velocity relationship:\[ 0.819 = A \cdot \pi\]Solve for \(A\):\[ A = \frac{0.819}{\pi} \approx 0.261 \text{ m} \approx 26.1 \text{ cm}\]

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes a type of periodic motion where an object moves back and forth along a path. In this scenario, the microphone attached to a spring is an excellent example. It swings up and down in repetitive cycles with a constant time period. SHM has key characteristics that include sinusoidal movement, a restoring force proportional to displacement, and a defined equilibrium position.

When the microphone is displaced from its resting position, the spring pulls it back, creating oscillation. The time it takes to complete one full cycle of motion is the period, which here is 2 seconds. This uniformity in oscillation allows us to predict how and where the microphone will be at any given time.

Frequency Variation

The Doppler Effect explains frequency variation and occurs when a sound source or observer moves relative to each other. As the microphone in our problem moves in its cycle, it alternates between moving towards and away from the stationary sound source.

This motion results in shifts in the frequency detected by the microphone. When the microphone approaches the source, the frequency increases; as it moves away, the frequency decreases. This change is quantified as a frequency shift and is given as 2.1 Hz in this scenario. The difference between the highest and lowest frequencies perceived is due to these relative movements.

Angular Frequency

Angular frequency (\(\omega\)) relates to how quickly an object rotates in a circle, even though SHM is linear. It is closely related to frequency but expressed in radians per second. For SHM, angular frequency is calculated using the formula \(\omega = \frac{2\pi}{T}\), where \(T\) is the period of oscillation.

In our case, with a period of 2 seconds, the angular frequency becomes \(\pi\) radians per second. This value of \(\omega\) helps in determining other characteristics of motion like maximum velocity and amplitude, making it a crucial component for calculating various aspects of SHM.

Amplitude Calculation

Amplitude is the peak height of an oscillation in SHM, representing the maximum distance from the equilibrium position. Knowing the amplitude provides insight into how far the microphone moves in its cycle.

In this exercise, the maximum velocity \(v_{max}\) helps us find the amplitude. The formula \(v_{max} = A \cdot \omega\) connects these elements, where \(\omega\) is the angular frequency. Given \(v_{max} = 0.819 \text{ m/s}\) and \(\omega = \pi\), we solve for \(A\), resulting in an amplitude of approximately 26.1 cm. This calculation shows the extent of the microphone's swinging motion.