Chapter 16: Problem 4
A loud factory machine produces sound having a displacement amplitude of 1.00 \(\mu\)m, but the frequency of this sound can be adjusted. In order to prevent ear damage to the workers, the maximum pressure amplitude of the sound waves is limited to 10.0 Pa. Under the conditions of this factory, the bulk modulus of air is 1.42 \(\times\) 10\(^5\) Pa. What is the highest-frequency sound to which this machine can be adjusted without exceeding the prescribed limit? Is this frequency audible to the workers?
Short Answer
Step by step solution
Understand the Relationship
Substitute Known Values
Solve for Frequency
Calculate the Maximum Frequency
Check Audibility
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Displacement Amplitude
Understanding displacement amplitude helps in grasping how sound waves propagate through air. When sound waves travel, they cause vibrations in the air particles. The displacement amplitude is directly related to how much these particles oscillate.
Key points about displacement amplitude:
- The higher the displacement amplitude, the louder the sound tends to be.
- It does not depend on the frequency of the sound.
- It's closely related to the energy transported by the wave.
Pressure Amplitude
Sound waves create alternating high and low pressures in the medium, and pressure amplitude quantifies the peak of these variations. It's an essential metric when it comes to environmental and auditory safety standards, as excessive pressure amplitudes can lead to hearing impairment.
Important aspects of pressure amplitude include:
- It is proportionally related to both displacement amplitude and the bulk modulus.
- A critical factor in determining whether sound levels meet occupational safety guidelines.
- Independent of frequency, but adjustments in frequency influence wave behavior and potential audibility.
Bulk Modulus of Air
When sound travels through air, it alternately compresses and rarefies the air particles. A higher bulk modulus indicates that the medium resists compression more, causing sound to travel more rapidly and efficiently.
Key insights about bulk modulus include:
- It is an intrinsic property that affects the speed of sound in the medium.
- Directly impacts the relationship between displacement amplitude and pressure amplitude.
- Higher bulk modulus materials typically transmit sound faster.
Frequency of Sound
In the exercise, by calculating and manipulating formulas, the maximum allowable frequency was found to be about 3,846 Hz. This frequency is comfortably within the range of human hearing, meaning all workers can hear it unless there are other environmental factors.
Core points about frequency:
- It is inversely related to wavelength—higher frequencies have shorter wavelengths.
- Affects the pitch of the sound—it determines how high or low a sound is perceived.
- While impacting audibility, frequency adjustments can also control the safety and health aspects of sound in environments.