Question

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

The maximum frequency is 3,846 Hz, which is audible.

Step-by-step solution

Understand the Relationship

Sound waves in air are described by their pressure amplitude \( P_{max} \), displacement amplitude \( s_{max} \), and frequency \( f \). The pressure amplitude is given by the formula: \[ P_{max} = B k s_{max} \]where \( B \) is the bulk modulus of air, and \( k \) is the wave number related to frequency \( f \) by:\[ k = \frac{2\pi f}{v} \]with \( v \) as the speed of sound in air. The problem is to find the maximum frequency such that \( P_{max} \) does not exceed 10 Pa.

Substitute Known Values

Substitute the given values into the pressure amplitude formula:\[ 10.0 = 1.42 \times 10^5 \times \frac{2\pi f}{v} \times 1.00 \times 10^{-6} \]Assume the speed of sound in air \( v \) is approximately 343 m/s, a common value at room temperature.

Solve for Frequency

Isolate \( f \) on one side of the equation:\[ 10.0 = 1.42 \times 10^5 \times \frac{2\pi f}{343} \times 1.00 \times 10^{-6} \]Rearrange to solve for \( f \):\[ f = \frac{10.0 \times 343}{1.42 \times 10^5 \times 2\pi \times 1.00 \times 10^{-6}} \]Now, calculate \( f \).

Calculate the Maximum Frequency

Substitute the constants into the equation:\[ f = \frac{10.0 \times 343}{1.42 \times 10^5 \times 6.2832 \times 1.00 \times 10^{-6}} \approx 3,846.33 \text{ Hz} \]Thus, the highest frequency allowed is approximately 3,846 Hz.

Check Audibility

The audible range for humans is approximately 20 Hz to 20,000 Hz. Since 3,846 Hz falls within this range, the frequency is indeed audible to the workers.

Key concepts

Displacement Amplitude

In sound waves, displacement amplitude refers to the maximum distance particles in the medium move from their rest position as the wave passes through. It is one of the key parameters defining the intensity and energy of a sound wave. For the exercise, the displacement amplitude is given as 1.00 \( \mu \text{m} \) (micrometer). This is a tiny displacement, indicating that while the factory machine is loud, individual air particles move only slightly back and forth.

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.

The given exercise limits the amplitude displacement to ensure that the sound stays within safe levels for workers, particularly because loud sounds at significant displacement amplitudes can have damaging effects if exposure is prolonged.

Pressure Amplitude

Pressure amplitude represents the maximum change in pressure from atmospheric levels induced by the sound wave. In the given scenario, it's capped at 10.0 Pa (Pascals) to avoid harm to workers. Pressure amplitude correlates directly with how loud a sound is perceived; higher values mean more intense sound, thus more risk to hearing.

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.

In this case, maintaining pressure amplitude under 10.0 Pa is crucial, ensuring the workplace remains safe in terms of sound exposure.

Bulk Modulus of Air

Bulk modulus is a fundamental property of materials, including air, and it describes their resistance to uniform compression. For sound waves, bulk modulus influences how pressure waves transmit through the medium. In the exercise, the bulk modulus of air is specified as 1.42 \( \times \) 10\(^5\) Pa.

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.

This concept helps explain why certain settings, like the dense atmosphere or different states of matter, can change how sound behaves. Understanding bulk modulus is vital in tailoring safe sound levels, as seen in industrial applications.

Frequency of Sound

Frequency in sound waves refers to the number of wave cycles that pass a point per second, measured in Hertz (Hz). The human ear can generally detect frequencies from 20 Hz to 20,000 Hz. For the factory machine, the frequency must not cause pressure amplitude to exceed 10.0 Pa to prevent auditory harm.

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.

When solving sound-related problems, considering frequency is crucial as it influences multiple facets of wave behavior, from energy transmission to human perception.