A loud factory machine produces sound having a displacement amplitude of 1.00 mm, 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 × 10⁵ 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?

Verified step by step guidance

Start by understanding the relationship between pressure amplitude and displacement amplitude in sound waves. The pressure amplitude (ΔP) is related to the displacement amplitude (s) by the formula: ΔP = B * k * s, where B is the bulk modulus of the medium, k is the wave number, and s is the displacement amplitude.

The wave number (k) is related to the frequency (f) and the speed of sound (v) in the medium by the formula: k = 2πf/v. The speed of sound in air at room temperature is approximately 343 m/s.

Substitute the expression for k into the pressure amplitude formula: ΔP = B * (2πf/v) * s. Rearrange this formula to solve for the frequency (f): f = (ΔP * v) / (2π * B * s).

Plug in the given values: ΔP = 10.0 Pa, B = 1.42 * 10^5 Pa, s = 1.00 mm (which is 0.001 m), and v = 343 m/s into the formula to find the highest frequency.

Determine if the calculated frequency is within the audible range for humans, which is typically between 20 Hz and 20,000 Hz. Compare the frequency you calculated to this range to see if it is audible to the workers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sound Waves

Sound waves are longitudinal waves that travel through a medium by compressing and rarefying the particles within it. The amplitude of these waves determines their loudness, while the frequency determines their pitch. Understanding the relationship between amplitude, frequency, and pressure is crucial for analyzing sound wave behavior in different environments.

Bulk Modulus

The bulk modulus is a measure of a material's resistance to uniform compression, defined as the ratio of pressure increase to relative volume decrease. In the context of sound waves, it helps determine how pressure changes with displacement amplitude in a medium like air. A higher bulk modulus indicates that the medium is less compressible, affecting sound wave propagation.

Frequency and Audibility

Frequency refers to the number of oscillations per second of a wave, measured in Hertz (Hz). Human hearing typically ranges from 20 Hz to 20,000 Hz, making frequencies outside this range inaudible. Determining whether a frequency is audible involves comparing it to this range, which is essential for assessing the impact of sound on human hearing.

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