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Chapter 15: Problem 22

Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly wallking toward the source. When you are 7.5 \(\mathrm{m}\) from it, you measure its intensity to be 0.11 \(\mathrm{W} / \mathrm{m}^{2} .\) An intensity of 1.0 \(\mathrm{W} / \mathrm{m}^{2}\) is often used as the "threshold of pain." How much closer to the source can you move before the sound intensity reaches this threshold?

Short Answer

Expert verified
You can move 5.35 m closer before reaching the threshold of pain.

Step by step solution

01

Understanding Intensity and Distance

The sound intensity \( I \) from a point source is given by \( I = \frac{P}{4\pi r^2} \), where \( P \) is the power of the source and \( r \) is the distance from the source. When you move closer to the source, \( r \) decreases, causing \( I \) to increase.
02

Compute Source Power

We start by using the known intensity at a specific distance to calculate the power \( P \). At \( r = 7.5 \, \mathrm{m} \), the intensity is \( 0.11 \, \mathrm{W/m}^2 \). Thus, \( P = I \times 4\pi r^2 = 0.11 \times 4\pi (7.5)^2 \).
03

Solving for Source Power

Calculate \( P \) using the formula: \[P = 0.11 \times 4\pi (7.5)^2 = 0.11 \times 4 \times 3.1416 \times 56.25 \approx 58.35 \, \mathrm{W}\]
04

Determine the New Distance

We now want to find the new distance \( r' \), when the intensity \( I' = 1.0 \, \mathrm{W/m}^2 \). Using \( I' = \frac{P}{4\pi r'^2} \), solve for \( r' \): \[1.0 = \frac{58.35}{4\pi r'^2}\].
05

Calculate the New Distance

Solve for \( r' \): \[1.0 \times 4\pi r'^2 = 58.35 \quad \Rightarrow \quad r'^2 = \frac{58.35}{4\pi} \approx 4.646 \] \[ r' \approx \sqrt{4.646} \approx 2.15 \, \mathrm{m} \]
06

Determine How Much Closer to Move

The current distance is \( 7.5 \, \mathrm{m} \) and the new distance is \( 2.15 \, \mathrm{m} \). The difference is \( 7.5 - 2.15 = 5.35 \, \mathrm{m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Threshold of Pain
The term "Threshold of Pain" is often used in acoustics to describe the intensity level of sound at which a normal human ear starts to feel pain. This level is typically around 1.0 W/m虏. When sound intensity reaches this threshold, it can be uncomfortable or even harmful to human hearing.
- It is crucial for understanding how sound waves interact with human perception, especially in situations involving loud noises.
- Knowing this threshold helps in assessing environments for safety and regulating exposure to sound.
In our problem, the threshold of pain is a key value as it dictates how close one can approach the sound source before it becomes painful. By calculating the distance at which the sound intensity reaches 1.0 W/m虏, we can understand the safety limits around the source.
Point Source
A point source in the context of sound is a theoretical concept where the sound is emitted from a single point and spreads uniformly in all directions. This simplification helps in mathematical modeling of sound propagation as it assumes no reflections or interruptions in the path of the sound waves.
- It simplifies the calculation of intensity at various distances from the source.
- Although idealistic, it serves as a foundational concept for understanding more complex sound propagation scenarios.
For the UFO exercise, assuming the sound comes from a point source allows us to use the formula for intensity, \( I = \frac{P}{4\pi r^2} \). This formula links the intensity \( I \) of the sound to the power \( P \) of the source and the distance \( r \) from it.
Distance and Intensity
Sound intensity decreases with distance from the source, which is a key principle in the study of acoustics. This relationship is mathematically represented as \( I = \frac{P}{4\pi r^2} \), meaning as you move away from a sound source, intensity decreases proportionally to the square of the distance (known as the inverse square law).
- Intensity and distance are inversely related; getting closer to the source increases the intensity.
- This principle is used to calculate safe distances from loud sound sources.
In our example, originally at a distance of 7.5 m the intensity is 0.11 W/m虏. As we approach the sound source, the intensity was calculated to potentially reach the threshold of pain at 2.15 m.** This illustrates how critical understanding the distance-intensity relationship is, especially to avoid harm from loud sounds.

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Most popular questions from this chapter

A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation \(y(x, t)=\) \((5.60 \mathrm{cm}) \sin\) \([(0.0340 \mathrm{rad} / \mathrm{cm}) x]\) sin \([(50.0 \mathrm{rad} / \mathrm{s}) t]\) where the origin is at the left end of the string, the \(x\) -axis is along the string and the \(y\) -axis is perpendicular to the string. (a) Draw a sketch that shows the standing wave pattern. (b) Find the amplitude of the two traveling waves that make up this standing wave. (c) What is the length of the string? (d) Find the wavelength, frequency, period, and speed of the traveling waves. (e) Find the maximum transverse speed of a point on the string. (f) What would be the equation \(y(x, t)\) for this string if it were vibrating in its eighth harmonic?

The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is 60.0 \(\mathrm{cm}\) long. and this length of the string has mass 2.00 g. The string sounds an \(\mathrm{A}_{4}\) note \((440 \mathrm{Hz})\) when played. (a) Where must the player put a finger (what distance \(x\) from bridge) to play \(\mathbf{a}\) Ds note \((587 \quad \mathrm{Hz}) ? \quad\) (See Fig. 15.36 . For both the A_{4} and \(D_{5}\) notes, the string vibrates in its fundamental mode. (b) Without retuning, is it possible to play a G \(_{4}\) note \((392 \mathrm{Hz})\) on this string? Why or why not?

Guitar String. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(\mathbf{B}_{3}\) (frequency 245 \(\mathrm{Hz} )\) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by \(1.0 \%,\) what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is 344 \(\mathrm{m} / \mathrm{s}\) , find the frequency and wavelength of the sound wave produced in the air by the vibration of the \(\mathrm{B}_{3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?

A wire with mass 40.0 \(\mathrm{g}\) is stretched so that its ends are tied down at points 80.0 \(\mathrm{cm}\) apart. The wire vibrates in its fundamental mode with frequency 60.0 \(\mathrm{Hz}\) and with an amplitude at the antinodes of 0.300 \(\mathrm{cm}\) . (a) What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire. (c) Find the maximum transverse velocity and acceleration of particles in the wire.

Adjacent antinodes of a standing wave on a string are 15.0 \(\mathrm{cm}\) apart. A particle at an antinode oscillates in simple harmonic motion with amplitude 0.850 \(\mathrm{cm}\) and period 0.0750 \(\mathrm{s}\) . The string lies along the \(+x\) -axis and is fixed at \(x=0 .(\text { a) How far }\) apart are the adjacent nodes? (b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern? (c) Find the maximum and minimum transverse speeds of a point at an antinode. (d) What is the shortest distance along the string between a node and an antinode?
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