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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 walking toward the source. When you are 7.5 m from it, you measure its intensity to be 0.11 W/m\(^2\). An intensity of 1.0 W/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 5.01 meters closer to reach the threshold of pain.

Step by step solution

01

Understand the Intensity Formula for Point Sources

The intensity of sound from a point source can be described by the formula \( I = \frac{P}{4\pi r^2} \), where \( I \) is the intensity, \( P \) is the power of the source, and \( r \) is the distance from the source.
02

Solve for the Source Power

Use the given intensity at 7.5 m to find the power of the source. Rearrange the intensity formula: \( P = I \cdot 4\pi r^2 \). Substitute \( I = 0.11 \text{ W/m}^2 \) and \( r = 7.5 \text{ m} \): \[ P = 0.11 \times 4\pi (7.5)^2 \].
03

Calculate Source Power

Calculate the power using the values: \[ P = 0.11 \times 4\pi \times 56.25 = 77.8217 \text{ W} \].
04

Determine New Distance for Intensity Threshold

Set the intensity to the threshold of pain: \( I = 1.0 \text{ W/m}^2 \). Use \( P = 77.8217 \text{ W} \) and solve for \( r \) using \( I = \frac{P}{4\pi r^2} \): \[ 1.0 = \frac{77.8217}{4\pi r^2} \].
05

Solve for New Distance \( r \)

Rearrange to find \( r^2 = \frac{77.8217}{4\pi} \) and then \( r = \sqrt{\frac{77.8217}{4\pi}} \).
06

Calculate and Compare Distances

Calculate \( r \): \[ r = \sqrt{\frac{77.8217}{12.5664}} \approx 2.49 \text{ m} \]. Determine how much closer you can get: 7.5 m - 2.49 m = 5.01 m.

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Key Concepts

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

Point Source
When studying sound or any wave phenomena, a point source is an essential concept. Imagine a tiny, single point in space from which sound radiates out uniformly in all directions. This is a point source. Although real-world sources might not be perfect points, many can be approximated as such for simplicity.
Point sources create sound waves that spread out spherically. That means as you move away, the sound becomes less intense because the energy spreads over a larger area. Think of it as ripples on water widening with distance.
Key features of point sources include:
  • Sounds decrease in intensity with distance.
  • The spread is uniform, like an expanding sphere.
  • Simple calculations with the surface area of a sphere equation.
In the exercise, the UFO's sound source is treated as a point source, making it easier to calculate changes in sound intensity.
Intensity Formula
Sound intensity is a measure of power per unit area, showing how strong or weak a sound is at a certain distance. The formula for intensity from a point source is given by:\[ I = \frac{P}{4\pi r^2} \]where
  • \(I\) is the sound intensity, measured in watts per square meter (W/m\(^2\)).
  • \(P\) is the power of the sound source, in watts (W).
  • \(r\) is the distance from the sound source, in meters (m).
This formula reflects how intensity decreases with the square of the distance from the source, known as the inverse square law.
In practical terms, if you double the distance, the sound intensity drops to a quarter of its original value.
To find out how close you can get before reaching a specific intensity level, like the threshold of pain, you can rearrange the formula to solve for \(r\) with a known power and intensity.
Threshold of Pain
The threshold of pain is a critical concept in acoustics. It's the maximum sound intensity level humans can typically tolerate without experiencing pain. The commonly accepted value for this threshold is about 1.0 W/m\(^2\).
Experiencing sounds above this level can cause discomfort or even damage to one's hearing over time. Therefore, knowing how sound intensity changes helps you gauge safe listening distances from loud sources.
In the context of our exercise, the threshold of pain acts as the limit for how close you can approach the sound source. Once this threshold is reached, it's crucial to maintain a safe distance to avoid discomfort, hence calculating the exact distance using the intensity formula is important.

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Most popular questions from this chapter

A jet plane at takeoff can produce sound of intensity 10.0 W/m\(^2\) at 30.0 m away. But you prefer the tranquil sound of normal conversation, which is 1.0 \(\mu\)W/m\(^2\). Assume that the plane behaves like a point source of sound. (a) What is the closest distance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

A wave on a string is described by \(y(x, t) = A \mathrm{cos}(kx - \omega t)\). (a) Graph \(y, v_y\), and \(a_y\) as functions of \(x\) for time \(t = 0\). (b) Consider the following points on the string: (i) \(x =\) 0; (ii) \(x = \pi/4k\); (iii) \(x = \pi/2k\); (iv) \(x = 3\pi/4k\); (v) \(x = \pi k\); (vi) \(x = 5\pi/4k\); (vii) \(x = 3\pi/2k\); (viii) \(x = 7\pi/4k\). For a particle at each of these points at \(t = 0\), describe in words whether the particle is moving and in what direction, and whether the particle is speeding up, slowing down, or instantaneously not accelerating.

At a distance of \(7.00 \times 10^{12}\) m from a star, the intensity of the radiation from the star is 15.4 W/m\(^2\). Assuming that the star radiates uniformly in all directions, what is the total power output of the star?

A 1750-N irregular beam is hanging horizontally by its ends from the ceiling by two vertical wires (\(A\) and \(B\)), each 1.25 m long and weighing 0.290 N. The center of gravity of this beam is one-third of the way along the beam from the end where wire A is attached. If you pluck both strings at the same time at the beam, what is the time delay between the arrival of the two pulses at the ceiling? Which pulse arrives first? (Ignore the effect of the weight of the wires on the tension in the wires.)

With what tension must a rope with length 2.50 m and mass 0.120 kg be stretched for transverse waves of frequency 40.0 Hz to have a wavelength of 0.750 m?

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