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Chapter 16: Problem 26

Compare the intensity of sound at the pain level, \(120 \mathrm{~dB}\), with that at the whisper level, \(20 \mathrm{~dB}\).

Short Answer

Expert verified
Answer: The intensity of sound at the pain level is 10^10 times greater than that at the whisper level.

Step by step solution

01

Know the decibel scale formula

The decibel scale formula is: \(I_{dB} = 10 \times \log_{10}(\frac{I}{I_0})\) where \(I_{dB}\) is the intensity level in decibels, \(I\) is the intensity of the sound, and \(I_0\) is the reference intensity (\(10^{-12}\) W/m²).
02

Find the intensity of sound at the pain level

We are given that the intensity level at the pain level is 120 dB. We will plug in the values to the decibel scale formula and solve for the intensity \(I\): \(120 = 10 \times \log_{10}(\frac{I}{10^{-12}})\) First, divide both sides by 10: \(12 = \log_{10}(\frac{I}{10^{-12}})\) Now, calculate the inverse logarithm to find the intensity: \(I = 10^{12} \times 10^{-12}\) \(I = 1\,\text{W/m}^2\)
03

Find the intensity of sound at the whisper level

Similarly, we will find the intensity at the whisper level, which is 20 dB: \(20 = 10 \times \log_{10}(\frac{I}{10^{-12}})\) Divide both sides by 10: \(2 = \log_{10}(\frac{I}{10^{-12}})\) Now, calculate the inverse logarithm to determine the intensity: \(I = 10^{2} \times 10^{-12}\) \(I = 10^{-10}\,\text{W/m}^2\)
04

Compare the intensities

Now that we have the intensities of both the pain level and the whisper level sounds, we can compare them: Intensity at pain level: \(1\,\text{W/m}^2\) Intensity at whisper level: \(10^{-10}\,\text{W/m}^2\) To show the difference, we can calculate the ratio of the intensities: \(\text{Intensity ratio} = \frac{1\,\text{W/m}^2}{10^{-10}\,\text{W/m}^2} = 10^{10}\) The intensity of sound at the pain level is \(10^{10}\) times greater than that at the whisper level.

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Most popular questions from this chapter

Standing on the sidewalk, you listen to the horn of a passing car. As the car passes, the frequency of the sound changes from high to low in a continuous manner; that is, there is no abrupt change in the perceived frequency. This occurs because a) the pitch of the sound of the horn changes continuously. b) the intensity of the observed sound changes continuously. c) you are not standing directly in the path of the moving car. d) of all of the above reasons.

On a windy day, a child standing outside a school hears the school bell. If the wind is blowing toward the child from the direction of the bell, will it alter the frequency, the wavelength, or the velocity of the sound heard by the child?

A bugle can be represented by a cylindrical pipe of length \(L=1.35 \mathrm{~m} .\) Since the ends are open, the standing waves produced in the bugle have antinodes at the open ends, where the air molecules move back and forth the most. Calculate the longest three wavelengths of standing waves inside the bugle. Also calculate the three lowest frequencies and the three longest wavelengths of the sound that is produced in the air around the bugle.

The sound level in decibels is typically expressed as \(\beta=10 \log \left(I / I_{0}\right),\) but since sound is a pressure wave, the sound level can be expressed in terms of a pressure difference. Intensity depends on the amplitude squared, so the expression is \(\beta=20 \log \left(P / P_{0}\right),\) where \(P_{0}\) is the smallest pressure difference noticeable by the ear: \(P_{0}=2.00 \cdot 10^{-5} \mathrm{~Pa}\). A loud rock concert has a sound level of \(110 . \mathrm{dB}\), find the amplitude of the pressure wave generated by this concert.

A sound level of 50 decibels is a) 2.5 times as intense as a sound of 20 decibels. b) 6.25 times as intense as a sound of 20 decibels. c) 10 times as intense as a sound of 20 decibels. d) 100 times as intense as a sound of 20 decibels. e) 1000 times as intense as a sound of 20 decibels.
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