Textbook Question
A guitar string produces 3 beat/s when sounded with a 350-Hz tuning fork and 8 beat/s when sounded with a 355-Hz tuning fork. What is the vibrational frequency of the string? Explain your reasoning.
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Ch. 16 - Sound
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179
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Table of contents
- Ch. 01 - Introduction, Measurement, Estimating10
- Ch. 02 - Describing Motion: Kinematics in One Dimension30
- Ch. 03 - Kinematics in Two or Three Dimensions; Vectors15
- Ch. 04 - Dynamics: Newton's Laws of Motion16
- Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces22
- Ch. 06 - Gravitation and Newton's Synthesis10
- Ch. 07 - Work and Energy21
- Ch. 08 - Conservation of Energy33
- Ch. 09 - Linear Momentum26
- Ch. 10 - Rotational Motion41
- Ch. 11 - Angular Momentum; General Rotation27
- Ch. 12 - Static Equilibrium; Elasticity and Fracture27
- Ch. 13 - Fluids28
- Ch. 14 - Oscillations14
- Ch. 15 - Wave Motion35
- Ch. 16 - Sound5
- Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law40
- Ch. 18 - Kinetic Theory of Gases18
- Ch. 19 - Heat and the First Law of Thermodynamics31
- Ch. 20 - Second Law of Thermodynamics32
- Ch. 21 - Electric Charge and Electric Field7
- Ch. 23 - Electric Potential20
- Ch. 24 - Capacitance, Dielectrics, Electric Energy, Storage15
- Ch. 25 - Electric Current and Resistance32
- Ch. 26 - DC Circuits22
- Ch. 27 - Magnetism21
- Ch. 28 - Sources of Magnetic Field24
- Ch. 29 - Electromagnetic Induction and Faraday's Law21
- Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits42
- Ch. 31 - Maxwell's Equations and Electromagnetic Waves25
- Ch. 32 - Light: Reflection and Refraction42
- Ch. 33 - Lenses and Optical Instruments21
- Ch. 34 - The Wave Nature of Light: Interference and Polarization26
- Ch. 35 - Diffraction24
- Ch. 36 - The Special Theory of Relativity29
- Ch. 38 - Quantum Mechanics1
- Ch. 40 - Molecules and Solids6
- Ch. 43 - Elementary Particles3
- Ch. 44 - Astrophysics and Cosmology9
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
Chapter 16, Problem 104b
Assuming that the maximum displacement of the air molecules in a sound wave is about the same as that of the speaker cone that produces the sound (Fig. 16–46), estimate by how much a loudspeaker cone moves for a fairly loud (105 dB) sound of 35 Hz.
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Understand the relationship between sound intensity and displacement amplitude. The sound intensity level (in decibels) is related to the displacement amplitude of the sound wave. The formula for sound intensity is: \( I = \frac{1}{2} \rho v \omega^2 s^2 \), where \( \rho \) is the air density, \( v \) is the speed of sound, \( \omega \) is the angular frequency, and \( s \) is the displacement amplitude.
Convert the given sound intensity level (105 dB) into actual intensity \( I \). Use the formula: \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) \), where \( L \) is the sound level in decibels, \( I \) is the intensity, and \( I_0 = 10^{-12} \, \text{W/m}^2 \) is the reference intensity. Rearrange to solve for \( I \): \( I = I_0 \cdot 10^{L/10} \).
Determine the angular frequency \( \omega \) of the sound wave. The angular frequency is related to the frequency \( f \) by the formula: \( \omega = 2 \pi f \). Substitute \( f = 35 \; \text{Hz} \) to calculate \( \omega \).
Rearrange the intensity formula to solve for the displacement amplitude \( s \). From \( I = \frac{1}{2} \rho v \omega^2 s^2 \), isolate \( s \): \( s = \sqrt{\frac{2I}{\rho v \omega^2}} \). Here, \( \rho \) is the density of air (approximately \( 1.21 \; \text{kg/m}^3 \)), and \( v \) is the speed of sound in air (approximately \( 343 \; \text{m/s} \)).
Substitute the known values for \( I \), \( \rho \), \( v \), and \( \omega \) into the formula for \( s \). Perform the calculations to estimate the displacement amplitude \( s \), which corresponds to the movement of the loudspeaker cone.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sound Intensity and Decibels
Sound intensity is a measure of the power per unit area carried by a sound wave. The decibel (dB) scale quantifies sound intensity logarithmically, where an increase of 10 dB represents a tenfold increase in intensity. A sound level of 105 dB indicates a relatively loud sound, which can be associated with significant energy transfer in the medium, affecting the displacement of air molecules.
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Sound Intensity Level and the Decibel Scale
Frequency and Wavelength
Frequency refers to the number of oscillations or cycles of a wave that occur in one second, measured in hertz (Hz). In sound waves, frequency determines the pitch of the sound; lower frequencies correspond to deeper sounds. The relationship between frequency and wavelength is inversely proportional, meaning that as frequency decreases, the wavelength increases, which is crucial for understanding how sound waves propagate.
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Amplitude and Displacement
Amplitude is the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In the context of sound waves, amplitude relates to the loudness of the sound; greater amplitude results in louder sounds. The displacement of the loudspeaker cone directly influences the amplitude of the sound wave produced, which in turn affects the maximum displacement of air molecules in the wave.
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