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Ch 16: Sound & Hearing
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
Chapter 16, Problem 9

An oscillator vibrating at 1250 Hz produces a sound wave that travels through an ideal gas at 325 m/s when the gas temperature is 22.0°C. For a certain experiment, you need to have the same oscillator produce sound of wavelength 28.5 cm in this gas. What should the gas temperature be to achieve this wavelength?

Verified step by step guidance
1
First, understand the relationship between frequency, wavelength, and speed of sound. The formula to use is: v=f×λ, where v is the speed of sound, f is the frequency, and λ is the wavelength.
Convert the given wavelength from centimeters to meters for consistency in units. Since 1 cm = 0.01 m, the wavelength of 28.5 cm is equivalent to 0.285 m.
Use the formula v=f×λ to find the new speed of sound required for the given wavelength. Substitute f = 1250 Hz and λ = 0.285 m into the equation.
Recognize that the speed of sound in a gas is related to the temperature of the gas. The formula for the speed of sound in an ideal gas is v=γRTM, where γ is the adiabatic index, R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas.
Solve for the new temperature T using the speed of sound calculated in step 3 and the formula from step 4. Remember to convert the temperature from Kelvin to Celsius if needed.

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Key Concepts

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

Wave Speed Equation

The wave speed equation relates the speed of a wave (v) to its frequency (f) and wavelength (λ) through the formula v = f * λ. This equation is crucial for understanding how changes in frequency or wavelength affect the speed of a wave, which is essential for solving problems involving sound waves in different conditions.
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Intro to Waves and Wave Speed

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, describes the relationship between pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. It is fundamental in determining how changes in temperature affect the properties of a gas, such as the speed of sound, which is influenced by the temperature and density of the gas.
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Ideal Gases and the Ideal Gas Law

Temperature and Sound Speed

The speed of sound in a gas is affected by the temperature of the gas, as higher temperatures increase the speed due to greater molecular activity. The relationship is given by v = sqrt(γRT/M), where γ is the adiabatic index, R is the gas constant, T is the temperature, and M is the molar mass. Understanding this concept helps in calculating the required temperature for a specific sound wavelength.
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Related Practice
Textbook Question

Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate. Typically, the diameter of this membrane is about 8.4 mm in humans. How much energy is delivered to the eardrum each second when someone whispers (20 dB) a secret in your ear?

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Textbook Question

(a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0 dB? (b) Explain why you don't need to know the original sound intensity

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Textbook Question

Consider a sound wave in air that has displacement amplitude 0.0200 mm. Calculate the pressure amplitude for frequencies of (a) 150 Hz; (b) 1500 Hz; (c) 15,000 Hz. In each case compare the result to the pain threshold, which is 30 Pa.

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Textbook Question

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?

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Textbook Question

A metal bar with a length of 1.50 m has density 6400 kg/m3. Longitudinal sound waves take 3.90 × 10-4 s to travel from one end of the bar to the other. What is Young's modulus for this metal?

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Textbook Question

What must be the stress (F/A) in a stretched wire of a material whose Young's modulus is Y for the speed of longitudinal waves to equal 30 times the speed of transverse waves?

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