Ch 16: Sound & Hearing

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 16: Sound & Hearing

Problem 12

Chapter 16, Problem 12

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?

Verified step by step guidance

Understand the relationship between the speed of longitudinal waves (v_l) and transverse waves (v_t) in a material. The problem states that v_l = 30 * v_t.

Recall the formula for the speed of longitudinal waves in a material: v_l = sqrt(Y / ρ), where Y is Young's modulus and ρ is the density of the material.

Recall the formula for the speed of transverse waves in a stretched wire: v_t = sqrt(T / μ), where T is the tension in the wire and μ is the linear mass density (mass per unit length) of the wire.

Since v_l = 30 * v_t, substitute the expressions for v_l and v_t into this equation: sqrt(Y / ρ) = 30 * sqrt(T / μ).

Square both sides of the equation to eliminate the square roots, and solve for the tension T in terms of Young's modulus Y, density ρ, and linear mass density μ. Then, use the relationship between tension and stress (T = F/A) to find the stress in the wire.

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Key Concepts

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

Young's Modulus

Young's modulus (Y) is a measure of the stiffness of a material, defined as the ratio of stress (force per unit area) to strain (deformation per unit length). It quantifies how much a material will deform under a given load, and is crucial for understanding the relationship between stress and wave speed in a material.

Longitudinal Waves

Longitudinal waves are waves in which the particle displacement is parallel to the direction of wave propagation. In solids, these waves travel through the material by compressing and expanding it, and their speed is influenced by the material's density and elastic properties, such as Young's modulus.

Transverse Waves

Transverse waves are waves where the particle displacement is perpendicular to the direction of wave propagation. In solids, these waves move through the material by shearing it, and their speed is determined by the material's shear modulus and density. Understanding the speed difference between longitudinal and transverse waves is key to solving the problem.

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Related Practice

Textbook Question

You are trying to overhear a juicy conversation, but from your distance of 15.0 m, it sounds like only an average whisper of 20.0 dB. How close should you move to the chatterboxes for the sound level to be 60.0 dB?

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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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?

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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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