Ch 15: Mechanical Waves

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 15: Mechanical Waves

Problem 36b

Chapter 15, Problem 36b

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of the second overtone?

Verified step by step guidance

First, understand that the second overtone in a stretched string corresponds to the third harmonic. In a string fixed at both ends, the number of antinodes corresponds to the harmonic number.

For the third harmonic, the string will have three antinodes and two nodes (excluding the endpoints). The wavelength \( \lambda \) of the third harmonic is given by \( \lambda = \frac{2L}{3} \), where \( L \) is the length of the string.

Substitute the given length of the rope \( L = 1.50 \text{ m} \) into the wavelength formula to find the wavelength of the third harmonic.

The frequency \( f \) of a wave is related to its speed \( v \) and wavelength \( \lambda \) by the equation \( f = \frac{v}{\lambda} \). Use the wave speed \( v = 62.0 \text{ m/s} \) and the wavelength calculated in the previous step to find the frequency of the third harmonic.

By following these steps, you will determine both the wavelength and frequency of the second overtone (third harmonic) on the rope.

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Key Concepts

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

Wave Speed

Wave speed is the rate at which a wave propagates through a medium, calculated as the product of frequency and wavelength. In this scenario, the wave speed is given as 62.0 m/s, which is crucial for determining the wavelength and frequency of the wave on the rope.

Standing Waves and Overtones

Standing waves occur when waves reflect back and forth between fixed points, creating nodes and antinodes. Overtones are higher frequency modes of vibration. The second overtone corresponds to the third harmonic, where the rope has three segments vibrating, affecting the wavelength and frequency calculations.

Wave Equation for Harmonics

The wave equation for harmonics relates the length of the medium to the wavelength of the standing wave. For a rope fixed at both ends, the wavelength of the nth harmonic is given by λ = 2L/n. Understanding this equation helps calculate the wavelength and frequency of the second overtone, which is the third harmonic.

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Equations for Transverse Standing Waves

Related Practice

Textbook Question

A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the antinodes of 0.300 cm. What is the speed of propagation of transverse waves in the wire?

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Textbook Question

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of the fundamental?

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Textbook Question

A piano tuner stretches a steel piano wire with a tension of 800 N. The steel wire is 0.400 m long and has a mass of 3.00 g. What is the frequency of its fundamental mode of vibration?

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Textbook Question

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of the fourth harmonic?

Textbook Question

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m² at a distance of 4.3 m from the source. What is the intensity at a distance of 3.1 m from the source?

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Textbook Question

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m² at a distance of 4.3 m from the source. How much sound energy does the source emit in one hour if its power output remains constant?

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