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 36c

Chapter 15, Problem 36c

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?

Verified step by step guidance

First, understand that the harmonics of a stretched rope are related to the standing wave patterns that can form on the rope. The fundamental frequency (first harmonic) has one antinode, the second harmonic has two antinodes, and so on.

For the fourth harmonic, the rope will have four antinodes. The formula for the wavelength of the nth harmonic on a rope of length L is:

λ = \frac{2 L}{n}

, where n is the harmonic number.

Substitute the given values into the wavelength formula:

λ = \frac{2 \times 1.50}{4}

. This will give you the wavelength of the fourth harmonic.

Next, use the wave speed formula to find the frequency. The formula is:

v = f \times λ

, where v is the wave speed, f is the frequency, and

λ

is the wavelength.

Rearrange the wave speed formula to solve for frequency:

f = \frac{v}{λ}

. Substitute the values for v (62.0 m/s) and the calculated wavelength to find the frequency of the fourth harmonic.

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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. It is determined by the tension and mass per unit length of the medium, such as a rope. In this scenario, the wave speed is given as 62.0 m/s, which is crucial for calculating the wavelength and frequency of harmonics.

Harmonics

Harmonics are the natural frequencies at which a system, like a stretched rope, vibrates. Each harmonic corresponds to a specific mode of vibration. The fourth harmonic means the rope vibrates in a pattern with four segments, affecting the wavelength and frequency calculations.

Wavelength and Frequency Relationship

The wavelength and frequency of a wave are inversely related through the wave speed: v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. For the fourth harmonic, the wavelength is determined by the rope's length and harmonic number, while the frequency is derived from the wave speed and wavelength.

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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 second overtone?

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Textbook Question

CALC. A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation $y (x, t) = (5.60 cm) \sin [(0.0340 rad/cm) x] \sin [(50.0 rad/s) t]$ , where the origin is at the left end of the string, the $x$ -axis is along the string, and the $y$ -axis is perpendicular to the string. Draw a sketch that shows the standing-wave pattern.

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

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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