Ch. 15 - Wave Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 15 - Wave Motion

Problem 47

Chapter 15, Problem 47

A particular string resonates in four loops at a frequency of 320 Hz. Name at least three other frequencies at which it will resonate. What is each called?

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Understand the concept of resonance in a string: Resonance occurs when the frequency of a wave matches one of the natural frequencies of the string. These natural frequencies are called harmonics, and they are integer multiples of the fundamental frequency.

Identify the given information: The string resonates in four loops at a frequency of 320 Hz. This means that 320 Hz corresponds to the fourth harmonic (n = 4). The fundamental frequency (n = 1) can be calculated as \( f_1 = \frac{f_4}{4} \).

Calculate the fundamental frequency: Using the relationship \( f_n = n \cdot f_1 \), divide 320 Hz by 4 to find \( f_1 \), the first harmonic or fundamental frequency.

Determine the other frequencies: The second harmonic (n = 2) is \( f_2 = 2 \cdot f_1 \), the third harmonic (n = 3) is \( f_3 = 3 \cdot f_1 \), and the fifth harmonic (n = 5) is \( f_5 = 5 \cdot f_1 \). These are the other frequencies at which the string will resonate.

Name the harmonics: The frequencies \( f_1 \), \( f_2 \), \( f_3 \), and \( f_5 \) are called the fundamental frequency, second harmonic, third harmonic, and fifth harmonic, respectively.

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Key Concepts

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

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system, such as a vibrating string, naturally resonates. In this case, the string resonates at 320 Hz, which is its fundamental frequency. This frequency corresponds to the first harmonic, where the string vibrates in a single loop.

Harmonics

Harmonics are integer multiples of the fundamental frequency. For a string fixed at both ends, the frequencies at which it resonates are given by the formula f_n = n * f_1, where n is the harmonic number and f_1 is the fundamental frequency. Thus, the second harmonic would be 640 Hz, the third 960 Hz, and the fourth 1280 Hz.

Resonance

Resonance occurs when an object vibrates at its natural frequency due to an external force. In the context of the string, it means that when the string is excited at certain frequencies, it will vibrate with greater amplitude. This phenomenon is crucial for understanding how musical instruments produce sound and how different frequencies relate to one another.

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Textbook Question

A guitar string is 91 cm long and has a mass of 3.2 g. The vibrating portion of the string from the bridge to the support post is ℓ = 64cm and the string is under a tension of 520 N. What are the frequencies of the fundamental and first two overtones?

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One end of a horizontal string is attached to a small-amplitude mechanical 60.0-Hz oscillator. The string’s mass per unit length is 3.9 x 10⁻ ⁴ kg/m. The string passes over a pulley, a distance ℓ = 1.50 m away, and weights are hung from this end, Fig. 15–38. What mass m must be hung from this end of the string to produce five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true.

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Suppose two linear waves of equal amplitude and frequency have a phase difference ϕ as they travel in the same medium. They can be represented by: D₁ = A sin (kx - ωt); D₂ = A sin ( kx - ωt + ϕ). Describe the resultant wave, by equation and in words, if ϕ = π/2.

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Suppose two linear waves of equal amplitude and frequency have a phase difference ϕ as they travel in the same medium. They can be represented by: D₁ = A sin (kx - ωt); D₂ = A sin ( kx - ωt + ϕ). What is the amplitude of this resultant wave? Is the wave purely sinusoidal, or not?

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The speed of waves on a string is 96 m/s. If the frequency of standing waves is 435 Hz, how far apart are two adjacent nodes?

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