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 72

Chapter 15, Problem 72

A guitar string is supposed to vibrate at 247 Hz, but is measured to actually vibrate at 262 Hz. By what percentage should the tension in the string be changed to get the frequency to the correct value?

Verified step by step guidance

The frequency of a vibrating string is related to the tension in the string by the formula:

f = \frac{v}{2 L}

, where

v = \sqrt{\frac{T}{μ}}

. Here,

f

is the frequency,

T

is the tension,

μ

is the linear mass density, and

L

is the length of the string.

Since the length and linear mass density of the string remain constant, the frequency is proportional to the square root of the tension:

\frac{f}{f} = \sqrt{\frac{T}{T}}

, where

f

is the desired frequency (247 Hz),

f'

is the actual frequency (262 Hz),

T

is the desired tension, and

T'

is the current tension.

Rearrange the equation to solve for the ratio of the desired tension to the current tension:

\frac{T}{T} = \frac{f^{2}}{f^{' 2}}

Substitute the given frequencies into the equation:

\frac{T}{T} = \frac{247^{2}}{262^{2}}

. This will give the ratio of the desired tension to the current tension.

To find the percentage change in tension, calculate the difference between the desired tension and the current tension, divide by the current tension, and multiply by 100:

\frac{T - T'}{T} × 100

. Use the ratio from the previous step to compute this percentage.

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Key Concepts

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

Frequency and Tension Relationship

The frequency of a vibrating string is directly related to the tension in the string. According to the formula for the fundamental frequency of a string, f = (1/2L)√(T/μ), where f is frequency, L is the length of the string, T is tension, and μ is the linear mass density. This relationship indicates that increasing the tension raises the frequency, while decreasing the tension lowers it.

Percentage Change Calculation

To determine the percentage change in tension required to adjust the frequency, the formula for percentage change is used: % Change = [(New Value - Original Value) / Original Value] × 100%. This calculation helps quantify how much the tension must be altered to achieve the desired frequency of 247 Hz from the measured 262 Hz.

Square Root Relationship

The relationship between frequency and tension involves a square root, meaning that a change in frequency does not result in a linear change in tension. Specifically, if the frequency increases, the tension must increase by the square of the ratio of the new frequency to the original frequency. This concept is crucial for accurately calculating the necessary adjustments to tension.

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Textbook Question

An earthquake-produced surface wave can be approximated by a sinusoidal transverse wave. Assuming a frequency of 0.60 Hz (typical of earthquakes, which actually include a mixture of frequencies), what amplitude is needed so that objects begin to leave contact with the ground? [Hint: Set the acceleration a > g. Why?]-

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Textbook Question

Destructive interference occurs where two overlapping waves are 1/2 wavelength or 180° out of phase. Explain why 180° is equivalent to 1/2 wavelength.

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Textbook Question

When you slosh the water back and forth in a tub at just the right frequency, the water alternately rises and falls at each end, remaining relatively calm at the center. Suppose the frequency to produce such a standing wave in a 45-cm-wide tub is 0.85 Hz. What is the speed of the water wave?

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Textbook Question

(II) For any type of wave that reaches a boundary beyond which its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle θ_iM corresponds to an angle of refraction equal to 90°. If θᵢ > θ_iM, all the wave is reflected at the boundary and none is refracted, because this would correspond to sin θᵣ > 1 (where is the angle θᵣ of refraction), which is impossible.

(a) Find a formula for θ_iM using the law of refraction, Eq. 15–19.

Textbook Question

A bug on the surface of a pond is observed to move up and down a total vertical distance of 0.10 m, lowest to highest point, as a wave passes. If the amplitude increases to 0.15 m, by what factor does the bug’s maximum kinetic energy change?

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Textbook Question

A longitudinal earthquake wave strikes a boundary between two types of rock at a 41° angle. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8. Assuming that the elastic modulus (Section 15–2)is the same for both types of rock, determine the angle of refraction.

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