Ch. 15 - Wave Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 15 - Wave Motion

문제 72

15장, 문제 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?

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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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주요 개념

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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.