Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 18, Problem 79PQ

(a)

To determine

The possible frequencies for tuning the guitar string.

(a)

Expert Solution
Check Mark

Answer to Problem 79PQ

The possible frequencies for tuning the guitar string either 192Hz or 200Hz.

Explanation of Solution

Write the expression for beat frequency

  fbeat=|f2f1|                                                                                          (I)

Here, f1 is the frequency of the tune in guitar string and f2 is the frequency of the reference tone.

Conclusion:

Substitute 196Hz for f2 and 4.00beats/s for f1 in equation (I) to find fbeat.

  fbeat=|196Hz4.00beats/s|=192Hz

  fbeat=|196Hz+4.00beats/s|=200Hz

Therefore, the possible frequencies for tuning the guitar string either 192Hz or 200Hz.

(b)

To determine

The frequency of the guitar string, if the musician loosens the guitar string.

(b)

Expert Solution
Check Mark

Answer to Problem 79PQ

The frequency of the guitar string, if the musician loosens the guitar string is 198Hz.

Explanation of Solution

If loosening the string lowers the wave speed and frequency, if the original frequency was 192Hz, the difference in frequencies would become larger and the beats would speeds up.

Write the expression for beat frequency, if the musician loosens the guitar string

  fbeat=|f2f1|                                                                                       (II)

Here, f1 is the frequency of the loosened guitar string and f2 is the original frequency of the guitar string.

Conclusion:

Substitute 200Hz for f2 and 2.00beats/s for f1 in equation (II) to find fbeat.

  fbeat=|200Hz2.00beats/s|=198Hz

Instead the frequency must have started at 200Hz to become 198Hz.

Therefore, frequency of the guitar string, if the musician loosens the guitar string is 198Hz

(c)

To determine

The percentage of the tension in the guitar string change for the string to be tuned to 196Hz.

(c)

Expert Solution
Check Mark

Answer to Problem 79PQ

The percentage of the tension in the guitar string change for the string to be tuned to 196Hz is reduced by 2.01%.

Explanation of Solution

Write the expression for fundamental frequency depends on the tension in the string.

  f1=12LFTμ                                                                                                    (III)

Here, f1 is the fundamental frequency, L is the length of the string, FT is the tension force, and μ is the linear mass density.

The ratio of the two frequencies then depends on the square root of the ratio of tensions.

  f2f1=FT1FT2

Here, f1 is the frequency of the guitar string, if the musician loosens the guitar string, f2 is the original frequency of the tone, FT1 is the tension due to frequency f1, and FT2 is the tension due to frequency f2.

Rearrange the above equation for FT2.

  (f2f1)2=FT1FT2FT2=(f2f1)2FT1                                                                                          (IV)

Conclusion:

The fractional change should be made in the tension is,

  fractional change=FT2FT1FT1

Rearrange the above equation and then compare with equation (IV).

  fractional change=FT2FT11=(f2f1)21

Substitute 198Hz for f1 and 196Hz for f2 in the above equation.

  fractional change=(196Hz198Hz)21=0.0201=2.01%

Therefore, the current tension should be reduced by a factor of 2.01% in order to bring the string in tune with the reference tone.

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Chapter 18 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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