Concept explainers
(a)
The linear mass density of the rope.
(a)
Answer to Problem 26PQ
The linear mass density of the rope is
Explanation of Solution
Write an expression for the wave speed.
Here,
Write an expression for the wave speed.
Here,
Compare equation (I) and (II).
Rearrange equation (III) to find
Write an expression for the tension.
Here,
Write an expression for the wave number.
Substitute equation (V) and (VI) in equation (IV).
Conclusion:
Substitute
Thus, the linear mass density of the rope is
(b)
The change in wave properties if the frequency of the wave is doubled.
(b)
Answer to Problem 26PQ
The tension and linear density will not vary with the frequency, the wave number is proportional to the angular frequency, and thus, the wave number doubles. The new wave number is
Explanation of Solution
Write an expression for the wave speed.
Write an expression for the wave speed.
Compare equation (I) and (II).
Thus, the wave number is proportional to the angular frequency. Thus, the frequency doubled the wave number doubles. The tension and linear density will not vary with the frequency.
Since the wavenumber and the frequency are proportional, write an expression for the new wave number.
Here,
Write an expression for the old wave number.
Substitute equation (VIII) in equation (VII).
Conclusion:
Substitute
(c)
The change in wave properties if the mass is doubled.
(c)
Answer to Problem 26PQ
As the mass doubled, the tension doubles, the linear mass density and the frequency remains constant. The wave number decreases by a factor of
Explanation of Solution
Write an expression for the new wave number.
Substitute
Substitute
Substitute
Conclusion:
Substitute
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Chapter 17 Solutions
Student Solutions Manual For Katz's Physics For Scientists And Engineers: Foundations And Connections, Volume 1
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