Concept explainers
(a)
The mass per unit length of the guitar string.
(a)
Answer to Problem 79PQ
The mass per unit length of the guitar string is
Explanation of Solution
Write the expression for volume of the wire.
Here,
The wire has a circular cross section.
Write the expression for cross sectional area of wire.
Here,
Use equation (II) in (I).
Write the expression for linear mass density of the wire.
Here,
Rewrite equation (I) to find
Use above equation in (IV).
Use equation (II) in (V).
Write the expression for density of the wire.
Here,
Use equation (VII) in (VI).
Write the expression to find radius of the wire.
Here,
Conclusion:
Substitute
Substitute
Therefore, the mass per unit length of the guitar string is
(b)
The tension in the guitar string.
(b)
Answer to Problem 79PQ
The tension in the guitar string is
Explanation of Solution
Write the expression for the fundamental frequency of vibration of the guitar string.
Here,
Write the expression to find the speed of the waves in terms of the tension and mass density of the wire.
Here,
Use above expression in equation (X).
Solve equation (XI) for
Conclusion:
Substitute
Therefore, the tension in the guitar string is
(c)
The tension in the guitar string if the temperature is increased to
(c)
Answer to Problem 79PQ
The tension in the guitar string if the temperature is increased to
Explanation of Solution
Write the expression to find the unstressed length of the guitar string at
Here,
Solve equation (XIII) to find
Write the expression to find the unstressed length at
Here,
Write the equation to find the scale length of the string.
Here,
Solve equation (XVI) for
Conclusion:
Substitute
Substitute
Substitute
Substitute
Therefore, the tension in the guitar string if the temperature is increased to
(d)
The fundamental frequency of guitar string at
(d)
Answer to Problem 79PQ
The fundamental frequency of guitar string at
Explanation of Solution
Write the expression for the ratio of frequency at
Here,
Solve equation (XVIII) for
Conclusion:
Substitute
Therefore, the fundamental frequency of guitar string at
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Chapter 19 Solutions
Physics For Scientists And Engineers: Foundations And Connections
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