Concept explainers
Review. A steel guitar string with a diameter of 1.00 mm is stretched between supports 80.0 cm apart. The temperature is 0.0°C. (a) Find the mass per unit length of this siring. (Use the value 7.86 × 103 kg/m4 for the density.) (b) The fundamental frequency of transverse oscillations of the string is 200 Hz. What is the tension in the string? Next, the temperature is raised to 30.0°C. Find the resulting values of (c) the tension and (d) the fundamental frequency. Assume both the Young’s modulus of 20.0 × 1010 N/m2 and the average coefficient of expansion α = 11.0 × 10-6 (°C)-1 have constant values between 0.0°C and 30.0°C.
(a)
The mass per unit length of the string.
Answer to Problem 19.73CP
The mass per unit length of the string is
Explanation of Solution
Given Info: The diameter of the string of the steel guitar is
Formula to calculate the radius of the wire is,
Here,
Substitute
Thus, the value of the radius is
The area of cross section of the steel string is,
Substitute
Thus, the area of cross section of the steel string is
Formula to calculate the mass per unit length of the steel string is,
Here,
Substitute
Conclusion:
Therefore, the mass per unit length of the string is
(b)
The Tension in the string.
Answer to Problem 19.73CP
The Tension in the string is
Explanation of Solution
Given Info: The diameter of the string of the steel guitar is
Formula to calculate the fundamental frequency is,
Here,
Rearrange the above expression for
Substitute
Conclusion:
Therefore, the Tension in the string is
(c)
The Tension in the string when the temperature is raised to
Answer to Problem 19.73CP
The Tension in the string when the temperature is raised to
is
Explanation of Solution
Given Info: The diameter of the string of the steel guitar is
Formula for the change in the length, when temperature varies is,
Here,
Substitute
Thus, the final length of the brass pendulum is
Formula to calculate the tension in the wire is,
Substitute
Conclusion:
Therefore, the Tension in the string when the temperature is raised to
is
(d)
The fundamental frequency of the string.
Answer to Problem 19.73CP
The fundamental frequency of the string is
Explanation of Solution
Given Info: The diameter of the string of the steel guitar is
Write the expression for the fundamental frequency.
Substitute
Conclusion:
Therefore, the fundamental frequency of the string is
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Chapter 19 Solutions
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