Concept explainers
(a)
The wave function for this vibration.
(a)
Explanation of Solution
Given:
Length of string is
Maximum amplitude is
Frequency of the string is
Formula used:
Write expression for general form of wave function for 3rd harmonic.
Write expression for
Write expression for wavelength of standing wave for string fixed at one end.
Write expression for wave number.
Calculation:
Substitute
Substitute
Substitute
Substitute
Conclusion:
Thus, the wave function for vibration is
(b)
Function for kinetic energy; time at which kinetic energy is maximum.
(b)
Explanation of Solution
Given:
Length of string is
Maximum amplitude is
Frequency of the string is
Formula used:
Write expression for general form of wave function for 3rd harmonic.
Write expression for angular velocity.
Write expression for wavelength of standing wave for string fixed at one end.
Write expression for wave number.
Write expression for kinetic energy.
Calculation:
Substitute
Substitute
Substitute
Substitute
Differentiate above expression with respect to
Substitute
Substitute
Write expression for condition of maximum kinetic energy.
Solve above expression.
Conclusion:
Thus the function for kinetic energy is
(c)
The maximum kinetic energy of string by integration.
(c)
Explanation of Solution
Given:
The function for kinetic energy is
Formula used:
Write expression for general form of wave function for 3rd harmonic.
Write expression for kinetic energy.
Calculation:
Differentiate equation (1) with respect to
Substitute
Integrate above expression from
Conclusion:
Thus, the value of maximum kinetic energy is
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Chapter 16 Solutions
Physics for Scientists and Engineers, Vol. 3
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