Concept explainers
(a)
To calculate: The period and frequency of the waves on string.
(a)
Answer to Problem 102P
The period
Explanation of Solution
Given:
Frequency =
Amplitude =
Linear mass density =
Tension =
Formula used:
Theperiod and frequency of the waves on string can be calculated as:
Where,
Calculation:
The frequency of the waves on the string is the similar as that of frequency of the tuning fork and their period is the reciprocal of the frequency.
The frequency of the wave given is:
The period of the wave on the wire is the reciprocal of their frequency:
Conclusion:
Thus, the period
(b)
To calculate: The speed of the wave.
(b)
Answer to Problem 102P
The speed of the wave
Explanation of Solution
Given:
Frequency =
Amplitude =
Linear mass density =
Tension =
Formula used:
For wave speed formula used is:
Where,
Calculation:
By using the tension and the linear density, wave speed can be calculated.
Relate the speed of the waves to the tension in string and linear density:
Conclusion:
Thus, the speed of the wave
(c)
To calculate: The wavelength and wave number.
(c)
Answer to Problem 102P
The wavelength
Explanation of Solution
Given:
Frequency =
Amplitude =
Linear mass density =
Tension =
Formula used:
Where,
Sound’s speed:
Frequency of wave:
The wavelength:
Calculation:
By using the frequency and the speed of the waves and the wave number The wavelength can be determined.
Relate the wavelength and wave no to the speed and frequency of the wave:
Where,
After substituting the values,
Therefore,
Now, evaluate the wave number using wave length:
Hence,
Conclusion:
Thus, the wavelength
(d)
To calculate: Suitable wave function for the wave on the string.
(d)
Answer to Problem 102P
The suitable wave function is
Explanation of Solution
Given:
Frequency =
Amplitude =
Linear mass density =
Tension =
Formula used:
For wave speed formula used is:
Where,
Calculation:
The general form of the wave function for waves on a string is
So, with the help of
Initially, find out the angular frequency of the waves:
Now, put
Conclusion:
Thus, the suitable wave function is
(e)
To calculate: max speed and acceleration point on the string.
(e)
Answer to Problem 102P
The max speed
Explanation of Solution
Given:
Frequency =
Amplitude =
Linear mass density =
Tension =
Formula used:
For max speed formula used is:
Where,
Calculation:
The max speed and acceleration ofa point on the string can be determined from the angular frequency and amplitude ofthe waves.
Relate the max speed of apoint on the string to the amplitude of the waves and tuning fork’s the angular frequency:
Now, expression for the max acceleration of string point in terms of the amplitude and angular frequency of the tuning fork is:
Put the values to get max acceleration:
Conclusion:
Thus, the max speed
(f)
To calculate: minimum average rate of energy supplied to fork.
(f)
Answer to Problem 102P
The minimum average rate of energy
Explanation of Solution
Given:
Frequency =
Amplitude =
Linear mass density =
Tension:
Formula used:
For minimum average rate of energyformula used is:
Where,
Calculation:
The expression for the minimum average power essential to keep the tuning fork oscillating at steady amplitude in terms of linear density of string, the amplitude of its vibrations and wave speed:
Where,
Now, substitute the values in the equation:
Conclusion:
Thus, the minimum average rate of energy
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