(a)
Find the rate at which
(a)
Answer to Problem 5.45P
The rate at which
And average power is
Explanation of Solution
Write the mathematical equation of force driven to the oscillator
Here
The equation for displacement of damped oscillation is given by
Here
Write the equation for amplitude of an oscillator driven by a sinusoidal force with variable frequency
Differentiate (II)
The rate at which force does work is equal to the product of force and velocity.
Then,
Since
The average rate of work done is given by
For one complete cycle,
But
So,
If number of cycles completed by the oscillator is
Then,
Conclusion:
The rate at which
And average power is
(b)
Verify that the average rate of work done over any number of complete cycles is equal to the average rate at which energy is lost to the resistive force.
(b)
Answer to Problem 5.45P
The average rate of work done over any number of complete cycles is equal to the average rate at which energy is lost to the resistive force.
Explanation of Solution
Average rate at which energy lost to the resistive force is,
Resistive force is proportional to the velocity
Substitute
Average rate at which energy lost to the resistive force is,
The value of integration
Then,
Conclusion:
The average rate of work done over any number of complete cycles is equal to the average rate at which energy is lost to the resistive force.
(c)
Show that
(c)
Answer to Problem 5.45P
Explanation of Solution
The average rate at which the force does the work is
Substitute for
For maximum value of
Conclusion:
Therefore,
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Chapter 5 Solutions
Classical Mechanics
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