54. Considering an undamped, forced oscillator (b 0), show that Equation 15.35 is a solution of Equation 15.34, with an amplitude given by Equation 15.36. A common example of a forced oscillator is a damped oscillator driven by an external force that varies periodically, such as F(t) = F, sin wt, where F, is a constant and w is the angular frequency of the driving force. In general, the frequency w of the driving force is variable, whereas the natural frequency w, of the oscillator is fixed by the values of k and m. Modeling an oscillator with both retarding and driv- ing forces as a particle under a net force, Newton's second law in this situation gives d'x EF = ma, F, sin wt- b dx - kx = m dt (15.34) dt2 Again, the solution of this equation is rather lengthy and will not be presented. After the driving force on an initially stationary object begins to act, the ampli- tude of the oscillation will increase. The system of the oscillator and the surround- ing medium is a nonisolated system: work is done by the driving force, such that the vibrational ener of the system (kinetic energy of the object, elastic potential energy in the spring) and internal energy of the object and the medium increase. After a sufficiently long period of time, when the energy input per cycle from the driving force equals the amount of mechanical energy transformed to internal energy for cach cycle, a steady-state condition is reached in which the oscillations proceed with constant amplitude. In this situation, the solution of Equation 15.34 is x = A cos (wt + 6) (15.35) where Amplitude of a > (15.36) bo (? - w,?) + driven oscillator and where w, = Vk/m is the natural frequency of the undamped oscillator (b = 0).
54. Considering an undamped, forced oscillator (b 0), show that Equation 15.35 is a solution of Equation 15.34, with an amplitude given by Equation 15.36. A common example of a forced oscillator is a damped oscillator driven by an external force that varies periodically, such as F(t) = F, sin wt, where F, is a constant and w is the angular frequency of the driving force. In general, the frequency w of the driving force is variable, whereas the natural frequency w, of the oscillator is fixed by the values of k and m. Modeling an oscillator with both retarding and driv- ing forces as a particle under a net force, Newton's second law in this situation gives d'x EF = ma, F, sin wt- b dx - kx = m dt (15.34) dt2 Again, the solution of this equation is rather lengthy and will not be presented. After the driving force on an initially stationary object begins to act, the ampli- tude of the oscillation will increase. The system of the oscillator and the surround- ing medium is a nonisolated system: work is done by the driving force, such that the vibrational ener of the system (kinetic energy of the object, elastic potential energy in the spring) and internal energy of the object and the medium increase. After a sufficiently long period of time, when the energy input per cycle from the driving force equals the amount of mechanical energy transformed to internal energy for cach cycle, a steady-state condition is reached in which the oscillations proceed with constant amplitude. In this situation, the solution of Equation 15.34 is x = A cos (wt + 6) (15.35) where Amplitude of a > (15.36) bo (? - w,?) + driven oscillator and where w, = Vk/m is the natural frequency of the undamped oscillator (b = 0).
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Transcribed Image Text:54. Considering an undamped, forced oscillator (b 0),
show that Equation 15.35 is a solution of Equation
15.34, with an amplitude given by Equation 15.36.
A common example of a forced oscillator is a damped oscillator driven by an
external force that varies periodically, such as F(t) = F, sin wt, where F, is a constant
and w is the angular frequency of the driving force. In general, the frequency w of
the driving force is variable, whereas the natural frequency w, of the oscillator is
fixed by the values of k and m. Modeling an oscillator with both retarding and driv-
ing forces as a particle under a net force, Newton's second law in this situation gives
d'x
EF = ma, F, sin wt- b
dx
- kx = m
dt
(15.34)
dt2
Again, the solution of this equation is rather lengthy and will not be presented.
After the driving force on an initially stationary object begins to act, the ampli-
tude of the oscillation will increase. The system of the oscillator and the surround-
ing medium is a nonisolated system: work is done by the driving force, such that
the vibrational ener
of the system (kinetic energy of the object, elastic potential
energy in the spring) and internal energy of the object and the medium increase.
After a sufficiently long period of time, when the energy input per cycle from the
driving force equals the amount of mechanical energy transformed to internal
energy for cach cycle, a steady-state condition is reached in which the oscillations
proceed with constant amplitude. In this situation, the solution of Equation 15.34 is
x = A cos (wt + 6)
(15.35)
where
Amplitude of a >
(15.36)
bo
(? - w,?) +
driven oscillator
and where w, = Vk/m is the natural frequency of the undamped oscillator (b = 0).
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