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    Oscillations. See attached file for full problem description.
    1) An undamped oscillator has period tau_0 = 1.000s, but I now add a little damping so that its period changes
    to tau1 = 1.001s. What is the damping factor Beta? By what factor will the amplitude of oscillation decrease
    after 10 cycles? Which effect of damping would be more noticeable, the change of period or the
    decrease of the amplitude?

    2) As the damping on an oscillator is increased there comes a point when the name "oscillator" seems
    barely appropriate. (a) Illustrate this, prove that a critically damped oscillator can never pass through
    the origin x = 0 more than once. (b) Prove the same for an overdamped oscillator.

    3) The solution for x(t) for a driven, undamped oscillator is most conveniently found in the form:
    Solve that the equation and the corresponding expression for x dot, to give the coefficients B1 and B2 in
    terms of A, ?, and the initial position and velocity xo and vo. Verify the expressions given in:

    4) We know that if the driving frequency omega is varied, the maximum response (A2) of a driven damped
    oscillator occurs at (if the natural frequency is ?0 and the damping constant beta << omega0). Show that
    A2 is equal to half its maximum value when so that the full width at half maximum is just

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    Solution Summary

    Different types of Oscillations are investigated, such as damped Oscillation, driven, undamped Oscillation. The solution is detailed and well presented.