Assume that the damping force for the damped harmonic oscillator is proportional to the sq...

Assume that the damping force for the damped harmonic oscillator is proportional to the square of its velocity; that is, it is given by . The equation of motion for such an oscillator is thus

where γ= c2/2m and . Find x(t) by numerically integrating the above equation of motion. Let γ = 0.20 m−1 and ω0 = 2.00 rad/s. Let the initial conditions be x(0) = 1.00 m and  m/s.

(a) Plot x(t) from t = 0 to 20 s. Also, on the same graph, plot the solution for the damped harmonic oscillator where the damping force is linearly proportional to the velocity; that is, it is given by . Again, let γ = c1/2m = 0.20 s−1 and ω0 = 2.00 rad/s.

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(b) For the case of linear damping, plot the log of the absolute value of the successive extrema versus their time of occurrence. Find the slope of this plot, and use it to estimate γ. (This method works well for the case of weak damping.)

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(c) Find the value of γ that results in critical damping for the linear case. Plot this solution from t = 0 to 5 s. Can you find a well-defined value of γ that results in critical damping for the quadratic case? If not, what value of γ is required to limit the first negative excursion of the oscillator to less than 2% of the initial amplitude?