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Problems

Harmonic Oscillators with Damping

Problem 1

Using a force of 4 newtons, a damped harmonic oscillator is displaced from equilibrium by 0.2 meters. At t = 0 it is released from rest. The resultant displacement of the oscillator, from the equilibrium position, as a function of time, is shown in the figure below. Estimate, as well as you can using the given information:

  1. The mass of the oscillator.
  2. The quality factor of the oscillator.

 Graph showing results of displacement on the y axis and time on the x axis. The oscillations gradually move from large displacement to small displacement in sixty seconds.

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Problem 2

An RLC circuit with switch open at time zero. 

The circuit shown above consists of a capacitor (C), an inductor (L) and a resistor (R). Initially the switch is open and the charge (Q) on the capacitor is Q(t \leq 0) =Q_0. At t = 0, the switch is closed.

  1. For what values of R, the initial sign of Q(t) will be reversed at some later times?
  2. Assuming that the value of R satisfies the condition in (1), find Q(t) and the current I(t) in terms of Q_0, R, L and C. Draw qualitative sketches of both Q(t) and I(t). (Make sure to indicate the behaviour around t=0).
  3. If it takes 10 cycles for the energy in the circuit to decrease to 1/e times its initial value, find the value of the resistance R in terms of L and C.

 

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