Problems

Driven Harmonic Oscillators

Problem 1

A torsional oscillator comprises a cylinder with moment of inertia, \(I\), hanging from a light rod with torsional spring constant, \(\kappa\). The cylinder also experiences a drag torque equal to \(-\mu \dot \theta\), when moving with angular velocity \(\dot \theta\). The top of the rod is driven with angular displacement \(\phi(t) = \phi_0 \cos{\omega t}\).

  1. Find the steady-state solution for \(\theta(t)\).
  2. Plot the amplitude \(A(\omega) \) and phase \(\delta(\omega)\) of your solution for \(\theta(t)\) in (1) as a function of \(\omega\). For your plot, assume that the natural frequency of oscillation of the system \(\omega_0 = 1\), and plot three curves on the same plot with \(\frac{\mu}{I} = 0.25\), 1 and 2. Label your curves to distinguish the three cases.

 

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Problem 2

A capacitor (of capacitance C), a resistor (of resistance R) and an inductor (of inductance L) are connected to an AC voltage source \(V = V_0 \sin(\omega t)\) starting at \(t=0\) as shown in the diagram below.

An RLC circuit connected to an AC voltage source.

Assuming that both the current and the charge of the capacitor are initially zero, determine the expression for \(V_C(t\ge0)\) with \(\omega=\omega_0=\dfrac{1}{\sqrt{LC}}\) and \(L < 4R^2C\).

 

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