Homework for the damped oscillator

  1. The current in a circuit is represented by the complex number \(I(t)=I_{0}e^{i\left( \omega t+\varphi\right)}\). Current cannot be complex, but we represent it this way with the understanding that the real part will represent the actual current in the circuit. (a) Represent the current on an Argand diagram at $t=0$ and your choice of \(\varphi\). (b) Represent on the same diagram at $t=0$ and the same choice of \(\varphi\). What is the phase of $I(t)$ relative to \(\dot{I}(t)\) at any time? (c ) Represent the charge $q(t)$ on a capacitor in that circuit on the same diagram at $t=0$ and the same choice of \(\varphi\). What is the phase of $I(t)$ relative to $q(t)$ at any time?
  2. A series LRC circuit is driven by a sinusoidal voltage that by convention we write as $V_0 \cos \left( \omega t\right)$. Draw phasor diagrams representing the driving voltage and each of the voltages across the capacitor VC, resistor VR, inductor VL in a driven LRC circuit for three different cases: (1) , (2) (resonance frequency), (3) . And example is done for you. For each of the 3 frequency regimes, explain whether and why the circuit as a whole behaves predominantly as a resistor, capacitor, or inductor.
  3. A forced oscillator characterized by a damping parameter $\beta$ and frequency $\omega_0$ has a generalized displacement response \[x(t)=\frac{F_0/m}{\sqrt{\left( \omega _0^{2}-\omega ^{2} \right)^{2}+4\omega ^{2}\beta ^{2}}}\cos \left( \omega t+\delta \right).\] This function is of the form \(x(t)=A\left( \omega \right)\cos \left( \omega t+\delta \right)\), where \(A\left( \omega \right)\) measures the amplitude of the response and it is a (strong) function of frequency.

(a) Sketch the form of \(A\left( \omega \right)\). (b) Find the exact frequency at which the amplitude \(A\left( \omega \right)\) is maximal. (c) If the damping is light ($\omega_0 \beta \ll 1$), show that the frequency of maximum response (resonance frequency) is approximately \(\omega_0\), and that it differs from \(\omega _0\) by a term of order \(\beta ^{2}\). You will need to use the approximation \[(1+z)^{p}\approx 1+pz+\frac{p\left( p-1 \right)}{2!}z^{2}+\ldots \] (d) In the light damping approximation, what is the value of the maximum value of \(A\left( \omega \right)\)? (e) In the light damping approximation, find the “width” of the \(A\left( \omega \right)\) function, by finding the two frequencies at which \(A\left( \omega \right)\) drops to \(\frac{1}{\sqrt{2}}\) of its maximum. Be careful not to make such a stringent approximation that the width is zero! (f) Define the difference between these two frequencies as $\Delta \omega$, and find an expression for \(Q\equiv \frac{\omega _{0}}{\Delta \omega }\) in terms of \(\beta\). [Note: Generally, it is the power that is measured (proportional to the square of the displacement), not the displacement itself. Thus $\Delta \omega$ as defined above corresponds to the full-width at half-maximum (FWHM) of the power response curve.]

  1. FM radio stations have broadcast frequencies of approximately 100 MHz. Assume that your radio uses a series LRC circuit similar to the one you used in the lab as part of the receiver electronics. The quality factor $Q$ of the receiver circuit determines the spacing of the broadcast frequencies of the stations your receiver pick up without interference from other stations. Estimate the spacing of the broadcast frequencies of FM stations if typical receivers have a $Q$ of 500 or better. Explain your reasoning, and include a graph.

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