:PROPERTIES: :ID: a67bd5a3-5cd8-4912-be74-33f4e905f28c :mtime: 20231024013706 :ctime: 20231024013705 :END: #+title: oscillation problems #+filetags: :public:project: * Oscillation Problem An Oscillation problem is a problem with a pendulum or a circle. * Oscillation Problems using [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][Complex Numbers]] The angular displacement of a pendulum can be modeled by \[x(t) = \alpha \cos(\omega t) + b\sin(\omega t)\] so \[x(t) = Re(A\exp(i\omega t))\] * Example Problems ** Example 1: Impedance of an AC Circuit #+BEGIN_SRC dot :file AC.png :cmdline -Kdot -Tpng :exports both digraph traffic { node [shape=Mrecord] C [label="{Capacitor}"] ; R [label="{Resistor}"] ; V [label="{Alternating Voltage}"] ; V -> R ; R -> C ; C -> V ; } #+END_SRC #+RESULTS: [[file:AC.png]] \[Q_{cap} = V_{cap}C\] \[Q_{cap} = Re(Q_{0} e^{i \omega t})\] \[V_{cap} = Re(\frac{Q_{0}}{C}e^{i \omega t})\] \[I = \frac{dQ_{cap}}{dt} = Re(i\omega Q_{0}e^{i \omega t})\] \[I = \frac{dQ_{cap}}{dt} = Re(I_{0} e^{i \omega t})\] \[V_{res} = IR = Re(i \omega Q_{0} e^{i \omega t})\] \[V_{tot} = V_{res} + V_{cap} = Re((i \omega R + \frac{1}{C}Q_{0} e^{i \omega t}))\] \[V_{tot = Re((R + \frac{1}{i \omega C})I_{0}e^{i \omega t})}\]