2.5: Chapter Checklist

Last updated
Save as PDF

Page ID: 34355

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

You should now be able to:

Solve for the free motion of the damped harmonic oscillator by looking for the irreducible complex exponential solutions;
Find the steady state solution for the damped harmonic oscillator with a harmonic driving term by studying a corresponding problem with a complex exponential force and finding the irreducible complex exponential solution;
Calculate the power lost to frictional forces and the phase lag in the forced harmonic oscillator;
Feel it in your bones!

Problems

2.1. Prove that an overdamped oscillator can cross its equilibrium position at most once.

2.2. Prove, just using linearity, without using the explicit solution, that the steady state solution to (2.16) must be proportional to \(F_{0}\).

2.3. For the system with equation of motion (2.14), suppose that the driving force has the form \[f_{0} \cos \omega_{0} t \cos \delta t\]

where \[\delta \ll \omega_{0} \quad \text { and } \quad \Gamma=0 .\]

As \(\delta \rightarrow 0\), this goes on resonance. What is the displacement for \(\delta\) nonzero to leading order in \(\delta / \omega_{0}\)? Write the result in the form \[\alpha(t) \cos \omega_{0} t+\beta(t) \sin \omega_{0} t\]

and find \(\alpha(t)\) and \(\beta(t)\). Discuss the physics of this result. Hint: First show that \[\cos \omega_{0} t \cos \delta t=\frac{1}{2} \operatorname{Re}\left(e^{-i\left(\omega_{0}+\delta\right) t}+e^{-i\left(\omega_{0}-\delta\right) t}\right) .\]

2.4. For the system shown in Figure \( 2.9\), suppose that the displacement of the end of the wire vanishes for \(t < 0\), and has the form \[d_{0} \sin \omega_{d} t \quad \text { for } \quad t \geq 0 .\]

Find the displacement of the block for \(t > 0\). Write the solution as the real part of complex solution, by using a complex force and exponential solutions. Do not try to simplify the complex numbers. Hint: Use (2.23), (2.24) and (2.6). If you get confused, go on to part b.
Find the solution when \(\Gamma \rightarrow 0\) and simplify the result. Even if you got confused by the complex numbers in a., you should be able to find the solution in this limit. When there is no damping, the “transient” solutions do not die away with time!

2.5. For the \(LC\) circuit shown in Figure \( 1.10\), suppose that the inductor has nonzero resistance, \(R\). Write down the equation of motion for this system and find the relation between friction term, \(m \Gamma\), in the damped harmonic oscillator and the resistance, \(R\), that completes the correspondence of (1.105). Suppose that the capacitors have capacitance, \(C \approx 0.00667 \mu F\), the inductor has inductance, \(L \approx 150 \mu H\) and the resistance, \(R \approx 15 \Omega\). Solve the equation of motion and evaluate the constants that appear in your solution in units of seconds.