MATH 597/GEOPH 697 Fall 2004





HOMEWORK DUE THURSDAY SEPTEMBER 9:
Chapter 1 - #4
Chapter 2 - #1
The following three questions:





A nonlinear damped pendulum satisfies the equation

\begin{displaymath}l\ddot{\theta } + k\dot{\theta } + g \sin(\theta )=0\end{displaymath}


\begin{displaymath}\theta (0) =0, \hspace*{.2in} \dot{\theta }(0)=\omega_0.\end{displaymath}

  1. By suitably non-dimensionalizing the equation, show that the model can be written in the form

    \begin{displaymath}\ddot{\theta } + \dot{\theta } + \epsilon \sin(\theta )=0\end{displaymath}


    \begin{displaymath}\theta (0) =0, \hspace*{.2in} \dot{\theta }(0)=\mu ,\end{displaymath}

    and give the definitions of $\epsilon $ and $\mu $.
  2. The pendulum is suspended in a bath of liquid (e.g. water). Why might this be consistent with a value of $\epsilon \ll 1$?
  3. By rescaling $t=\tau / \epsilon $, find an approximate equation satisfied by $\theta $ over this longer time scale, and explain why a suitable initial condition for $\theta $ as $\tau \rightarrow 0$ is $\theta \approx \mu $.