\documentclass{article}
\usepackage{exam2}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{epstopdf}
\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 11}
\time{Start of class on Thurs. March 31}
\semester{Spring}
\year{2011}                     % <- default is the current year
%\campus{City}
%\note{ Be sure to show your work and explain what you are doing. (Correct answers, for which we cannot follow the work, are worth no credit). }
\begin{document}

\begin{questions}

\question Consider a simple  pendulum of length $L=10 m$.
\begin{enumerate}
\item In an ideal world (assuming no damping, and making the small angle approximation) determine the period of oscillation. 

Now imagine we  take into account air friction  and determine that it causes the period to change  by $0.1\%$. Using Taylor's notation introduced in Eq. 5.28, what is the damping factor $\beta$? By what factor will the amplitude  of oscillation decrease after 10 cycles?
\item Which effect of damping would be  more noticeable - the change of the period or the decrease of the amplitude? Explain.
\end{enumerate}


%\question  The phase space trajectory of an undamped oscillator is shown below left.
%In the diagram, each division along the position axis corresponds to 0.1 m; along the velocity axis, 0.10 m/s.
%\begin{figure}[htc]
%\centering
%{\includegraphics[width=2.3in]{Hom10Fig1.pdf}}\caption{}
%\end{figure}
%\begin{enumerate}
%\item  What is the angular frequency $\omega_0$ of the undamped oscillator? Explain how you can tell.
%\item A retarding force is now applied to the oscillator for which the damping constant
%is equal to $\beta=0.069\omega_0$. By what factor does the amplitude change after a single oscillation? Show all work.
%\item  On the basis of your results above, carefully sketch the phase space plot for the first cycle of the motion of the damped oscillator, starting at point P.
%\end{enumerate}

\question
Imagine two concentric cylinders, centered on the vertical z axis, with radii $R \pm \epsilon$, where $\epsilon$ is very small.  A small {\it frictionless} object of radius $\epsilon$ is inserted between the two cylinders, so that it can be considered a point mass that can move freely at a fixed distance from the vertical axis.  At time $t=0$ the puck is released at height $h$ with a purely angular initial velocity $\omega_0$.

\begin{figure}[htc]
\centering
{\includegraphics[width=2.in]{Hom11Fig1.pdf}}\caption{}
\end{figure}

\begin{enumerate}
\item Write down Newton's second law first in terms of cartesian coordinates and then in terms of cylindrical polar coordinates.  Which form would be easiest to use to solve for the motion of the ball described above?   Why? (\textit{If you need a reminder about what cylindrical polar coordinates are, problem 1.47 on p. 40 of Taylor might help. If you need a reminder about Newton's law in other coordinate systems, you might look back to Taylor's discussion that led up to Eq 1.48}) 
\item Describe and/or sketch a prediction of what the motion of the ball will be for $t>0$ .  You can assume that $h$ and $\omega$ are positive, and that the ball is subject to the force of gravity (on earth) which points in the $-\hat{z}$ direction.  
\item Use the equations you wrote in part (a) to solve for the motion of the ball.  Does it match what you predicted?  If not, explain how to reconcile any differences.
\end{enumerate}

\question Consider the motion of a 2D non-isotropic oscillator, in which the
spring constant in the x direction ($k_x$) is not equal to the spring constant in the y direction
($k_y$). Each trajectory below depicts the possible motion of a unique non-isotropic
oscillator. All three oscillators, share the property that the angular frequencies $\omega_x$ and
$\omega_y$ for the motions along the x- and y-axes are commensurate, i.e., that the angular
frequencies satisfy the following relationship: $\frac{\omega_x}{n_1}=\frac{\omega_y}{n_2}$
where $n_1$ and $n_2$ are integers.
For each case below,
\begin{figure}[htc]
\centering
{\includegraphics[width=3.in]{Hom11Fig3.pdf}}\caption{}
\end{figure}
 \begin{enumerate}
\item determine whether $\omega_x$ is greater than, less than, or equal to
$\omega_y$, and determine the values $n_1$ and $n_2$  that satisfy the condition. Explain.
\item Using Mathematica generate the three Lissajous plots shown above (You can download the Mathematica code from the course webpage). What  relative phase angle (the difference between x and y phases) is needed for the three trajectories (and what flexibility is there in this choice)? 
%\item  If  $\omega_x=\omega_y$  what is  the relative phase required for  the trajectory to be  an ellipse? What determines the semimajor and semiminor axes?
\end{enumerate}


\question  Fig 2 shows  the phase space plot for a simple harmonic oscillator
(dashed) and  the same oscillator with a retarding force applied (solid). Point P
represents the initial conditions of the oscillator in both instances. In the diagram,
you may assume each division along the position axis corresponds to 0.1 m; along the
velocity axis, 0.10 m/s.
\begin{figure}[htc]
\centering
{\includegraphics[width=2.in]{Hom11Fig2.pdf}}\caption{}
\end{figure}
\begin{enumerate}
\item Explain how you can tell that the damped oscillator is not underdamped.
\item Is the damped oscillator critically damped or overdamped? Explain how you can
tell.
\item  If you said in part b that the oscillator is \{critically damped, overdamped\}, then
draw how the phase space plot would be different if the oscillator (starting at point
P) were instead \{overdamped, critically damped\}. Explain your reasoning
\end{enumerate}




\question Energy considerations in a 1D simple harmonic oscillator:
 \begin{enumerate}
\item Consider a simple harmonic oscillator with period $\tau$. Let $\langle f \rangle$ denote the average value of a function $f(t)$ averaged over one complete cycle:
    \begin{equation}
    \langle f \rangle=\frac{1}{\tau}\int_0^\tau f(t) dt.
    \end{equation} Prove that $\langle T\rangle$=$\langle U\rangle=E/2$, where $E$ is the total energy of the oscillator, $T$ the  kinetic energy and $U$ the potential energy.
    
    {\it Hint} Start by proving the more general, and extremely useful, result that $\langle \sin^2(\omega t-\delta)\rangle = \langle \cos^2(\omega t -\delta) \rangle = 1/2$. Explain why these two results are almost obvious, then prove them by using trig identities to rewrite $\sin^2\theta$ and $\cos^2\theta$ in terms of $\cos(2\theta)$.
    
    \item If an additional damping force $F_{damp}=-b \dot x$ is added, find the rate of change of the energy $E = {1\over 2} m {\dot x}^2 + {1\over 2} k x^2$ by straightforward differentiation (getting a simple result in terms of $x$, $\dot x$, and/or $\ddot x$) and then with the help of Taylor 5.24, show that it is exactly minus the rate at which energy is dissipated  by $F_{damp}$.
 \end{enumerate}   
    
{\it The rest is pure extra credit, worth up to 6 points!} 

The position of an overdamped oscillator is given by Eq. 5.40 in Taylor's text. 
 Find the constants $C_1$ and $C_2$  in terms of the initial position $x_0$ and velocity $v_0$.
  Then, sketch the behavior of $x(t)$ for the two separate cases $x_0=0$,  and $v_0=0$. 
Finally,  show that if you let 
    $\beta=0$,  your solution for x(t) matches the correct solution for undamped motion.  
    
    {\it (I find this rather remarkable, since the solution you started from is for the overdamped case, it wasn't supposed to work for the undamped case?!) As Taylor puts it, the math is sometimes cleverer than we are!}  
   



\end{questions}
\end{document}

