\documentclass{article}
\usepackage{exam3}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{epstopdf}
\usepackage[colorlinks=true]{hyperref}
\usepackage{enumerate}
\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 11}
\time{Start of class on Thurs. Apr 5}
\semester{Spring}
\year{2012}                     % <- 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}



%  I tried to incorporate some of this physics into the 2nd question below (the one from UMaine) 
%\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. Then,  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 in practice - 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}

{\bf IMPORTANT!} As part of this homework (i.e. before Apr 5) ONE member of your final project team should go to \url{ http://www.colorado.edu/physics/phys2210/phys2210_sp12/preflights/project_v1.html} and fill in the brief form. If you have partners, please coordinate with each other - everyone on the team gets one homework problem credit for this.  
\begin{questions}
\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.
%\begin{figure}[htc]
%\centering
\begin{center}
{\includegraphics[width=3.5in]{Hom11Fig3.pdf}}%\caption{}
%\end{figure}
\end{center}
 \begin{enumerate}
\item determine the (simplest possible) ratio of $\omega_x$ /$\omega_y$.  Explain your reasoning.
\item Use your favorite computational environment to generate the three Lissajous plots shown above (ParametricPlot may be handy in Mathematica) We said in class that you can always choose your ``x" phase angle to be zero. Can you do that for all 3 figures?  Briefly, explain why or why not. 
%\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 The phase space trajectory of an undamped oscillator is shown below.  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
{\Large MISSING FIGURE}
%%%%\begin{center}
%%%%{\includegraphics[width=1.5in]{hw11_umaine_fig1.pdf}}%\caption{}
%%%%%\end{figure}
%%%%\end{center}
\begin{enumerate}
\item  First, what is the angular frequency $\omega_0$ of the undamped oscillator?  Explain how you can tell.   Next - a retarding force is now applied to the oscillator for which the damping constant (using Taylor's notation, introduced in Eq. 5.28) is equal to $\beta = 0.0644 \omega_0$.   By what factor does the amplitude change after a single oscillation?  After 10 cycles?   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. {\it There is a larger figure on the last page of this homework set that you can print out to do your sketching.} 
\item  If this system corresponded to a real pendulum, which effect of damping would be more noticeable - the change of the period or the decrease of the amplitude? Justify your opinion.  

\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, each division along the position axis corresponds to 0.2 m; along the velocity axis, 0.20 m/s.
%\begin{figure}[htc]
%\centering
%{\includegraphics[width=1.7in]{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 {\it 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 we add a damping force $F_{damp}=-b \dot x$, 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 the rate at which work is done  by $F_{damp}$.
 
    
\item {\it The rest is pure extra credit, worth up to 4 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 overdamped, it wasn't supposed to work for the undamped case?! As Taylor puts it, the math is sometimes cleverer than we are!}  
    \end{enumerate}
   

\question {\bf In the following pair of problems} you will compare our model for a damped harmonic oscillator to real data collected from a video of a spring-mass system immersed in viscous oil. The video from which this position vs time data is collected is here: \url{http://www.youtube.com/watch?v=ZYFVKZPut9w}.
Data was obtained using free video tracking software (Tracker) and saved in a comma-separated variables (CSV) file.

 First, you will describe a few properties of a damped oscillator to assist you in your analysis of the data.

\begin{enumerate}[(a)]
\item Write down the differential equation for a damped harmonic oscillator with undamped natural frequency, $\omega_0$, and damping parameter, $\beta$.  Write down the general solution for this differential equation for the underdamped case ($\beta < \omega_0$). Then, consider the following sketch of a solution to this differential equation for a particular choice of $\beta$ and $\omega_0$ when the oscillator was displaced and let go. 

\begin{center}
\includegraphics[width=0.7\linewidth]{sample_oscillator.pdf}
\end{center}

You can obtain an estimate for $\omega_1 = \sqrt{\omega_0^2 - \beta^2}$ using the time between successive zero crossing. Estimate  $\omega_1$ in the figure above and explain how you obtained your result.

\item You can also obtain an estimate for $\beta$ using successive maxima or minima from the plot. Estimate  $\beta$ and $\omega_0$. Explain how you obtained your result.

\item If $\omega_0$ were kept constant but $\beta$ were increased (but still below $\omega_0$), sketch how the plot above would change. What physical change to a spring-mass immersed in oil can be done to increase $\beta$? Answer both questions for the case where $\omega_0$ is kept constant but $\beta$ is decreased. What property of the sketch does $\beta$ appear to control?

\item If $\beta$ were kept constant but $\omega_0$ were decreased (but still above $\beta$), sketch how the plot above would change. What physical change to a spring-mass immersed in oil can be done to decrease $\omega_0$? Answer both questions for the case where $\beta$ is kept constant but $\omega_0$ is increased. What property of the sketch does $\omega_0$ appear to control?

\end{enumerate}

\question You will now analyze a real spring-mass system immersed in viscous oil for which position versus time data was collected for the oscillator.

\begin{enumerate}[(a)]

\item The position versus time data appears in a comma-separated variables (CSV) file on our course home page and also our D2L page. Importing this data is straight-forward in Mathematica using the \texttt{Import} command. You'll need to store the data in a variable, so you can plot it.\\ 

For example, \texttt{data = Import["/file/to/datafile.csv"]} will import the values in the CSV file into the variable {\tt data}.
 (You will use Insert $\rightarrow$ Filepath to generate the right file path on your computer - if you need a little more help, a video describing how to import data into Mathematica is located here: \url{http://youtu.be/MmS3JNk7JE4} .)

Plot this data using the {\tt ListPlot} command,\\{\tt experiment = ListPlot[data[[All, \{1, 2\}]], PlotRange -> All]}.\\
You are storing this plot (as {\tt experiment}) for later, so you can display the experimental data and the results from your model on the same plot.\\

\item Using the plot of the data, estimate values for $\omega_1$, $\beta$, and $\omega_0$. Determine the initial position and initial velocity for the oscillator. This oscillator was displaced and released from rest.\\

\item Plot the particular solution to damped harmonic oscillator for your model parameters ($\omega_1$, $\beta$, and $\omega_0$) and initial conditions. You should store this plot (e.g., {\tt model = Plot[...]}).

\item Plot the result of your model and the experimental data on the same axes using the {\tt Show} command. This is why you stored the plots earlier (e.g., {\tt Show[experiment, model]}). How does your model match the experimental data? Can you tweak your model parameters to make the fit better? 

\item Why did we do this? Experimental physicists collect data and often attempt to fit a model of that system to their data. It's likely that you got pretty good but not great agreement with the data that was collected. Can you identify at least 3 aspects of the physical system (spring mass immersed in oil) that might have improved the model? Can you identify at least 2 aspects of the data collection procedure (video tracking software) that might have helped you better estimate your model parameters ($\omega_1$, $\beta$, and $\omega_0$)?

\end{enumerate}

\end{questions}

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{\Large MISSING FIGURE}
%%%\begin{center}
%%%{\includegraphics[width=6.5in]{hw11_umaine_fig1.pdf}}%\caption{}
%%%\end{center}

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