\documentclass{article}
\usepackage{exam3}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{epstopdf}
\usepackage{hyperref}
\usepackage{color}
\usepackage{enumerate}

\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 12}
\time{Start of class on Thurs. Apr 12}
\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}

\begin{questions}




%\question On the last exam, we had a problem with a flat half-disc of radius $R$, with uniform mass per unit area of $\sigma$. We asked for the gravitational potential energy of mass m at position $z$ on the central axis, and the answer was

 %  \begin{equation}
 %   U(z) =  -\pi G \sigma m (\sqrt{R^2+z^2} - \sqrt{z^2})
   % \end{equation}

%     \begin{enumerate}
   %  \item  We asked people to ``check" their answer in two ways. Many of you did fine on this, but I get the sense that a lot of the class still isn't seeing what we mean by checking.       If you are far from the disk (on the $z$ axis), what do you {\it expect} for the formula for U(z)? (Don't say ``0'' -  as usual, we want the functional form of $U(z)$ as you move far away. Also, explicitly state what we mean by ``far away''. (Please don't compare something with units to something without units!)
%    \item Show explicitly that the formula above does indeed give precisely the functional dependence you expect.
    
    
    \question
    
 Go back to midterm \#2, and pick any ``lettered" portion (like 7-i or whatever) where you lost more than $\sim$1/3 of the points. Look it over, redo the problem, and explain  what pieces you were missing and why you missed them. Solutions are posted, so we're not so much interested in ``the right answer" as your reflections about what you did wrong and why. (For a couple of you, there were no sub-parts where you lost  this much credit - just let the grader know.)

\question 
We often ask you to {\it check} your answer. ``Check"  (in this sense) doesn't mean ``redo the problem carefully", nor to invent and solve a related problem. It means  think about a basic physical result you know must be true {\it independent } of working out the formula, and then you check that your formulaic  answer agrees with that expectation. (E.g, the units are as expected, or the behavior at extreme values of some parameter does what you expect...)   To be useful, you must {\it already know} what that behavior should be (e.g. if friction goes up, terminal velocity should go down. Or, if you get far away, a force should vanish. That sort of thing)  


{\it Let's practice! } In each case below, I will pose an abbreviated problem and a proposed answer. Your task is NOT to solve the problem(!!) but rather, to simply CHECK the given answer. If you can think of more than one way, all the better. In each case, briefly comment! If the check does NOT work out, does it suggest to you what MIGHT be wrong, where you might look more carefully if you needed to get the answer right? 

\begin{enumerate}
%\item A star consists of an inner spherical core of radius $R_1$ and uniform density $\rho_1$,
%surrounded by a spherical gaseous shell of outer radius $R_2$ and uniform density $\rho_2$.
%A point mass, $m$, is placed within the gaseous shell, a distance $r$ from the center of the star
%(so $R_1<r<R_2$) .  What is the gravitational force on $m$?


%An alien death star is built with a thick unobtanium hull surrounding a hollow inner working space. The (nearly) solid hull has inner radius $R_1$, and  outer radius $R_2$. There is a narrow crawlway through this hull. What is the gravitational force on a test mass ``m" in this crawlway (i.e. deep within the massive hull), located a distance r from the center of the ship?

%The student's response after several lines of careful calculations was:
  % \begin{equation}
  % \vec  F_{grav}=  -{8\over 3} \pi m G \rho( {{R_1^3}\over r^2} -  r)\hat r
  %  \end{equation}



\item  In a tragic accident, a car (mass $m$) plunges into an icy lake. The water produces a linear drag force on the car $\vec f_d=-b\vec v$. Find the velocity of the car as a function of time. The student's response after several lines of careful calculations was:
   \begin{equation}
    v(t) =  v_0 e^{b t/m}
    \end{equation}
       (Remember, you are NOT being asked to solve this problem!! You only need to make two {\bf checks} of this answer to see if it is reasonable.  If your checks indicate it is not reasonable, do they suggest any possible fix, or something you might look into?)

%\item A buoy bobs in the Boulder reservoir. It is cylindrical in shape, with a circular base of radius $r$, and height $h$. Small periodic wavelets (of amplitude $A$ meters, frequency $\omega$) drive the buoy. (Water has density $\rho$)  What is the resulting period of oscillation, averaged over a very long time? The student's response after several lines of careful calculations was:
  % \begin{equation}
   % T= 2 \pi / \omega- 2 \pi \sqrt{m/(\rho g (2 \pi r h)) }
   % \end{equation}
  %  (Again, don't solve the problem,  just check the solution - and then, can you make any suggestions about what might  be wrong even without having solved the problem? )

  \item  A formula for potential energy U(x,y,z) was given, and the question asked for the resulting force. The student's response after several lines of careful calculations was:
   \begin{equation}
\vec F(x,y,z) = c x\  \hat j
    \end{equation}
with the constant $c  = 4.9 /s^2$    (Again, don't solve the problem,  just check the solution - and then, can you make any suggestions about what might  have gone wrong even without having solved, or in this case even seen, the original problem?)

\item A space platform is in the shape of an annulus: a flat ring of  uniform mass per unit area of $\sigma$, with different inner and outer radii given in terms of some distance $R$.   A student on an exam works out the gravitational potential energy of a test mass m at position $z$ above the central axis, and  gets 
   \begin{equation}
    U(z) =  -2\pi G \sigma  (\sqrt{4R^2+z^2} + \sqrt{R^2+z^2})
   \end{equation}
   One final time, don't redo the problem to check it - just check this given solution. (I can think of at least two obvious ways, each of which catches a different small but important mistake.) 
   
\end{enumerate}


\question
 \begin{enumerate}
\item  Solve Boas Problem 8.6.12 (That's chapter 8, section 6, problem 12, on page 423).  We just want the general solution here. 

Please also tell us {\it  explicitly} if the associated (complementary) homogenous problem is (choose one) \{over-damped, under-damped, undamped, critically damped\} How do you tell? 


\item Now, given that the system starts with $y(0)=0$, $y'(0)=0$, solve that problem completely. There should be no undetermined coefficients left!

\item Sketch the homogeneous {\it piece} of the solution in part b) above (by itself). Also sketch the particular solution {\it piece} (by itself). Finally add them to sketch the full solution - noting any important aspects of the sketch.  (Don't worry about precision, this is a sketch, not a plot. We want to know the {\it features}.) Then, use a computer to plot the actual solution. Print it out and discuss - in particular, discuss if it did not exactly match with your sketch.  (The differences may or may not be subtle, depending on the care of your initial sketch)

\item Let's change the problem slightly. Suppose the middle term (the``4D" term) is {\it quadratic} rather than {\it linear} drag.  Write down the new ODE (don't use the ``D" notation, just write it out as an ODE) This is no longer analytically solvable! But that needn't stop us - keeping all numerical coefficients the same as Boas had, just replacing linear with quadratic drag, use your favorite computational environment to solve and plot the new solution (with the same initial conditions). Did changing linear for quadratic drag change any qualitative aspects? Briefly, comment.  

\item Invent two {\it physics} problems, for which the ODEs are the ones used in parts a and d respectively. Be explicit - what are the values of all relevant physical parameters, in SI units? What physics would induce you to choose the ODE of part d instead of part a? 
 \end{enumerate}


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


\question A series LRC circuit (Taylor Fig 5.10) is connected across the terminals of an AC power supply that produces a voltage $V(t) = V_0 e^{i \omega t}$.  The ``equation of motion'' for the charge $q(t)$ across the capacitor is as follows:
\begin{equation}
L\ddot q + R \dot q + q\frac{1}{C} = V_0 e^{i\omega t}
\end{equation}


The above differential equation has a steady-state solution which can be written as:

$q(t) = q_0 e^{i (\omega t + \delta)}$
[Note:  The parameters $q_0$ and $\delta$ are actually functions of $\omega$, the frequency of the AC power supply.  However, in this problem you will {\it not}  have to solve for or write out these functions in full!]
\begin{enumerate}
\item	In terms of $q_0, \omega, \delta$ and the relevant coefficients from the differential equation, write down the following functions:

i.	the potential difference $\Delta V_C(t)$  across the capacitor\\

ii.	the potential difference $\Delta V_R(t)$  across the resistor\\

iii.	the potential difference $\Delta V_L(t)$ across the inductor\\

iv.    Finally, determine the smallest positive values of $\alpha, \beta$, and $\gamma$  (in radians) that satisfy the following Euler relations:
$e^{i \alpha} = i $,  $e^{i \beta} = -1 $,      $e^{i \gamma} = -i $

\item Using your results from part iv, rewrite the functions in part i-iii so that each function can be written as a positive real number times a complex exponential function.  Use your rewritten functions to answer the following questions.

i.	What is the phase difference between $\Delta V_C(t)$ and $\Delta V_R(t)$?  

Do the peaks of $\Delta V_C(t)$ come just {\it before} (``leads") or just {\it after} (``lags") the peak of $\Delta V_R(t)$?  (Show/explain your reasoning)\\
ii.	What is the phase difference between $\Delta V_R(t)$ and $\Delta V_L(t)$?  

Does $\Delta V_R(t)$ lead or lag $\Delta V_L(t)$?

\item Assuming for simplicity that $\delta=0$, make a crude sketch on one graph of $\Delta V_R(t), $ and $\Delta V_C(t)$  being very careful to identify which is which. Does your sketch agree with your claim in part b about ``leading" or ``lagging"? (Briefly, comment) 

\end{enumerate}

\question
 In section 5.6, Taylor states that the amplitude of an
  oscillator subject to a sinusoidal driving force is
\begin{equation*}
\\A^2=\frac{f_0^2}{(\omega_0^2-\omega^2)^2+4\beta^2 \omega^2}
\end{equation*}

\begin{enumerate}
\item Find an expression for $\omega$ such that the amplitude $A$ is
  maximized. That is, derive our text's Equation 5.73. ({\it Hint: } Rather than differentiate and extremize the entire expression for $A(\omega)$, make your job easier by judiciously picking which part of that expression to differentiate and extremize.)
  
  
 Then, use your expression to find the exact maximum amplitude,
  $A_{max}$. 
  
  Write a second expression for the amplitude $A_0$ which corresponds to $\omega=\omega_0$. 
  
  Briefly, comment on the difference between $A_{max}$ and $A_0$.
\item Open up the resonance PhET sim, found at

\url{http://phet.colorado.edu/en/simulation/resonance}
Play around to see what the sim can do. 
  
  %\item What should change if you now double the damping coefficient? Careful: you will want to describe changes in \emph{all} variables that depend on $b$ for this one. Now actually do this on the PhET sim. Explain how what you notice is (or is not, if you've made a   mistake!)  consistent with what you expect.

%\item Now turn the damping as low as it can go and stop the driver  until the mass comes to rest (you can stop it quickly by maxing the damping, but be sure to return it to the lowest possible value  afterwards). Decrease the driving frequency of the oscillator by  about 10\% and turn the driver back on. Describe the motion of the  mass.

%\item To understand this phenomenon mathematically, refer to Taylor Equation 5.68. Explain in 1-2 sentences how this equation describes what you are seeing.

%  Now suppose that you want to set up a system in which the oscillator will undergo 10 cycles of oscillation before coming to rest again. Find values for $\omega$ and $\omega_1$ that satisfy this condition. Verify this on the PhET sim by setting your values and  starting the motion from rest.

%If you set the damping at a moderate value (say, 1 N m/s) and then decrease the driving frequency of the oscillator to roughly 10\% below the natural frequency, what do you predict will happen?  Focus first on simply position as a function of time, and predict, based on Taylor 5.68 and the formula for A above what you should see (just qualitatively is fine.) Then, watch the sim carefully.  Explain how what you see is or is not consistent with what you predicted. How long does it take for the motion to settle down?  (you will only be able to crudely approximate this, but if you are way off, chances are that something went wrong)

%After waiting for any transients to die out in the system you have set up, focus on the relationship  between the phase of the driver and the mass.  What does Taylor's Fig 5.19 predict? What does it look like on the sim? (It's hard to tell exactly, just be approximate. Note that there is a ``slo motion" slider, you can put rulers on the screen, and you can set up multiple oscillators with different parameters if that helps you)


%\item Keeping all other values the same, now try changing the damping coeffient to $\beta \geq 2$ AND start the driver (and mass) from rest again. What aspects of the motion are different than what you  found for a very small $\beta$? Explain mathematically why this  is the case.

%\item Now  increase the frequency of the driver to about 10\% \emph{higher} than the resonant frequency.  What relationship between the phase of the driver and the mass do you expect? Does the PhET sim confirm this?  How does the time it takes to reach this final value compare to what it took  for the case of 10\% \emph{lower} driving frequency? (Briefly, comment)

  %Again starting from rest, how many cycles does it take for the mass to reach a stand-still. Did this number change from part (f)? Explain.



%\item Check out the case of driving the mass at $\omega_0$. What is the phase difference between driver and mass?
  
By investigating the sim in a variety of ways, clearly explain Taylor's
  Figure 5.19 in your own words, {\it referring explicitly to ``experiments" or setups you made with the sim}  to help you connect the figure to the behavior of the objects.  (Please discuss briefly what effects "transients" have, and how you dealt with them in the sim)  
  


\end{enumerate}


{\it Extra credit}.  Play with the sim a little more, and write down one additional (real, interesting) question that  you have about something you notice!  

%\textcolor{red}{From Planning Doc: Modify Alysia HW problem to investigate resonance and reduce scaffolding. That question appears below.}

%\question
%This problem is based on the system described in Taylor Problem 5.38.

%\begin{enumerate}[(a)]
%\item Do Taylor Problem 5.38. (Note that you must plot the result using your favorite computational tool).
%\item Redo problem 5.38 with $\omega \approx \omega_0 + \beta = 1.1$. Once again find $A$, $\delta$, $B_1$,
%and $B_2$. Make a plot of $x(t)$ for the first ten or so periods. In the previous problem, you proved that $A^2$ is equal to half its maximum value at $\omega \approx \omega_0 \pm \beta$. Show that indeed for this value of $\omega$, $A^2$ is roughly half of the value that you found in part a.
%\item Now plot your solutions for part a and b on the same axes. How does the solution to part b compare to the solution for part a? What differences do you notice? (Hint: In Mathematica you can plot two functions together like this {\tt Plot[\{f1[t],f2[t]\},\{t,0,10\}, PlotRange-$>$All]}. Just make sure that you do not reuse the same variables names for both functions.
%\end{enumerate}


\end{questions}
\end{document}

