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

\title{Homework 13}
\time{Start of class on Thurs. Apr 14}
\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}

{\it As always, lettered parts are worth 4 points unless otherwise stated}
\begin{questions}



\question {\it Since not all of you had a chance to look at the recent tutorial extra-problem, we decided to assign  it  as a homework problem!} 

A harmonic oscillator with a restoring force $25m\alpha^2x$ is subject to a damping
force $3m\alpha v$ and a sinusoidal driving force $F_0 \cos(10 \alpha t)$.

  \begin{enumerate}
     \item 	Write down the differential equation that governs the motion of this oscillator.
   	Show that if the driving force were removed the oscillator would become underdamped, and express the frequency of the oscillator in terms of the given quantities. Explain your reasoning.


     \item 	 Each x vs. t graph below illustrates the actual motion (transient plus steady state) of a damped, driven oscillator starting at $t = 0$. For each case, is the frequency of the steady-state motion greater than, less than, or equal to that of the transient motion? Explain.
         
          \begin{figure}[htc]
\centering
{\includegraphics[width=6in]{Hom13Fig1.pdf}}\caption{}
\end{figure}
     \item (2 pts) Identify which graph (1 or 2) would better correspond to the damped, driven  oscillator described in parts a)  of this problem. Explain your reasoning.
    \end{enumerate}

%\question
%For the periodic function $f(x)=x$  for $-1<x\leq 1$
%\begin{enumerate}
%\item Compute the Fourier coefficients $a_n$ and $b_n$
%\item Use Mathematica to plot   the sum of the first 3 terms in the Fourier series,  the sum of the first 10 terms  and  the sum of the  first 100 terms.
%\item  Compare these partial sums with  the function itself. When you add up lots of terms, where is the series convergence worst? Can you explain why?
%\end{enumerate}

\question
 Do you know the precise value of the summation of the reciprocals of the squares of the natural numbers,
 $\sum_{n=1}^\infty \frac{1}{n^2}$? This  is a famous problem in mathematical analysis with relevance to number theory. It was first posed by Pietro Mengoli in 1644 and solved by  Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Let's solve it using Fourier series (a tool that wasn't yet available to Euler!) 
\begin{enumerate}
\item Define the periodic function $f(x)=x^2$ for $-\pi<x\leq \pi$ and compute the Fourier series.
\item Evaluate the function at $x=\pi$ and use the Fourier series to evaluate  $\sum_{n=1}^\infty \frac{1}{n^2}$.

{\it The answer is  $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$. Your job is to show how this comes about!) }
\end{enumerate}



\question  Any periodic function, $f(t)=f(t+T)$ can also be Fourier expanded as 

\begin{equation}
f(t)=\sum_{n=-\infty}^{\infty} c_n e^{i\omega n t}
\end{equation} with $\omega=2\pi/T$.

 {\it  This is very similar to our usual expansion in terms of sin's and cos's, but combines them more elegantly using complex exponentials, and is the more common form used by physicists. Let's work out the details!  }
\begin{enumerate}
\item Show that $\frac{1}{T}\int_{-T/2}^{T/2} e^{i(m-n) t\omega}dt =0$ if $m\neq n$ and $\frac{1}{T}\int_{-T/2}^{T/2} e^{i(m-n) t\omega}dt =1$ if $m = n$. 

Here $m,n$ are integers.

    \item Use (a) to show that $c_m =\frac{1}{T}\int_{-T/2}^{T/2}  e^{-i m t \omega} f(t) dt$, where $m$ is just a dummy index. 
    
    {\it Hint!  Look at the use of {\it Fourier's trick} from my lecture on Apr 7 (see concept test slides), or online lecture notes on p.\ 11a and 11b, or Boas Ch 7.7,  and figure out how to apply it here.}
    
\item  Use Euler's formula $e^{ix}=\cos x+i \sin x$ to show that 
\begin{equation}
c_n=\left \{\begin{array}{c}
              \frac{1}{2} ( a_n-ib_n)  \quad n>0\\
            % \frac{a_0} {2} 
            a_0\quad  \quad \quad \quad  n=0\\
                \frac{1}{2} ( a_{-n}+ib_{-n}) \quad n<0
            \end{array}\right. \notag
\end{equation}
Where the a's and b's are the usual Fourier coefficients we've defined in class and Taylor. 
\end{enumerate}
{\it Note that, when expanding in complex exponentials like this, we ``win" by having just one simple formula for the $c_n$'s (and integrals of exponentials are often easier). 

The price we pay is that we have to run our sum over negative n's }
\question %A ball is thrown with initial speed $v_{0}$ directly up and over an inclined plane. The plane is inclined at an angle $\alpha$ above the horizontal, and the ball's initial velocity is at an additional angle $\beta$ above the plane's surface. 

An inclined plane is tilted at an angle $\alpha$ above the horizontal. A ball is launched (initial velocity $v_0$) into the air directly up the inclined plane, launched at an {\it additional} angle $\beta$ above the plane's surface. 


\begin{enumerate}
\item Draw the situation here, and roughly sketch a predicted path of the ball for some initial conditions that you choose. 
\item   (\textit{Hint: before you start to work on this question, read the whole thing and then think carefully about what coordinate system would be easiest here.})  Find the ball's position as a a function of time (until it hits the plane).  Show that the ball lands a distance $R = 2v_{0}^{2}sin\beta cos(\beta + \alpha)/(g cos^{2}\alpha)$ from its launch point. Show that for given $v_{0}$ and $\alpha$, the maximum possible range up the inclined plane (i.e.\  the maximum value of the distance between landing and launch point) is $R_{max} = v_0^2/[g(1 + sin\alpha)]$.

\item Think of two specific combinations of $\alpha$, $\beta$, and $v_{0}$ for which you can easily predict the outcome (either the ball's position as a function of time, or the range/maximum range) and use these combinations to check your calculations in the previous step (or to check that the final formulas we gave you are OK)
\end{enumerate}
\question  Evaluate the following integrals: (2 pts each)
\begin{enumerate}
\item $\int_0^\pi dx \sin(x)\delta(x -\pi/2)$
\item $ \int_0^3(5t - 2)\delta(2 - t)dt$
\item $\int_0^5(t^2 + 1)\delta(t + 3)dt$
\item $ \int_{-\infty}^\infty e^x \delta (3 x)$
%\item If  a particle feels a force  $F$ of the form   $F=A\delta(x)$ with $x$ the position of the  particle, what are the units of $A$?
\item If a particle feels a force $F(t)$ of the form $F=A\delta(t)$, with $t$ the time, what are the units of $A$? 

Give a brief  physical interpretation to this formula - what sort of physical force are we trying to represent here?

\end{enumerate}


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{\it \bf Extra credit}  Go back to problem 1, and ``reverse engineer it" - write a Mathematica code which produces the plot which corresponds to the oscillator described in part a of the problem. (Taylor's Eq's 5.68 and 5.70 will help! )


\end{questions}
\end{document}

