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

\title{Homework 9}
\time{Start of class on Thurs. March 15}
\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
Check out \url{ http://www.colorado.edu/physics/phys2210/phys2210_sp12/project_info.html}  (also available directly from our course website) For {\it this homework}, please simply indicate the names of the people you plan to work with. (Groups of two or three are suggested, please no more than that.)

%\question Collisions are {\it elastic} if the total kinetic energy before and after the collision is the same. Consider an elastic collision between two identical atoms.  The incoming atom moves with velocity $\vec{v}_1\neq 0$ towards a``target" which is initially at rest ($\vec{v}_2=0$) Let the prime variables  $\vec{v}'_{1,2}$ denote corresponding velocities {\it after} the collision.  Write down the vector equation representing  conservation of momentum and the scalar equation representing the conservation of kinetic energy in an elastic collision. Use this to prove that the angle between  $\vec{v}'_{1}$ and  $\vec{v}'_{2}$ is $\pi/2$.  Describe a situation where this fact might be useful or relevant.



\question  Assuming that asteroids have roughly the same mass density as the moon, make an estimate of the largest asteroid that an astronaut could be standing on, and still have a chance of throwing a small object (with their arms, no machinery!)  so that it completely escapes the asteroid's gravitational field. (This minimum speed is called ``escape velocity")  Is the size you computed typical for asteroids in our solar system?
%\question You are stranded on the surface of the  asteroid Vesta. If the mass of the asteroid is $M$ and its radius is $R$,
% \begin{enumerate}
%\item 
%how fast would you  have to jump off its surface to be able to escape from its gravitational field? (Your estimate should be based on parameters that characterize the asteroid, not parameters that describe your jumping ability.) 
   % \item 
 %Given your formula, look up the approximate mass and radius of the asteroid Vesta 3 and determine a numerical value of the escape velocity. Could you escape in this way?  (Briefly, explain) If so, roughly how big in radius is the maximum the asteroid could be, for you to still escape this way? If not, estimate how much smaller an asteroid you would need, to escape from it in this way?
   % \end{enumerate}
 %\begin{figure}[htc]
%\centering
%{\includegraphics[width=1in]{Hom9Fig1.pdf}}\caption{}
%\end{figure}

\question {\it Determining the gravitational force between two rods:}

\begin{minipage}{0.6\linewidth}
\begin{enumerate}
\item Consider a thin, uniform rod of mass m and length $L$ (and negligible other dimensions) lying on the x axis (from x=-L to 0),  as shown in fig 1a.  Derive a formula for the gravitational field ``g" at any arbitrary point x to the right of the origin (but still on the x-axis!) due to this rod.
\item Now suppose a second rod of length $L$ and mass m sits on the x axis as shown in fig 1b, with the left edge a distance ``d" away. Calculate the mutual gravitational force between these rods. 
\item  Let's do some checks!  Show that the units work out in parts a and b. 

Find the magnitude of the force in part a, in the limit $x>>L.$ What do you expect?  Briefly, discuss!  

Finally, verify that your answer to part b gives what you expect in the limit $d >>L$. {\it ( Hint: This is a bit harder! You need to consistently expand everything to second order, not just first, because of some interesting cancellations)}
\end{enumerate}
\end{minipage}
\begin{minipage}{0.4\linewidth}
\begin{center}
{\includegraphics[width=2in,clip,trim=20mm 50mm 137mm 70mm]{Hom9_fig_rod.pdf}}
{\includegraphics[width=2in,clip,trim=142mm 50mm 15mm 70mm]{Hom9_fig_rod.pdf}}
\end{center}
\end{minipage}

\vskip -.1in


\question Two large solid spheres, each with mass M, are fixed in
place a distance 2l apart, as shown. A small ball of mass m
is constrained to move along the x-axis, shown as a dashed line. (Let x = 0
represent the point on the axis directly between the spheres,
with +x to the right.)

\begin{minipage}{0.5\linewidth}
\begin{center}
{\includegraphics[width=2.8in]{Hom9Fig3.pdf}}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{enumerate}
\item Compute the net force on the ball by both large spheres, and write down a differential equation that governs the
motion of the small ball.
\item Is the net force exerted on the small ball a restoring force? Explain!  Under what limiting conditions for x can we say the motion of m can be
{\it approximated} as simple harmonic motion? (For large x? For small x? Large or small compared
to what?) In this limiting case, find the period of motion in terms of the
given parameters. Explain your reasoning.
\end{enumerate}
\end{minipage}

\newpage


\question %Let's look again at the simple pendulum problem.
%Remember it consists of a point mass m fixed to the end of a massless rod (length l), whose other end is pivoted from the ceiling to let it swing freely in the vertical plane. The pendulum's position can
%be identified by its angle $\phi$ from the equilibrium position. 
Last week you showed that
a pendulum's potential energy (measured from the equilibrium level ) is $U(\phi) = mgL (1 -\cos\phi)$ (where L is the pendulum length) and that  under the small angle assumption the motion is periodic with period $\tau_0=2\pi\sqrt{L/g}$. In this problem we will find the period for large oscillations as well:
% \begin{figure}[htc]
%\centering
%{\includegraphics[width=1.2 in]{Hom9fig3c.pdf}}\caption{}
%\end{figure}
\begin{enumerate}
\item Using conservation of energy, determine $\dot{\phi}$ as a function of $\phi$, given that the pendulum starts at rest at a starting angle of $\Phi_0$.  Now use this ODE to find an expression for the time the pendulum takes  to travel from $\phi=0$ to its maximum value $\Phi_0$. (You will have a formal integral to do that you cannot solve analytically! That's ok, just write down the integral, being very explicit about the limits of integration) Because this time is a quarter of the period,  write an expression for the full period, as a multiple of $\tau_0$. 

% Show it is given by
  %  \begin{equation}
    %\tau=A \tau_0 \int_0^{\Phi} \frac{d\phi}{\sqrt{\sin^\alpha [\Phi/2]-\sin^\beta [\phi/2]}}\end{equation}
   % Determine $A$, $\alpha$ and $\beta$.
   % \item  Use the substitution $\sin(\phi/2)= K u$ and show that \begin{equation}
   % \tau=2A \tau_0 \int_0^1 \frac{d\phi}{\sqrt{1-u^\alpha}\sqrt{1-K^\beta u^\beta}}\end{equation}
   
   
   
 \item  Use your favorite computational environment to evaluate the integral and plot $\tau/\tau_0$ for $0\leq \Phi_0\leq 3$ rad. For small $\Phi_0$, does your graph look like that you expect? (Discuss - what value DO you expect?)   What is $\tau/\tau_0$ for $\Phi_0=\pi/2$ rad? (Give us several significant figures)  
 \item  What happens to  $\tau$ as the amplitude of the oscillation approaches $\pi$? Discuss the physics here!  What can you say about how well simple harmonic motion approximates the behavior of a real pendulum? 
 
{\it Note: You may see some curious numerical pathologies if you use MMA. If your plot has little gaps, it's because MMA is generating tiny imaginary terms added to the result. You might just try plotting the real part of the integral, using Re[ ].  If it gives you symbolic results, you might fix that by using N[ ], which forces MMA to evaluate the expression as a number, if it can! By the way, this integral is called the ``complete elliptic integral of the first kind''.}

\item A real grandfather clock has a length of about half a meter, and you pull the pendulum, oh, a few cm to the side (let's say $\sim 10$ cm, for a decent sized clock). Use your code from the previous part to compute $\tau/\tau_0$ for this clock.   When designing the clock for practical use, would the clockmaker be safe in making the ``small angle" approximation? Don't just look at the number you got and say ``well, that looks pretty close to 1, so we should be ok". What we mean here is to {\it justify your answer with a calculation} that tells you whether this clock (calibrated assuming the small angle approximation) would be annoying or useful to have in your house!  Can you be concrete about what you would consider ``annoying" about a clock? (I was surprised by the result - are you?)  
\end{enumerate}

\question  {\it Practicing with complex numbers. (2 pt each). See Boas Ch 2.4-12 for more review, or further practice!) }
\begin{enumerate}
\item If $z_1=1-\sqrt{3}i$, draw $z_1$ in the complex plane.  Compute its real and imaginary parts and magnitude.  Write $z_1$ as $A e^{i\theta}$ and determine $A$ and $\theta$.

\item If $z_2=1/(1-i)$, draw $z_2$ in the complex plane.  Compute its real and imaginary parts and magnitude.  Write $z$ as $A e^{i\theta}$ and determine $A$ and $\theta$.
\item  If $z_3=0.5e^{-i \pi/6}$, what are the real and imaginary parts of $1/z_3^2$?  {\it (Note that this is just a simple square, not an absolute value squared, so the answer does not need to come out purely real!) } 

\item Compute  $Z = z_1* z_3$ with $z_1$ and $z_3$ given in parts (a) and (c). Draw $Z$ in the complex plane.  Compute its real and imaginary parts and magnitude.  Write $Z$ as $A e^{i\theta}$ and determine $A$ and $\theta$.

%\item If $z_4 = \sinh(i x) $, draw $z$ in the complex plane.  Compute its real and imaginary parts and magnitude.  Write $z$ as $A e^{i\theta}$ and determine $A$ and $\theta$.

\item  Pick either one of the following two items to prove - whichever you prefer!

-  Use Euler's theorem to rewrite $\sin x$ in terms of complex exponentials, and then use that to show that $\int_0^{2 \pi} \sin(n \theta) \sin(m \theta) d\theta$ vanishes if n and m are unequal integers. {\it (We'll make good use of this, soon!)}

- Euler's theorem says that $e^{i \theta}=\cos\theta + i \sin\theta$. Call $z=e^{i \theta}$ and evaluate $z^2$ two ways to show the (very handy!) trig identities $\cos 2\theta = \cos^2\theta - \sin^2\theta$ and $\sin 2\theta = 2 \sin\theta \cos\theta$. 



\end{enumerate}



\end{questions}

\end{document}

%\question An ideal (massless) spring with force constant $k$ is used to hang a crate from the ceiling, as shown (left, below). Suppose that the spring were treated as two separate springs (1 and 2) connected end-to-end.  (See Figure) Treat each spring as having its own spring constant ($k_1$ or $k_2$) given by Hooke's law, $|F_i| = k_ix_i$, where $|F_i|$ is the magnitude of the force exerted on (or by) spring $i$ when the length of that spring {\it changes} by an amount $x_i$.
%\begin{enumerate}
%\item  Relate the magnitudes of the forces $F_1$ and $F_2$ exerted by each spring individually, and the magnitude of the force $F$ exerted on the crate. Explain. (Hint: Sketch separate free body diagrams for the springs and the crate.) Then, find the  relationship between the stretches $x_1$ and $x_2$ of the individual springs and the stretch $x$ of the original (single) spring. Explain your reasoning.
%Based on these results, determine an expression for the spring
%constant of the original spring (k) in terms of the spring constants ($k_1$ and $k_2$) of the two individual `` spring pieces".  Based on this problem, if you chop an ideal spring in half, what happens to its spring constant? Show all work.

%\begin{figure}[htc]
%\centering
%{\includegraphics[width=2.3in]{Hom9Fig4a.pdf}}\caption{}
%\end{figure}

%\item Now consider the case in which you replace the original spring with two new springs (3 and 4)
%that connect directly to the crate and to the ceiling. (See Figure.)  Assume that the crate ends up at the same equilibrium height in this new situation as when there was just a single spring ($k$). Following the same logic as you did above, determine an expression for the spring constant k of the original spring in terms of the spring constants $k_3$ and $k_4$ of the  new springs. Explain your reasoning.
%\end{enumerate}
%\begin{figure}[htc]
%\centering
%{\includegraphics[width=2.3in]{Hom9Fig4.pdf}}\caption{}
%\end{figure}

%\question Alysia HW9 \#7\\
%We often consider the simple pendulum to be an example of simple harmonic motion. In this problem will will see how accurate this is.

%\begin{enumerate}[(a)]
%\item Imagine a vertical pendulum of length $l$ and mass $m$. By considering the forces on the pendulum and applying Newton�s second law, obtain differential equation for the forces on the pendulum in terms of $\theta$, the angle with respect to the vertical axis. (Hint: It might be easiest to work in polar coordinates.)
%\item Show that in the limit where $\theta$ is small that you obtain a second-order linear differential equation with constant coefficients. What is the general solution to this equation?
%\item Consider $m = 1$ kg, and $l = 1$ m. If the pendulum is released at rest at $t = 0$ from $\theta = 0.05$ radians, what is the particular solution for $\theta(t)$ in your expression from part b? What is the period of this solution?
%\item If instead the initial condition is that $\theta = 1.25$ radians at $t = 0$, what happens to the period of the motion?
%\item Now use Mathematica to solve the exact equation that you found in part a (keeping the $\sin\theta$ in the expression) for the case where the pendulum is released from rest at $t = 0$ from $\theta = 0.05$ radians. Make sure that you figure out how to apply initial conditions. Is there an exact solution if you use DSolve or do you have to use NDSolve?
%\item Make a plot of the solution. Now use the FindRoot command to figure our where the solution is zero.
%(You should have some idea of where to look for the roots from the plot that you made.) Use the locations of two roots to determine the period. How does this compare the to the period of the approximate small angle solution found above in part c?
%\item Now use Mathematica to determine a numerical solution for the case where the pendulum is released from rest at $t = 0$ from $\theta = 1.25$ radians. Plot the solution. Once again, use the FindRoot command to determine the locations of two roots to determine the period. How does this compare to the case where it was released from $\theta = 0.05 $ in part f?
%\item Based on these results, what can you say about how well simple harmonic motion approximates the behavior of a pendulum?
%\end{enumerate}
%\end{questions}

%\textcolor{red}{DC: My notes have ``combine Steve HW9 \#4 (clocks and small angle) and and Alysia HW9 \#7 (accuracy of small angles); further fade computational hand-holding''. I added Alysia's HW9 \#7 as the last question for this set.}


