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

\title{Homework 9}
\time{Start of class on Thurs. March 10}
\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 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  Consider two identical uniform rods of length $L$ and mass m lying along the same line and having their closest points separated by a distance d as shown in the figure
\begin{enumerate}
\item Calculate the mutual force between these rods, both its direction and magnitude.
\item  Now do several checks. First, make sure the units worked out (!) The, find the magnitude of the force in the limit $L \to 0$. What do you expect?  Briefly, discuss.   Lastly,  find the magnitude of the force in the limit $d\to \infty $ ? Again, is it what you expect? Briefly, discuss. 
\end{enumerate}
 \begin{figure}[htc]
\centering
{\includegraphics[width=3in]{Hom9Fig2.pdf}}\caption{}
\end{figure}


\question  {\it Practicing with complex numbers. (1 pt each)}
\begin{enumerate}
\item If $z_1=-\sqrt{3}+i$, 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 $z$ 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  What are the real and imaginary parts of $1/z_3^2$ with $z_3=0.5e^{-i \pi/3}$.

\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 $\theta$.
\end{enumerate}

\question Consider again 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. 
In homework 7 you showed that
the 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 use Mathematica to 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 MMA 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 $\Phi_0=\pi/8$ rad? How about $\pi/2$ rad? What happens to  $\tau$ as the amplitude of the oscillation approaches $\pi$? Explain.
 
{\it Note:  The integral has some curious numerical pathologies. If your plot has little gaps in it, 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 the command Re[ ].  Similarly, if it gives you symbolic results, you might fix that by using the command 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, about 10 cm to the side. When designing the clock for practical use, would the clockmaker be safe in making the ``small angle" approximation? (What we mean here is to justify your answer with a calculation that tells you whether this clock would be annoying or useful to have in your house!  Can you be concrete about what you would consider ``annoying" about a clock?)
\end{enumerate}

\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{figure}[htc]
\centering
{\includegraphics[width=1.9in]{Hom9Fig3.pdf}}\caption{}
\end{figure}
\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 how you can tell from your
differential equation above?  Under what limiting conditions for x is it possible to say that the motion of the mass m can be
approximated as simple harmonic motion? (For large x? For small x? Large or small compared
to what?) In this limiting case, determine an expression for the period of motion in terms of the
given parameters. Explain your reasoning.
\end{enumerate}

\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}
{\bf NOTE: } There is an extra-credit survey online - just go to our course main page, you'll see it linked prominently on our ``Special notes" section, under the current fun picture. Or go directly to

{\it http://www.colorado.edu/physics/phys2210/phys2210\_sp11/preflights/survey\_midterm\_2210.html} 

We really appreciate your feedback, it helps us make this a better course!

\end{questions}
\end{document}

