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

\title{Homework 8}
\time{Start of class on Thurs. March 8}
\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  Look at the figure below, which shows a possible design for a child's toy.  The designer (you!) wants to build a ``weeble'' (which wobbles, but doesn't fall down). It is a single, solid object, the base shape is a perfect hemisphere (radius R) on the bottom, connected to the rest of the body (here, a penguin). The CM is shown, it is ``a'' above the base. You don't need to try to compute ``a", assume it is a given quantity! (As the designer, how might you control/change the position of a?)
\begin{enumerate}
\item Write down the gravitational potential energy when the weeble is tipped an angle $\theta$ from the vertical, as a function of given quantities (m, a, R, g, and of course $\theta$).
%For that you need to find the position of the center of mass as a function of $\theta$.

%It might help to think about the height of the hemisphere's center O as the toy tilts.
\item  Assuming the toy is released from rest, determine the condition(s) for any equilibrium point(s).   Then consider stability: what values of (or relations between) ``a" or ``R" ensure that the weeble doesn't fall down? Explain. Can this toy work as desired?
\end{enumerate}

\begin{center}
{\includegraphics[width=3.5in,clip,trim=15mm 15mm 20mm 20mm]{Hom8fig1a.pdf}}
\end{center}

\question  A particle is under the influence of a force $\vec{F}=(-ax+ bx^3) \hat{x}$, where $a$ and $b$ are constants.
\begin{enumerate}
\item What are the units of $a$ and $b$? Assuming a and b are positive, find $U(x)$ (assuming $U(0)=0$) and sketch it.  (Sketch means hand-drawn, not plotted with Mathematica. Include all features you consider interesting in your sketch, including e.g. zero crossings, behavior at large x, ``scales" of your axes, etc)
\item  Find all equilibrium points and determine if they are stable or unstable.
\item Qualitatively describe the motion  of an object  in this force field released from rest at  $x=(1/2) \sqrt{a/b}$. 

Then, qualitatively describe the motion  of an object  in this force field released from rest at  $x=\sqrt{2a/b}$.
\item  Now suppose a and b are both negative.  Repeat parts b and c, commenting on what has changed. 
\item Setting a=b=-1 (with appropriate units), Taylor expand the  potential of part d {\it around the outermost stable equilibrium point} (not around x=0!!) Truncate your series after the first ``nontrivial" term. Plot the resulting approximate potential that a particle near that point feels, on top of the ``real" potential.   (Note this time I ask you to ``plot", not ``sketch", so use a computer) Use your plot to help you discuss the range of x for which you think this approximation would be reasonable.

 %{\it (Hint: the  ``Series" command in Mathematica can generate the Taylor series expansion for you, feel free to use it for this part. ) }


\end{enumerate}
 
 {\it Note that this potential, or small variants of it, occurs in various physics situations. This is a more accurate form of the true force on a pendulum  than our usual approximation, can you see why? Part d is the (famous!) case of the ``Higgs potential" in particle physics, which is being actively investigated at CERN's  LHC collider.}
\question  Consider a very (infinitesimally!) thin but massive rod, length L (total mass M), centered around the origin, sitting along the x-axis. (So the left end is at (-L/2, 0,0) and the right end is at (+L/2,0,0) Assume the mass density $\lambda$ (which has units of kg/m)is {\it not} uniform, but instead varies linearly with distance from the origin,  $\lambda(x) = c|x|$. 
\begin{enumerate}
\item What is that constant ``c" in terms of M and L? %Why did I put an absolute value in the formula? 
What is the  direction of the gravitational field generated by this mass distribution at a point in space a distance z above the center of the rod, i.e. at $(0,0, z)$  Explain your reasoning for the direction carefully, try not to simply ``wave your hands." (The answer is extremely intuitive, but can you justify that it is correct?)
\item  Compute the gravitational field, $\vec g$, at the point $(0,0, z)$  by directly integrating Newton's law of gravity, summing over all infinitesimal ``chunks" of mass along the rod.
\item Compute the gravitational potential at the point $(0,0,z)$ by directly integrating $-Gdm/r$, summing over all infinitesimal ``chunks" dm along the rod. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part.
\item In the limit of large z  what do you {\it expect} for the functional form for gravitational potential? (Hint: Don't just say it goes to zero! It's a rod of mass M, when you're far away what does it look like? {\it How } does it go to zero?) What does ``large z" mean here? Use the binomial (or Taylor) expansion to verify that your formula does indeed give exactly what you expect. (Hint: you cannot Taylor expand in something BIG, you have to Taylor expand in something small.) 

\item Can you use Gauss' law to figure out the  gravitational potential at the point $(0,0,z)$? (If so, do it and check your previous answers. If not, why not?)
\end{enumerate}

\question  
\begin{enumerate}
\item Imagine a planet of total mass M and radius R which has a nonuniform mass density that varies just with r, the distance from the center.  For this (admittedly very unusual!) planet, suppose the gravitational field strength {\it inside the planet} turns out to be {\bf independent} of the radial distance within the sphere.  Find the function describing the mass density $\rho = \rho( r) $ of this planet.  (Your final answer should be written in terms of the given constants.)  
\item Now, determine the gravitational force on a satellite of mass m orbiting  this planet at distance $r>R$.    (Use the easiest method you can come up with!)  Explain your work in words as well as formulas. For instance, in your calculation, you will need to argue that the magnitude of $\vec g(r,\theta, \phi)$ depends only on $r$.  Be explicit about this - how do you know that it doesn't, in fact, depend on $\theta$ or $\phi$? 
\item As a final check, explicitly show that your solutions inside and outside the planet (parts a and b) are {\it consistent} when $r=R$.
Please also comment on whether this density profile strikes you as physically plausible, or is it just designed as a mathematical exercise?  Defend your reasoning. 
\end{enumerate}

\question Now lets consider our (real) planet Earth, with total mass M and radius R which we will approximate as a {\it uniform} mass density, $\rho(r )=\rho_0$.  
\begin{enumerate}
\item Neglecting rotational and frictional effects, show that a particle dropped into a hole drilled straight through the center of the earth all the way to the far side will oscillate between the two endpoints. (Hint: you will need to set up, and solve, an ODE for the motion) 
\item Find the period of the oscillation of this motion. Get a number (in minutes) as a final result, using data for the earth's size and mass. (How does that compare to flying to Perth and back?!) 
\end{enumerate}
%\item Determine the gravitational force on a satellite of mass m orbiting  this planet.  In words, please outline the method you plan to use for your solution. (Use the easiest method you can come up with!) In your calculation, you will need to argue that the magnitude of $\vec g(r,\theta, \phi)$ depends only on $r$.  Be very explicit about this - how do you know that it doesn't, in fact, depend on $\theta$ or $\phi$?  
%\item Determine the gravitational force felt by a rock of mass m inside the planet, located at radius $r<R$. (If the method you use is {\it different} than in part b, explain why you switched. If not, just proceed!) 

%Explicitly check your result for this part by considering the limits $r\to 0$ and $r\to R$.
%\end{enumerate}

%\question  Imagine (in a parallel universe of unlimited budgets) that NASA digs a straight tunnel  through the center of the earth (see figure). 
%A robot place a rock in the  tunnel at position $r=r_0$ from the center of the earth, and releases it (from rest). Use  Newton's  second law  to write the equation of motion of the rock and solve for r(t). Explain in words the rock's motion. Does the rock  return to its initial position at any later time? If so, how long does it takes to return to it? (Give a formula, and a number.)  Assume for simplicity the earth's density is uniform throughout its volume, and ignore the earth's rotation (not to mention any temperature effects that would otherwise melt that rock!) 

{\it Extra Credit:  OK, even with unlimited budgets, digging a tunnel through the center of the earth is preposterous. But, suppose instead that the tunnel is a straight-line ``chord" through the earth, say directly from New York to Los Angeles. Show that your final answer for the time taken does not depend on the location of that chord! This is rather remarkable - look again at the time for a free-fall trip (no energy required, except perhaps to compensate for friction)  How long would that trip take? Could this work?! }
%{\it Hint: If you have forgotten how to solve the ODE you got, you might revisit Taylor's example 1.2.}
%\begin{center}
%{\includegraphics[width=1.6 in]{Hom8fig2b.pdf}}
%\end{center}


%\question  Consider a very (infinitesimally!) thin but massive loop, radius R (total mass M), centered around the origin, sitting in the x-y plane. Assume it has a uniform linear mass density $\lambda$ (which has units of kg/m) all around it. (So, it's like a skinny donut that is mostly hole, centered around the z-axis)
%\begin{enumerate}
%\item What is $\lambda$ in terms of M and R? What is the  direction of the gravitational field generated by this mass distribution at a point in space a distance z above the center of the donut, i.e. at $(0,0 , z)$  Explain your reasoning for the direction carefully, try not to simply ``wave your hands." (The answer is extremely intuitive, but can you justify that it is correct?)
%\item  Compute the gravitational field, $\vec g$, at the point $(0,0, z)$  by directly integrating Newton's law of gravity, summing over all infinitesimal ``chunks" of mass along the loop.
%\item Compute the gravitational potential at the point $(0,0,z)$ by directly integrating $-Gdm/r$, summing over all infinitesimal ``chunks" dm along the loop. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part.
%\item In the two separate limits $z <<R$ and $z>>R$, Taylor expand your g-field (in the z-direction)out only to the first non-zero term, and convince us that both limits make good physical sense.
%\item Can you use Gauss' law to figure out the  gravitational potential at the point $(0,0,z)$? (If so, do it and check your previous answers. If not, why not?)

%\end{enumerate}



\end{questions}
\end{document}

