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

\title{Homework 8}
\time{Start of class on Thurs. March 3}
\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  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 shape is a perfect hemisphere (radius a) on the bottom, glued to a solid cylinder on top. The CM is shown, it is ``h'' above the base. You don't need to try to compute h, assume it is a given quantity.  (As the designer, how might you control/change the position of h?)
\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, h, 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 ``h" ensure that the weeble doesn't fall down? Explain. Can this toy work as desired?
\end{enumerate}
 \begin{figure}[htc]
\centering
{\includegraphics[width=1.5in]{Hom8fig1.pdf}}\caption{}
\end{figure}

\question  A particle is under the influence of a force $\vec{F}=k^2( -x+ x^3/\alpha^2) \hat{x}$, where $k$ and $\alpha$ are constants.
\begin{enumerate}
\item What are the units of $k$ and $\alpha$? Now,  determine $U(x)$ 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 the equilibrium points, if any, and determine if they are stable or unstable.
\item Qualitatively explain the motion  of an object  in this force field released from rest at  $x=\alpha/2$. Then, qualitatively explain the motion  of an object  in this force field released from rest at  $x=3\alpha/2$
\end{enumerate}
{\it 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?) Another (famous!) case, albeit with signs flipped, gives  the ``Higgs potential" in particle physics, which is being actively investigated at CERN's  LHC collider.}
\question A planet of mass M and radius R has a nonuniform density that varies with r, the distance from the center according to $\rho=A r$ for $0\leq r\leq R$.
\begin{enumerate}
\item What is the constant A in terms of M and R?  Does this density profile strike you as physically plausible, or is just designed as a mathematical exercise? (Briefly, explain)
\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 moon (see figure). %(The tunnel crosses the vertical axis a distance ``a" from the origin. But, it will turn out, somewhat remarkably, that %your final answers will not depend on this distance!)
A robot place a rock in the  tunnel at position $r=r_0$ from the center of the moon, 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 the moon's density is uniform throughout its volume, and ignore the moon's rotation.

%{\it Hint: If you have forgotten how to solve the ODE you got, you might revisit Taylor's example 1.2.}
 \begin{figure}[htc]
\centering
{\includegraphics[width=1.6 in]{Hom8fig2b.pdf}}\caption{}
\end{figure}
\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?)

{\it Extra credit: If you place a small mass a small distance z away from the center, use your Taylor limit for $z<<R$ above to write a simple ODE for the equation of motion. Solve it, and discuss the motion}

\end{enumerate}



\end{questions}
\end{document}

