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\begin{document}
\lhead{PHYS 2210   Fall 2011   Homework Set 7}
\chead{}
\rhead{ Due at 4 PM on \textbf{Oct 13th, 2011}}

 \begin{enumerate}

\item (1 point)  Consider a spherical shell that extends from $r=R$ to $r=2R$ with a  \textbf{non-uniform density} $\rho(r)=\rho_{0}r$.  What is the total mass of the shell?

\item  (2.5 points)  Imagine that NASA digs a straight tunnel  through the center of the moon (see figure) to access the Moon's $^{3}$He deposits.  An astronaut places a rock in the  tunnel at the surface of the moon, and releases it (from rest).  Show that the rock obeys the force law for a mass connected to a spring.  What is the spring constant? Find the oscillation period for this motion if you assume that Moon has a mass of 7.35$\times 10^{22}$ kg and a radius of 1.74$\times10^{6}$ m.  Assume the moon's density is uniform throughout its volume, and ignore the moon's rotation.  
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{\includegraphics[width=1.6 in]{Home7.pdf}}\caption{}
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\item (3 points) Taylor 4.24

\item   Consider a very (infinitesimally!) thin but massive circular 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 (0.5 points)  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. 
\item (1 points) 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 find the graviational field.
\item  (1 points) 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. Check that you agree with your result from the previous part.

(please turn over!)
\item (0.5 points) 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 (0.5 points) 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{enumerate}

\end{document}  