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\title{Spring Break Problem Set for Parker Emmerson}
\author{Spring 2009\\
Instructor: Jonathan Franklin}
\date{March 14th, 2009}
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\textbf{Here are the rules...}
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1.  You have until March 30th to work on the test.  That way I will be able to review it before our tutorial on April 2nd.

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2.  The test is open-book and open-notes.  That means you can use pretty much whatever physical resources you want, including any textbooks, class notes, calculators, etc. 

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3.  Cite your sources.  If you are following an example that's worked out in the text, just say in your answer, "I'm following the example from page so and so of the book."  If you find one of the test problems worked out in some other textbook and use that to help answer the question, tell me that you've done so.  If you use something from a website, tell me what and where.  You don't have to go to extremes here; this isn't about getting commas in the right places in footnotes.  Just make a point of noting the sources you use, if/when you use any.  You were a little sloppy on this rule last time -- unless you solved the problem without any outside resources at all then you should have citations clearly linked to specific equations you looked up, or approaches you did not come up with on your own.

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4.  When answering the questions, don't forget to clearly explain your reasoning, and to show all of your work.  You should pretend you are teaching the problem in front of a class and write what you would say to your audience in such a setting.  If you are not sure how to do a given problem, aim for some partial credit by telling me what you do know, even if that seems embarrassingly minimal, e.g., why a certain approach to the problem doesn't work.  I'm serious about this -- this is how you impress upon me just how solid your general physics background is.  Without this I have no idea if you actually understand what you have written.   I'm expecting complete sentences here.  I know you are familiar with Mathematica at this point -- feel free to write up this whole problem set in Mathematica.  That would be great experience towards one possible method of writing up your Plan.  If you are familiar with LaTeX that would also be a great route to go.  Of course, you can still hand write it if you wish.   But please don't turn in all of your scrap paper -- submit a super clean final draft with full explanations of your methods.


Email: frankjon@marlboro.edu

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\section{Problem One}
Show that the moment of inertia of a uniform disc of mass $M$ and radius $R$ about an axis through its center and perpendicular to its surface is equal to $MR^2/2$.  

A record turntable is accelerated at a constant rate from 0 to 33 1/3 revolutions per minute in two seconds.  It is a uniform disc of mass 1.5 kg and radius 13 cm.  What torque is required to provide this acceleration and what is the angular momentum of the turntable at its final speed?

A mass of 0.2 kg is dropped vertically and sticks to the freely rotating turntable at a distance of 10 cm from its center.  What is the angular velocity of the turntable after the mass has been added?
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\section{Problem Two}
A  cylinder of mass $m$, and radius $r$ is released from rest and rolls without slipping down a plane inclined at an angle $\alpha$ to the horizontal.  Use the conservation of energy or an alternative method to derive an expression for the acceleration of the cylinder down the slope due to gravity.
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\section{Problem Three}
If gravitational forces alone prevent a spherical, rotating neutron star from disintegrating, estimate the minimum mean density of a star that has a rotation period of one second.

Assuming typical values of 10 km for the radius, and 2 solar masses for the mass, find the shortest rotation period possible without the neutron star flying apart.
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\section{Problem Four}
Using angular momentum conservation, find the angular velocity at which the Earth and Moon will both move, long in the future when they are finally tidally-locked, face to face.  How long will the Earth day be then?  How long will the "month" (the period of the Moon's orbit) be

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\section{Problem Five}
Stars on the (apparent) fringes of our galaxy, the Milky Way, appear to orbit the center with a speed of about 225 km/sec.  For a star whose galactic radius is 17 kiloparsec (about twice as far from the galactic center as the Sun), what does this imply about the total mass of the Milky Way galaxy?  How does this compare to the results of statistical studies which show that the Milky Way contains about 50 Billion stars comparable to the Sun?   Based on the numbers given here, what fraction of the Milky Way's total mass is dark matter?
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