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Cell[CellGroupData[{
Cell["Phys 261 Hwk #9", "Section",
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Cell["\<\
The agenda is to work through various homework problems.  Your task is (a) to \
evaluate each cell in order,
(b) do a few Plots and things as asked for below, and (c) to play with it as \
much as you have time/desire.
There is nothing to hand in, but you may want to print out a few cells for \
your own reference.\
\>", "Text",
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 "As a warm-up, why not ask ",
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 " to do the integral to find the area of a circle?\nThat is, breaking it up \
in to sticks of width dx and length 2 y = 2 Sqrt[R^2-x^2], \nwe want to \
integrate x from -R to R:"
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Cell["\<\

As an example of a double integral, here is the integral for a (hemi-)sphere.
Note the syntax for a double integral; the order of integration matters!\
\>", "Text",
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Cell["\<\
Try taking off the Assumptions option.
And try it with the x and y ranges out of order to see what happens when you \
make a mistake:\
\>", "Text",
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Cell["\<\
And how do you ask for the volume of a cone?  Try it the \"brute force\" way, \
by writing a triple integral over x y and z.\
\>", "Text",
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Cell["\<\
 Try it yourself before double clicking on this cell\[CloseCurlyQuote]s \
little arrow bracket over there ----->>\
\>", "Subsubsection",
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Cell["\<\
So z will vary from 0 to h.  Then given z, y ranges between the limits \
defined by the appropriate lines, and given y and z, x ranges over the limits \
given by the appropriate circle:\
\>", "Text",
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Cell["Morin 8.42 -- Falling Quickly", "Subsection",
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Cell["\<\
To fall quickly, i.e. with maximal angular acceleration, one wants to \
maximize the ratio of torque to moment of inertia.  Whether talking about the \
first moment in time or any other t, I claim that you will encounter the \
function\
\>", "Text",
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Cell["\<\
Plot[] this function, and find its maximum, e.g. by Solve[]ing for when \
f'[x]==0, or by using the FindMaximum function.\
\>", "Text",
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Cell["\<\
Morin 8.18 -- Ball hitting stick\
\>", "Subsection",
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Cell["\<\
So here is a rather generic collision between a ball and a stick, ignoring \
any external forces.
A ball of mass M moving at v0 down the x-axis collides with a stick of length \
L, mass m, and moment
of inertia I0 = \[Beta] m L^2 about its CM (BTW if the stick has uniform mass \
density, what is \[Beta]?).  The collision takes place a distance d from the \
CM (which is 
at the middle of the stick), and afterward the ball continues one way or the \
other along the x-axis.\
\>", "Text",
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Cell["\<\
Your task: finish these conservation equations and then Solve:\
\>", "Text",
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Cell["\<\
Choosing an origin at the middle of the stick the incoming L is:\
\>", "Text",
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Cell["\<\
And since we are told this is elastic ...\
\>", "Text",
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Cell["\<\
Note that there are various special cases you can \
\[OpenCurlyDoubleQuote]check\[CloseCurlyDoubleQuote] here.
For example, if m-->0, one expects that the ball keeps its original velocity.
Your task: think of other cases and check the limits.  For example, what if \
d=0 and m=M?
Or what if the \[OpenCurlyDoubleQuote]stick\[CloseCurlyDoubleQuote] is a \
massless rod protruding from one mass m (i.e. what is \[Beta], and what \
happens)?\
\>", "Text",
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Cell["\<\
Challenge: choose an origin at the point of impact and rewrite eqnLz \
appropriately.
Verify that the system of equations gives the same solution as before.\
\>", "Text",
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A coin (m,R) is launched at velocity v0 and with spin \[Omega]0 across a \
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Animate!\
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 Choosing an origin at the height of the CM, there is a torque  \
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and an angular momentum I0 \[Omega] where\
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This state of affairs continues until the spin rubs off and we roll w/o \
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In this one a cylinder of mass m and radius R sits atop a board of mass m.  A \
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We want the equations of motion, and are asked in particular for \"the \
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If we call the board coordinate x and the cylinder coordinate y (even though \
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Where f[t] is the friction force between the cylinder and the board\
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Choosing e.g. an origin on the board, there is no torque whatsoever, so \
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And there is a constraint: the distance \[OpenCurlyDoubleQuote]rolled\
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These equations we can Solve algebraically and then read off the frequency of \
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Cell["\<\
Try changing the origin to a point which coincides with the center of the \
cylinder and rewrite eq3 to reflect the appropriate torque and angular \
momentum.
Note that with that origin the only torque comes from friction at the ground, \
and that the \"orbital\" (center of mass RxP) term in L vanishes automatically.
Verify that the solution to the modified system of equation coincides with \
the solution we just got.\
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