Prep problems for the Nov 23 midterm:
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New one:

Consider a spinning oscillator box for which the natural position of the 
mass is displaced with respect to the center of the spinning.
  e.g., How does the equilibrium position depend on spinning frequency?
Calculate andf PLOT it!
 or, how does the oscillation frequency depend on spinning rate? (C & P vs omega)

More new stuff:  Think about degeneracies.  To describe the degeneracies of a 
system you need to describe how many linearly independent vectors there are for
each eigenvalue (eigenfrequency). 

Class I
 
Understand the eigenmodes and frequencies of coupled oscillator systems:

especially the four-corner oscillator, which is an extension of the triangle
oscillator*,

Four-corner:
                   |M| xxxxxxx |M|
                    x           x
                    x           x
                    x           x
                    x           x
                   |M| xxxxxxx |M|

where all springs and masses are the same;
and each mass moves only in 1-dimension along the diagonal.


*triangle oscillator:
                           m


                     m           m

Also consider a 3 mass oscillator system where the three masses are 
constrained to move on a circular ring and connected by indentical spings.  
possible extensions:
Consider the nature of larger (more masses) ring systems.
Discuss combining modes to get different types of motion and especially
motion which recovers the original symmetry of the system.
Include illustrations in terms of cartoon pictures.


Also don't forget:

          /|  k    _   k    _
          /|zzzzzz|m|zzzzzz|m|               Asymmetric
          /|       -        -


          /|  k    _    k2    _    k  |/
          /|zzzzzz|m|zzzzzzzz|m|zzzzzz|/        Symmetric
          /|       -          -       |/


          /|  k    _   k    _    k       k   |/
          /|zzzzzz|m|zzzzzz|m|zzzzzz|m|zzzzzz|/        Symmetric
          /|       -        -                |/


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Class II:

Consider a wire bent in the shape ax^2 + bx^4
Consider a wire bent in the shape ax^2 + bx^6
Consider a wire bent in the shape ax^2 + bx^8
...
Consider a wire bent in the shape ax^2 + bx^(2n)

Where are the equilibrium positions as it spins around its axis?
How do they depend on the rate of spinning?
which one(s) is/are stable?

class III:

e.g calculate the cirular orbit radius and frequency for small oscillations for:

the central potential U=kr^2.

how about the central potential U=kr.



   more to follow soon!

(PS.  You must understand eigenvectors!!)