1. For the "mass on the table top connected by a string through a hole in the
table top to another mass" problem, (U=mgh, no friction) that we discussed
last thursday,
a) write lagrangian and calculate Lag. eq. of motion.
(as well as eqn for cons. of ang. mom.)
b) determine equilibrium "states". How are they defined?
Is there a qualitative difference between r=constant and theta=constant motion?
c) are circular orbits stable? Can you define or calculate a frequency for
small departures from circular motion?
(hint: you cannot treat theta-dot as a constant, moreover, it is not independent
of r. what is a constant?)
d) can you use that to draw what slightly non-circular orbits look like?
2. a) Can you figure out the "normal mode" oscillations frequencies for the mass
spring system.
|wall|xxxxxxxx|m|xxxxxxxxx|m|xxxxxxx|wall|
You can assume that the two masses are equal, and that the two outer springs
have the same "k". (the middle one can be different)
b) Can you draw a time elapse cartoon of the motion of a normal mode?
c) what is you inderstanding of the relationship between the order of the frequencies and the motion.
3. a) Determine the linearized lagrangian eqn's of motion for the
double pendulum (like the thing in the lobby). (eqn's that are useful
for small deviations from equilibrium.)
b) can you find a frequency of motion for this system.
4) Find the trace, the e.v.'s (eigenvalues) and the eigenvectors of the
following matrices:
a) | 1 0 | b) | 1 1 | c) | 2 2 | d) | 3 1 |
| 0 2 | | 1 1 | | 2 2 | | 1 3 |
e) | 2 1 | f) | 2 1 | g) | 1 0 0 0 |
| 1 2 | | 1 1 | | 0 2 0 0 |
| 0 0 3 0 |
| 0 0 0 4 |
h) | 1 1 1 | i) | 1 1 1 1 |
| 1 1 1 | | 1 1 1 1 |
| 1 1 1 | | 1 1 1 1 |
| 1 1 1 1 |
[Hint for h) and i): find the eigenvectors BEFORE the eig. values (by guessing). Then get the
e.v. that belongs to each (by using the definition). Which is harder, h) or i) ? why?
question to ponder(and answer): What is degeneracy??