1. Consider a symmetrical mass spring system on a turntable spinning at a frequency Omega, like we did thursday in class, for which the potential energy of each spring is: U=kx^2/4 + kx^4/(2a^3)

a) what are the units of a?

b) What is the Lagrangian of this system?

c) Use the Lagrangian eqn of motion to determine the "effective potential" for this system. Plot it and calculate the position of all equilibrium points. Which are stable?

2. What is the Lagrangian for a bead of mass m that move without friction on a circular wire loop aligned vertically in gravity (g).


3. Consider a very lightly damped oscillator, characterized by sqrt(k/m)=1 sec-1 , and a damping, beta=0.01 sec-1 , Suppose it is driven by a "square wave" force such that,
F(t)=F_0 for -d < t < d,
and F(t)=0 for d < t < T, etc.
with d=T/100
and that F(t) is periodic in time with a period T=4x(3.14) sec, and,


a) Plot F(t).

b) calculate the first 3 terms of the Fourier series of F(t), and use them to obtain x(t). Is this a good approximation for F(t)?, for x(t)? why or why not?

c) plot x(t) paying attention to its relationship to F(t).

d) Show on a plot of D(w) what frequencies your 3 terms (in the Fourier series) are "sampling".


4. problem 7.13


5. Calulate the response x(t) for a lightly damped oscillator hit by successive delta-function impulses at t=0 and t=T. (Use the Green's function method.) What value(s) of T is/are kind of interesting...?



Reading, chapter 7