Oscillators and Classical Spring Concepts

See attachment for better formula representation.

1) Consider the two-dimensional anisotropic oscillator with motion given by:
x(t) = Ax cos (ωxt)
y(t) = Ay cos (ωyt - δ).
(a) Prove that if the ratio of frequencies is rational (that is ωx / ωy = p/q where p and q are integers) then
the motion is periodic. What is the period? (b) Prove that if the same ratio is irrational, the motion
never repeats itself.

2) Verify that the function x(t) = te-t
, is indeed a second solution of the equation of motion, x double dot +
2 x dot + ωo 2
x = 0 for a critically damped oscillator ( = ωo).

3) Verify that the decay parameter  - 2
ο
2 β − ω for an overdamped oscillator ( > ωo ) decreases with
increasing . Sketch its behavior for ωo <  < ∞.

4) Consider four identical springs attached in the middle by a circular mass. Each spring has force constant
k and unstretched length lo, and the length of each spring when the mass is at its equilibrium at the origin
is a. When the mass is displaced a small distance to the point (x, y), show that its potential energy has
the form (1/2)k'r2
appropriate to an isotropic harmonic oscillator. What is the constant k' in terms of k?
Give an expression for the corresponding force.

5) A massless spring is hanging vertically and unloaded, from the ceiling. A mass is attached to the bottom
end and released. How close to its final resting position is the mass after 1 second, given that it finally
comes to rest 0.5 meters below the point of release and that the motion is critically damped?