HW 6 Classical MechanicsSeptember 25, 2007 Due October 4

1. A bead of mass m is constrained to move on a hoop of radius R. The hoop rotateswith constant angular velocity Ω around a vertical axis (along which gravity acts).

(a) Set up the Lagrangian and obtain the equations of motion for the bead.

(b) Show that for Ω less than a critical angular velocity Ω0 the bead has a stableequilibrium position at the bottom of the hoop.

(c) Find the stable equilibrium position for Ω > Ω0.

(d) Find the frequency of small oscillations about equilibrium for Ω > Ω0.

2. Consider a double plane pendulum for which the lengths are equal (`1 = `2 = `)but different masses (the upper mass m1 differs from the lower mass m2).

(a) Find the normal modes.

(b) Suppose that originally the double pendulum is stationary, with the lowerpendulum vertical and the upper at a small angle θ0 to the vertical. Find themotions at subsequent times.

(c) Show that when the lower mass is small compared to the upper mass the tworesonant frequencies are almost equal. Examine (b) in this case (you will find‘beats’).

3. In a linear triatomic molecule the initial condition is that the center atom is atrest but displaced by an amount a0 from equilibrium while the other two are attheir equilibrium points. Find the amplitudes of the longitudinal small oscillationsabout the center of mass.

4. The equilibrium configuration of a molecule is represented by three atoms of equalmass at the vertices of a 45 right triangle connected by springs of equal forceconstant. Find the frequencies of oscillation (including three zero modes).

5. Consider the two-dimensional harmonic oscillator with frequencies ω1 and ω2.

(a) For ω1/ω2 = 2/5 describe and plot some Poincare maps.

(b) Repeat for ω1/ω2 = 199/500. Try to do this numerically.

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