Unformatted text preview: ng on the value left-hand side of the above equa˜
tion there can be 0, 1, or 2 stationary points. 11 30 Ω2 R2 gΑ 25
0.0 0.2 0.4 0.6 0.8 1.0 1.2 r0 R Figure 1: The frequency of oscillation about the equilibrium orbit.
(c) Which of these orbits is stable under small impluses along the surface transverse to the direction of motion? An orbit is stable if the second
derivative of the eﬀective potential is positive,
∂ 2 Ueﬀ
− mg 2 sin
This leads to the condition that a stationary point is stable if
> r0 sin(˜0 ).
m2 R2 gα
Substituting our result from the stationary point analysis into this equation
3 > r0 tan(˜0 ).
Solving this numerically, we see that a stationary point is stable if r0 < 1.19.
(d) If the orbit is stable, what is the frequency of oscillation about the
equilibrium orbit? The frequency ω is given by
mω 2 = ∂ 2 Ueﬀ
r=r0 Because there are transcendental equations in the problem it will be easier
to proceed numerically. Figure 1 shows ω 2 R2 /gα as a function of r0 /R in
the region where the stationary point is stable. 12...
View Full Document