Homework 1 Solutions

C which of these orbits is stable under small

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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 20 15 10 5 0 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 3L2 α r = − mg 2 sin . ∂r2 mr4 R R This leads to the condition that a stationary point is stable if 3L2 > r0 sin(˜0 ). ˜4 r m2 R2 gα Substituting our result from the stationary point analysis into this equation yields 3 > r0 tan(˜0 ). ˜ r 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 ∂r2 . 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...
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