Solved: If we assume that a damping force acts in a direction opposite to the motion of
Chapter 11, Problem 16(choose chapter or problem)
Miscellaneous Nonlinear Models
If we assume that a damping force acts in a direction opposite to the motion of a pendulum and with a magnitude directly proportional to the angular velocity \(d \theta / d t\), the displacement angle \(\theta\) for the pendulum satisfies the nonlinear second-order differential equation
\(m l \frac{d^{2} \theta}{d t^{2}}=-m g \sin \theta-\beta \frac{d \theta}{d t}\) .
(a) Write the second-order differential equation as a plane autonomous system, and find all critical points.
(b) Find a condition on m, l, and \(\beta\) that will make (0, 0) a stable spiral point.
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