Solved: The rod of a pendulum is attached to a movable joint at point P and rotates at

Chapter 11, Problem 21

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The rod of a pendulum is attached to a movable joint at point P and rotates at an angular speed of \(\omega\) (radians/s) in the plane perpendicular to the rod. See FIGURE 11.R.1. As a result, the bob of the pendulum experiences an additional centripetal force and the new differential equation for \(\theta\) becomes

\(m l \frac{d^{2} \theta}{d t^{2}}=\omega^{2} m l \sin \theta \cos \theta-m g \sin \theta-\beta \frac{d \theta}{d t}\) .

(a) Establish that there are no periodic solutions.

(b) If \(\omega^{2}<g / l\), show that (0, 0) is a stable critical point and is the only critical point in the domain \(-\pi<\theta<\pi\). Describe what occurs physically when \(\theta(0)=\theta_{0}\), \(\theta^{\prime}(0)=0\), and \(\theta_{0}\) is small.

(c) If \(\omega^{2}>g / l\), show that (0, 0) is unstable and there are two additional stable critical points \((\pm \hat{\theta}, 0)\) in the domain \(-\pi<\theta<\pi\). Describe what occurs physically when \(\theta(0)=\theta_{0}\), \(\theta^{\prime}(0)=0\), and \(\theta_{0}\) is small.

(d) Determine under what conditions the critical points in parts (a) and (b) are stable spiral points.