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A pendulum on a rigid rod oscillates according to the equation 8²0 Ət² where is the angle of the pendulum from the vertical. ㅠ 1. Show that 0 = 0 (the pendulum hanging straight down) and 0 = (the pendulum balanced vertically up) are possible equilibrium solutions of the equation. + sin 0 = 0, 2. By linearisation show that the solution is unstable, while the solu- tion 0 = 0 is stable; at least, show that it is not unstable.

A pendulum on a rigid rod oscillates according to the equation 8²0 Ət² where is the angle of the pendulum from the vertical. ㅠ 1. Show that 0 = 0 (the pendulum hanging straight down) and 0 = (the pendulum balanced vertically up) are possible equilibrium solutions of the equation. + sin 0 = 0, 2. By linearisation show that the solution is unstable, while the solu- tion 0 = 0 is stable; at least, show that it is not unstable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A pendulum on a rigid rod oscillates according to the equation 8²0 Ət² where is the angle of the pendulum from the vertical. + sin 0 = 0, 1. Show that 0 = 0 (the pendulum hanging straight down) and 0 = (the pendulum balanced vertically up) are possible equilibrium solutions of the equation. 2. By linearisation show that the solution is unstable, while the solu- tion 0 = 0 is stable; at least, show that it is not unstable.
Transcribed Image Text:A pendulum on a rigid rod oscillates according to the equation 8²0 Ət² where is the angle of the pendulum from the vertical. + sin 0 = 0, 1. Show that 0 = 0 (the pendulum hanging straight down) and 0 = (the pendulum balanced vertically up) are possible equilibrium solutions of the equation. 2. By linearisation show that the solution is unstable, while the solu- tion 0 = 0 is stable; at least, show that it is not unstable.
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