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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17176ae9-b253-442d-89f9-517b049b4c59%2F732c9e18-1510-4998-a12f-adcc8f604111%2Fjrtooqg_processed.png&w=3840&q=75)
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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