We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction or air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the undamped motion of the pendulum of length Lis: dx dt -=y de where x = 0 is the angular position, y = at is the angular velocity, and g is the acceleration due to gravity (see Figure (left)). (i) Determine all of the critical points for the system. (ii) Determine the linearised system for each critical point in part (i), and find the corresponding eigenvalues of the general solution. (iii) Consider the critical points at the origin and (7, O): ) dy = -9₁ sinx i. State the expected type and stability of each critical point, given the eigenvalues of the linearized system ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system

We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction or air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the undamped motion of the pendulum of length Lis: dx dt -=y de where x = 0 is the angular position, y = at is the angular velocity, and g is the acceleration due to gravity (see Figure (left)). (i) Determine all of the critical points for the system. (ii) Determine the linearised system for each critical point in part (i), and find the corresponding eigenvalues of the general solution. (iii) Consider the critical points at the origin and (7, O): ) dy = -9₁ sinx i. State the expected type and stability of each critical point, given the eigenvalues of the linearized system ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system

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Description: Please show steps We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction orair resistance to slow down the motion of the pendulum. In this case, the non-linear system of differe

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter12: Oscillatory Motion

Section: Chapter Questions

Problem 11OQ

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