The differential equation which represents the motion of a simple pendulum is 6+ (g/l) sin 0...

 

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The differential equation which represents the motion of a simple pendulum is 6+ (g/l) sin 0 = 0, where g is acceleration due to gravity, is the length of the pendulum, and is the angular displacement. (a) Convert the equation to a system of two equations of first order. (Note that = w is called the angular velocity.) (b) Find all the equilibrium points of the system. (c) Plot the phase field of the system using g = 9.8 m/s and = 1.96 m. In the plot you should include several equilibrium points and several trajectories of different shapes. (d) In your plot in part (c), you will observe two very different families of trajectories. One family consists of closed trajectories, and one consists of non-closed, unbounded ones. Explain what they indicate about the motion of the pendulum. The differential equation which represents the motion of a simple pendulum is 6+ (g/l) sin 0 = 0, where g is acceleration due to gravity, is the length of the pendulum, and is the angular displacement. (a) Convert the equation to a system of two equations of first order. (Note that = w is called the angular velocity.) (b) Find all the equilibrium points of the system. (c) Plot the phase field of the system using g = 9.8 m/s and = 1.96 m. In the plot you should include several equilibrium points and several trajectories of different shapes. (d) In your plot in part (c), you will observe two very different families of trajectories. One family consists of closed trajectories, and one consists of non-closed, unbounded ones. Explain what they indicate about the motion of the pendulum.