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The Nonlinear Pendulum - pgccphy.net
The Nonlinear Simpson, of Physical Sciences and EngineeringPrince George s Community CollegeDecember 31, 20101 The Simple Plane PendulumAsimple plane pendulumconsists, ideally, of a point mass connected by a light rod of lengthLto a frictionlesspivot. The mass is displaced from its natural vertical position and released, after which it swings back and are two major questions we would like to answer:1. What is the angle of the Pendulum from the vertical at any timet?2. What is the period of the motion?For such a simple system, the simple plane Pendulum has a surprisingly complicated solution. We llfirstderive the differential equation of motion to be solved, thenfind both the approximate and exact Differential Equation of MotionTo derive the differential equation of motion for the Pendulum , we begin with Newton s second law in rotationalform: DI DId2 dt2;(1)where is the torque,Iis the moment of inertia, is the angularacceleration, and is the angle from the the case of the Pendulum , the torque is given by D mgLsin ;(2)and the moment of inertia isIDmL2:(3)Substituting
The Nonlinear Pendulum D.G. Simpson, Ph.D. Departmentof PhysicalSciencesandEngineering Prince George’s Community College December31,2010 1 The Simple Plane Pendulum
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