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244lab3

# 244lab3 - 640:244 FALL 2009 Lab 3 The Pendulum This Maple...

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640:244 FALL 2009 Lab 3: The Pendulum This Maple lab is closely based on earlier versions prepared by Professors R. Falk and R. Bumby of the Rutgers Mathematics department. Introduction. In this lab we use Maple to examine differential equations modeling free os- cillations. In particular, we shall consider a linear model in part 1 and a nonlinear model of a pendulum in part 2, comparing the effect of damping in these models. For the linear equation, there are exact solutions that allow these properties to be studied via formulas. For the nonlinear equation, numerical solutions will be used. Please obtain the seed file from the web page and save it in your directory on eden. Then prepare the Maple lab according to the instructions and hints in the introduction to Lab 0. Turn in only the printout of your Maple worksheet, after removing any extraneous material and any errors you have made. 0. Setup. As usual, the seed file begins with commands which load the required Maple packages: with(plots): and with(DEtools): . The seed file also includes the definitions of some variables which will be used throughout the lab, as discussed in section 1 below. 1. The linear model. Consider the motion of a pendulum, which consists of a mass m attached to one end of a rigid rod of length L . The other end of the rod is fixed at a point O and the rod is free to rotate, within a fixed vertical plane, about O . The position of the pendulum at time t is described by the angle θ ( t ) between the rod and the downward vertical direction, with the counterclockwise direction taken as positive (see Figure 9.2.2 on page 498 of the text). The differential equation governing the motion of the pendulum is d 2 θ dt 2 + c mL dt + g L sin θ = 0 It is expected that small changes in the equation will lead to small changes in the solution. Two modifications designed to approximate the equation by one that is more easily solved are to ignore damping by setting c = 0 or to replace sin θ by θ to get a linear equation (or both). Note that θ is just the first term of the Taylor series for sin θ about θ = 0. As long as θ is small , so that the difference between sin θ and θ is very small , the solutions to this linear equation should give a good approximation to those of the general equation. This project will show that, even when θ is not

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