CSC 210

Week 7


Topics:


Assignment:

  1. Lunar Pendulum:

  2. Over the top: In the pendulum equation in Phaser set m = 1.4, l = 1.3, and g = 1. Assume that pendulum is at its stable equilibrium position. How much minimal initial velocity do you need to impart on the bob so that the pendulum goes over the top? Does this initial velocity depend on the length of the pendulum? Does it depend on the mass of the pendulum?

  3. Escape velocity: For this investigation use the Kepler ODE in the ODE Equation Library of Phaser. Suppose that a particle of mass = 1.1 is positioned at the coordinates (2.15, 0). For this problem, first you should load the equation defaults for Kepler ODE. Then set the appropriate initial conditions to follow the shapes of the orbits.

  4. Onset of chaos in the Lorenz Equations: Load the Lorenz ODE in the ODE library of Phaser. Load the Equation Defaults. Set the parameter value r = 12, leave the other two parameters as they are. Notice that the two solutions approach an equilibrium value. You should be able to see this in Xi Vs. Time and Phaser Portrait views. Now, increase the value of the parameter r gradually. What is the smallest value of r for which the solutions become chaotic? A chaotic solution is one that is not equilibrium or periodic. Although there are sophisticated tests for chaotic solutions, at this point you can decide by inspection if a solution is chaotic. If necessary, you can take longer times.

  5. R: Go to the web site http://www.r-project.org/ and download the appropriate version of R for your personal computer.