- Modeling continuous systems using Ordinary Differential Equations (ODE).
- Application: Pendulum.

Pendulum Equation

Pendulum history

**Lunar Pendulum:**- In the pendulum equation in PHASER set m = 1.2, l = 1.4, and g = 1. Note that x1 is the angular position and x2 is the angular velocity of the pendulum. What is the period of the pendulum motion starting with initial angle of 1.2 radians and zero velocity? To increase the period by 12% how much do you need to change the length of the pendulum? Use the Xi vs Time view to determine the period.
- If you take your grandfather's clock to the moon, does it slow down or speed up? By how much? (You need to look up the value of g for the moon relative to earth.)

**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?-
**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.- Use the initial velocity components (0.0, 0.35) and admire the elliptical orbit. Double the mass to m=2.2. Notice that the orbit gets flattened. Find the value of the vertical velocity component (0.35) that gives the previous elliptical orbit with m = 1.1.
- Set m =1.1 and find the minimum vertical velocity so that the particle escapes to infinity; that is, its orbit ceases to be an ellipse. You might need to use a fairly long time and a big window size to get a good estimate of the vertical escape velocity.
- Next, double the mass of the particle. What is the new escape velocity?

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