G3: Stable and Unstable Fixed Points

Joe Rogers, 1996

In this problem, you will use a computer simulation of a pendulum to look for conditions where simple harmonic motion occurs.

Getting started

When you run the simulation, a window will open on your screen with a picture of the pendulum and a graph allowing you to plot variables of the motion. First, enter your name in the top left space of the window. Try running the simulation by clicking on the Run button. You'll see the pendulum move according to its equation of motion. You can read the positions of points on the graph by holding down the mouse button (the left one if you have more than one button) while moving the cursor over the graph. Try it! (You may want to play a little with the buttons and graphs to see how the pendulum moves under different initial conditions. Instructions for the program are available by clicking the ``Assigment, and Help'' button).

(G3a) Equilibrium points

Start with the initial Theta equal to -4, and the initial Theta Dot equal to one, and hit "Run". (Leave the other settings at what they began with: restart the program if you've enjoyed fiddling around for a while). The pendulum should spin around like mad. Set the graph to plot Torque vs. Theta. You'll see a graph of the torque on the pendulum bob as a function of the angle Theta of the pendulum in radians (Theta=0 when the pendulum is straight down). At which angles Theta is the torque equal to zero? Describe in words what those pendulum positions are. Make a table that lists the angles Theta where the torque equals zero and the slope of the torque (d Tau / d Theta) at each of these angles. Leave one blank column at the end of your table.

(G3b) Stable and unstable equilibria

Start the pendulum out from rest (Theta Dot = d Theta / dt = 0) at a small initial angle Theta = 0.01, and display Theta vs. time on the right. Now set the angle ``Theta Initial'' to the first angle in your table. Run the simulation. Did the pendulum move? Was the motion a small oscillation around the initial angle, or did it run away from that angle? Try this for all the angles you listed in your table and in the last column, write ``stable'' or ``unstable'', depending on the behavior you observe. Explain the stable or unstable behavior in terms of the slope of the torque (d Tau / d Theta) where the torque is zero.

Links Back

  1. Exact vs. Approximate Solutions
  2. The Damped Pendulum
  3. Stable and Unstable Fixed Points

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).