Physics 321 -- Spring 2003

Homework #7, due Wednesday March 12

1.         (4) Marion & Thornton: problem 4-1.  You can solve this problem by calculating either the force on the particle or the potential energy.  To get the answer in the form shown in the problem, you must perform a Taylor series expansion of (1+y)^-1/2 where y = (x/L)².  Note that the rest length of the springs is (l-d), not just l.

2.         (4) Marion & Thornton, problem 4-2.  Draw the phase space diagram (xdot vs. x) right below the potential energy plot, as is done in Figure 4-5 in the book.  This way one can see the correspondence between the two diagrams.

3.         (2) Marion & Thornton, problem 4-6.  Use conservation of energy to calculate theta-dot as a function of theta.

4.         (15) Marion & Thornton, problem 4-10. To do this problem you will need to integrate the nonlinear differential equation using a computer. Download my Mathematica file from the course website to get started:  Mathematic File for Nonlinear Oscillator Simulation  It contains one simulation showing chaotic behavior and another one showing periodic behavior, both for a drive frequency of 0.5 in dimensionless units.  The only difference between the two simulations is the value of the drive amplitude. 

After you finish the specific simulations asked for in problem 4-10, I encourage you to play around with the three adjustable parameters: drive amplitude, drive frequency, and damping. The driven damped pendulum is a very rich system with some surprising behavior.

Note the following points:

1) The differential equation and initial conditions are enclosed together inside a curly bracket.

2) The differential equation and initial condition equations contain double equal signs.

3) The funny set of symbols in the Evaluate function /.% tells Mathematica to evaluate the function calculated in the previous step. (The % sign always refers to the previous step.)

4) Notice that I plot the results for t between 900 and 1000.  That is to ensure that the transient behavior due to the initial conditions has decayed away, and all we are left with is the particular solution.  If you ask Mathematica to go to very long times, it may give you an error message. I circumvented this by increasing the maximum number of steps allowed, with the MaxSteps option.  The number of steps needed varies depending on the parameters in the equation.

5) To make the phase space plots, I defined both the solution y[t] and its first derivative y'[t].  To do this I gave the solution to the differential equation a name ("solution"), then used the Evaluate function.  I’ve forgotten why I needed the “[[1]]” at the end of that line – but I know it’s important!