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Applications of Differential Equations

The Simple Pendulum

(continued from last page...)

An Example Involving Friction

Reminder: we're dealing with the initial value problem:

Friction Aware Initial Value Problem

y'' + k y' + (g/L) sin(y) = 0
y(0) = yo
y'(0) = vo

We'll now put in some actual numbers for the constants in our initial value problem. Taking k = 0.5, and as before L = 1, g = 9.8, and yo = 0 the initial value problem becomes:

y'' + 0.5 y' + 9.8 sin(y) = 0
y(0) = 0
y'(0) = vo

Now it's up to you to analyze this new problem in exactly the same way as we did earlier for the frictionless model. Find both:

  • a linsol solution representing the solution of the linearization of this problem. (You must do the linearization yourself, of course, but it should be easy if you look at what we did before.)
  • an actsol solution, found using NDSolve (use the interval 0 ≤ t ≤ 10 this time, instead).

Make colored graphs of both solutions on the same plot, and do the whole analysis twice:

  • once for vo = 1
  • once for vo = 10.

Carefully consider both graphs and decide whether they give the type of information you would expect physically.

OK, time to cut you loose. You can always return here if you forget any of the instructions, or you can look back at what you did earlier in your Mathematica session.

We'll now go look at the results that you should have found...


Compass If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.
 
 

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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