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Applications of Differential Equations

The Simple Pendulum

(continued from last page...)

The Initial Value Problem

On the last page we carefully derived the differential equation governing the motion of the simple pendulum. Once again we need to remind ourselves that solving this equation by itself will lead to a family of solutions. If we introduce some initial conditions, however, then we'll have an initial value problem with a unique solution.

As usual, for a second order initial value problem the initial conditions will be in terms of y and y', an initial angle and an initial angular velocity.

Our new initial value problem might look something like this:

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

As a reminder:

  • y = angle of displacement (a function of time, t, yet to be determined)
  • L = length of the pendulum
  • g = acceleration due to gravity
  • yo = initial angle of displacement
  • vo = initial angular velocity

Also, y = 0 corresponds to hanging straight down, and a counterclockwise displacement from vertical is considered positive.

Linearizing the Problem

This equation is non-linear and second order. It cannot be solved by standard methods. The usual classroom approach is the linearize the problem. This consists of replacing the non-linear term, sin(y), by a linear approximation.

So what would make a good linear approximation for sin(y)?

There are at least two common approaches that may be used to justify our next step:

The Graphical Approach:Use Mathematica to plot the graphs of both sin(y), and y itself on the same coordinate system. Use a domain of -0.5≤y≤0.5. The command that you'll need is:

Plot[{Sin[y],y},{y,-0.5,0.5}]

Notice the braces enclosing the set of functions to be plotted. These are always necessary when more than a single function is to be plotted.


Mathematica Icon It's now time to actually try the command. You can flip back here when needed in order to check the command. (Actually, you could even copy and paste it into Mathematica.) Click on the button provided on the left to switch. Don't forget to come back here when you're done!

Welcome back! Let's go discuss what you got...


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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