## Mathematics & Science
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## Applications of Differential Equations## The Simple Pendulum## Theoretical IntroductionOur problem in this laboratory involves the derivation and analysis of the equation governing the position of a pendulum as a function of time. A Before we can derive an equation that gives us the position of the pendulum as
a function of time we must first decide upon a coordinate system to use. It may
seem that the usual x, y, and
t for time. If we can avoid this, we would be better off. After all, two
variable problems can be bad enough, without making things worse.So what should we use to track the pendulum bob? Mathematicians would
describe the pendulum as a system that only exhibits .single variable
Likewise our pendulum has only one degree of freedom, so only one variable
is needed to give its position. Taking our cue from the example of
navigating a ship, it doesn't take a great leap of genius to realize that the
pendulum's position might be described in terms of the With me still? So the two variables we'll use in this problem will be time,
denoted by Pictorially we have this so far: Now that we've chosen our variables we start to look for the pendulum's equation. As usual with physics based problems the derivation depends upon an analysis of the forces involved in the system. ## Tangential ForceWe're going to look at the tangential force upon the pendulum bob from two different points of view.
which can soon be rearranged to give:
As we said a few seconds ago, acceleration is found by taking the second derivative of arclength. Applying the second derivative to the last equation gives us:
(Remember,
Almost there now. We have the angular acceleration, but we said we needed
to look at the
which immediately becomes:
thanks to Newton. So we now have one version of the
Note that the lower angle can also be labelled So there we have it, another name for the tangential component of force is, from our picture:
The force being ## Equating the ForcesNow we get the pay-off for the work we did deriving the tangential
component of force in two different ways. From our
and from our
These can be equated to give us:
Dividing both sides by
and moving the right hand side over to the left we get:
This is the equation we were seeking all along--the differential equation
governing the motion of the simple pendulum. Now we'll look at how we can
solve this differential equation using the computer. Let's move on and see how we can implement these ideas in |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2009 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |