## Mathematics & Science
Learning Center |
|||

## Applications of Differential Equations## The Simple Pendulum
## A Real Pendulum—Facing Up to FrictionFinally, then, what about a real pendulum? We've been leaving out friction. Now it's time we considered it. First we need to decide how this might be done realistically. It seems reasonable to assume that the Now let's see...angular velocity is given by the derivative of angular
position—in this case Back in the introduction to this lab we derived the following formulas for force:
which was found by analyzing the angular acceleration, and:
which came about when we analyzed the gravitational acceleration. It is
this second force into which we must incorporate our frictional force—after
all, friction, like gravity, acts
As we did in the introduction, we equate the two formulations of force to get:
or, moving all the terms to the left-hand side:
This could use a little more work. We could divide through by
But that's a bit too ugly. Hmmm! Hang on a second.
which is where I'm going to stop messing with it. Finally, then, if we
bring back our old initial conditions we have a new,
Let's go on and look at an actual example using this new initial value problem... |
|||

ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |