Physics 101 Introduction to Mechanics 

April 29 SHO & Pendulums  



Chaos
in Pendulums
Chaotic systems exhibit irregular, unpredictable
behavior. A classic example of this is what's known as the butterfly
effect due to nonlinearities in weather processes, a butterfly flapping
its wings in Tahiti can, in theory, produce a tornado in Kansas.
Here we don't have any external forces other than gravity, yet we see chaotic motion.
Now we pull the bob to one side and let go.
Here are graphs of the two angles versus time.
If we look at the actual equation for the pendulum,
no small oscillation approximations, we see that it is very nonlinear.
Okay, now what if there is a torque that pushes on the pendulum in a periodic manner?
Now what if there is a dampening force that is proportional to the angular velocity? (air drag)
Wow. This is a mess, but that's the beauty of it. This is a much more correct model than our simple harmonic oscillations, consequentially it also means that our motion prediction is closer to the actual motion. From the equations one sees that we get chaos.
The motion is no longer periodic.
Let's look at a few different systems, each with
different driving and dampening torques.
Notice that by making just slight changes to our
ideal system, we produced complicated, even chaotic, motion.


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