Mathematics & Science Learning Center
Computer Laboratory

Applications of Differential Equations

The Simple Pendulum

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Your picture should look like this, this time:

Don't misinterpret what you are seeing here in the picture. The red graph shows y getting larger and larger as time goes by, whereas the blue graph shows the value of y oscillating. In order to see more clearly the finer details of the oscillatory behaviour of the blue graph, let's zoom in a little. Replotting our previous graph with one simple difference, i.e. changing the domain to 0 ≤ t ≤ 1, we get the following image (try it yourself!)

Let's discuss the implication of these two totally different looking functions:

• It looks like the actual pendulum (the upper, red graph) makes an angle with the equilibrium position, y = 0, that just keeps on increasing forever. Physically this means that the pendulum just keeps spinning in a circle, generating a larger and larger positive angular displacement.

• On the other hand, (don't be fooled by your instincts,) the linearized model (the blue graph) gives a totally unbelievable solution! It shows a positive displacement of as high as 3.2 radians at first (that's more than 180 degrees), but then it actually comes back and does a similar displacement in the opposite direction. It repeats this forever. Think about it—this is physically impossible.

Now you may say that the actual solution is impossible too—after all, a real pendulum can't just keep spinning forever. However, remember, our pendulum model is frictionless, so in theory it really should do exactly what the red graph predicts!

We now move on to consider a more realistic model for the pendulum...

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