## Applications of Differential Equations

### The Simple Pendulum

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Your first plot, using the initial condition of vo=1, should look like this:

As we saw earlier in the frictionless model, for small values of y the linearized model does a very good job of approximating the actual solution. (The values of y are small, notice, on the order of 0.3 radians, and shrinking thereafter.)

Your second plot, using the initial condition of vo=10, should look like this:

Here we get a big difference between the two solutions. The reason, of course, is that the y-values involved here are much larger than in the last problem—on the order of more than 2.8 radians, or 160 degrees. The linearized model tells us that the pendulum manages to swing around a full 160 degrees counter-clockwise, and then comes back. This may be possible, but it sounds a little doubtful. Somehow, it seems that an initial velocity of 10 radians/second should manage to knock the pendulum around a little further than this.

The actual solution indicates that the angle achieved by the pendulum is as high as about 14.2 radians, or 814 degrees, and then it started oscillating about 12.6 radians or so. The 814 degree angle would indicate that the pendulum swung around about two full circles before it lost enough energy to the friction that it settled down to a more steady oscillation.

Both models nicely exhibit the presence of friction in that their amplitudes get smaller and smaller as time goes by. Such motion is described as damped oscillatory motion. Pendulums are not the only physical systems that exhibit damping. Spring systems also demonstrate this effect.

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ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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