# A good theoretical approximation for a magnetically damped pendulum

In a laboratory course we had to perform an experiment with a pendulum (just an iron weight on a wire) and play around for some time with its wire's length and so on.

This was quite boring and we decided to make something more interesting:

We took two magnets (like this one) an tied one on the previously mentioned wire (which is made of plastic), and placed the other one on the surface of the table (I've tried to create a schematic).

We tried to create a regular damped oscillator but the lab's staff told us it's not a good approximation, the problem is that we've only had one semester of mechanics and we've just began electricity.

So my question is what what be a good relatively easy to understand theory which we actually apply to our measurements (we've measured the amplitude , the angle the time we've just video taped the whole motion).

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## 1 Answer

Your idea is a good one. When I taught this course we actually used a metallic (but non-magnetic) pendulum which moved through a magnetic field during its whole swing. It turns out that to a good approximation the damping is proportional to the velocity. What causes the damping is this: by moving a metal around in a magnetic field you create currents in the metal which create their own magnetic field. These two fields interact and can create weird effects -- usually resistance.

For example, if you drop a magnet down a copper tube, the magnet will fall very slowly. The concept is the same.

The reason your particular experiment is not a good approximation is because the damping agent is only present during a part of the swing. Also, it's not proportional to velocity, but to how close the magnets are. Luckily this last part isn't so bad because the velocity is roughly proportional to how close the magnets are (because the pendulum moves fastest when its at equilibrium.)

As far as I know there isn't really a good closed form solution to your problem. But this "answer" (if you can call it that) should give you a good idea of how to make an actual damped pendulum.

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