The project only makes sense to me if you have all strings the same length and want to "magically" adjust the swing frequency different for each pendulum to get something like the video shows. I do not think however you can change the frequency that much for each pendulum.
So I suggest you do a single pendulum experiment first, time the frequency (see how long 20 swings take) and then see how much you can influence it with your electromagnet. The distance ball/magnet should be only a few mm, the magnetic field strength falls off quite fast with distance (quick google; I cant remember - dipole strength by distance from pole is a ... inverse cube.)
For sensor I would suggest a reflection/proximity sensor, two if you want to know where the pendulum comes from (well, you should know in the Arduino software, it comes from each side alternativly 
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If you use MOSFETs make sure you (a) use a diode to absorb the turnoff/backpuls, and (b) use a "Logic level" MOSFET (fully turns on at 3 or 5V, not the usual 10V).