I just this morning noticed that the fan on my Prusa i3 was not running. Going further up the wires, I saw that the electrical tape at a splice had caught on the extruder gear, pulling it loose and letting the wires short. I tried several different fans on D9, but nothing works. Everything else on the printer is running fine. (except for an unrelated extruder grip issue) Have I fried a component on my RAMPS 1.4? or do I just need to reset something?

You might have just fried the D9 MOSFET. The MOSFET is the upright transistor that has a metal tab to which you can attach a heat sink. There are 3 on a RAMPS, one for D8, D9, and D10.If you're handy with a soldering iron, it's not that difficult to replace, and the component itself is relatively inexpensive. If there's nothing else obviously wrong, this is the approach I would take.

Ah, that might be it. I'll get a new MOSFET, replace it, and try that. Better to replace a single component than the whole board. Thanks!

While waiting for it.....
Have you got 12v on D9+ ? If so, check the bottom of the Ramps board for a damaged track

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Waitaki 3D Printer