What determines the rate at which a planet spins?

Well it's a very good question. The interesting part about this is how do you actually stop a planet from rotating?

The way planets are actually formed, the only way planets can be formed, is in these very large spinning disks [around a developing star]. As young stars collapse [in on themselves as they form], they begin to to lose angular momentum and the dust clouds themselves are naturally spinning.

It's much easy to make things spin than it is actually to have things stationary. So as the planet system forms, the disk spins faster and faster to lose angular momentum. The material can then form and fall towards the star, and out of this collapsing disk you get these sort of lumps and bits and pieces, which become planets.

These planets are intrinsically spinning from the angular momentum of the overall formation disk. Then, as the planets evolve and as they begin to clear gaps and stuff, if they bump into other things these can either slow them down or in some cases even spin them up. It becomes a very complicated interaction. Sort of like if you spin a whole bunch of marbles in a bucket and just let them go, they'll all start bumping into each other.

That essentially is what happens in the early planetary systems. At the end of all these process, as more and more material gets swept away and the planets become more and more isolated, they're basically left with whatever spin they last had at their last interaction. That could be the intrinsic angular momentum of the disk or angular momentum that they actually gained or lost as they bumped and smashed into each other.

The moon is actually a very interesting case because this is what's called tidal locking. You can get to a certain point where, if you're spinning at a certain rate, the angular momentum of the Earth and the moon begin to interact and they can actually lock if they are the right distance apart and going at roughly at the right speed.

This is in fact what happened with the moon. It became tidally locked with the Earth and is now rotating in this locked sense so that you only ever see one face of the moon at any one time. The only thing that will now change the rate of the moon is if something hits it. Well it will actually all begin to slow down a little bit because it's losing its angular momentum because the tides and the Earth are actually putting a little bit of drag on the moon's rate.

So over billions and billions of years, things may go out of lock up again. So things do change, but it can take a very long time. But in the very early solar system, these things were happening very very fast...