A hypothetical question on mechanics Being located in a tropical region, I am quite acquainted with the Ceiling fan. I have a question about it.
If the top, that is, the axle (I'm not sure of the terminology: I mean the part which is thin, rodlike, attached to ceiling)...is rigidly fixed, then when the fan is turned on, the blades spin.
On the other hand, when the local mechanic brings the fan down and puts it on the floor, then when it is powered on the blades are fixed and the axle spins.
My question is this:
If, neglecting friction, I keep both top and bottom of the fan free to move (like maybe in outer space), and I turn on the fan, what will happen?
 A: If your fan is not connected to anything, and the blades do not encounter any air drag (outer space) then conservation of momentum means that

the blades will turn in one direction, and the motor assembly in the other direction

If you know the moment of inertia of the blades, call it $I_b$, and of the motor, $I_m$, then the ratio of the angular velocity of the blades vs motor will be given by the conservation of angular momentum:
$$I_b \omega_b = -I_m \omega_m$$
In other words, if the moment of inertia for the blades is the same as for the motor, they will rotate in opposite directions with the same speed (with their relative speed equal to the speed that the motor can reach, given the applied voltage).
Note that the above assumes the following layout:

I can't quite figure out from your description whether the motor is rigidly attached to the blades (or maybe half the motor is) - so you might need to interpret my answer accordingly... bits that are fixed to the blade add to the inertia of the blade, and bits that are attached to the axle/motor add to their combined moment of inertia.
A: 
If, neglecting friction, I keep both top and bottom of the fan free to move (like maybe in outer space), and I turn on the fan, what will happen?

Some satellites use something very much like this to keep the satellite pointing in the right direction, and do so without using rockets. It's called a reaction wheel. The motor is rigidly connected to the satellite body. What about the fan blades? There's no air to move in space, and moving air isn't the point of a reaction wheel. The point of a reaction wheel is to make the satellite rotate a bit so it points in the right direction. Reaction wheels use a solid flywheel instead of fan blades.
When your fan breaks down you just call your electrician or mechanic. That's a bit tougher in space. Astronauts replaced one of the reaction wheels on the Hubble Space Telescope during one of the Hubble repair missions.  The Kepler satellite on the other hand is out of the reach of any repair mission. The breakdown of two of its reaction wheels pretty much put Kepler out of the planet finding business.
