The Basic Physics Behind Helicopters

The in class demonstration on the conservation of angular momentum inspired me to look at the physics behind how helicopters fly.

The rotation of the propellers creates a force that pushed down on the air. Given Newton's third law, the air pushes back with an equal and opposite force. When that force is greater than the force of gravity, the helicopter can move up.

But, given the conservation of angular momentum, why doesn't the helicopter spin in the opposite direction of its propellers? It has to do with the spinning tail motor that opposes the rotational force of the blades.

Here is an example of what happens to a helicopter that loses its tail motor.

—http://www.youtube.com/watch?v=hnK9bGCvYtU