Physics Q & A - How can a helicopter fly?
Cheung Kai-chung (Translation by Janny Leung)    

How can a helicopter fly up the sky? Why does it need an upright tail motor?

If the tail motor does not exist...
Fig. 1   If the tail motor does not exist, the motor blades and the fuselage would spin in opposite direction.
You feel cool standing in front of a fan. If you switch to a higher speed, the wind will become stronger, and the fan itself will even tilt slightly backward. You could imagine a helicopter as a huge fan producing wind strong enough to lift itself up! The rapidly spinning main motors are like blades of a fan. When a motor blade spins, it presses the air downward (action force). According to Newton's third law, the air will provide it with a reaction force in return, lifting the helicopter up.

What is the use of an upright tail motor then? Let us firstly consider a situation without the tail motor. According to the principle of conservation of angular momentum, when there is no external force, the total sum of angular momentum of the helicopter will be zero. Suppose the motor blades spin clockwise, the fuselage should spin in an opposite direction, that is, anticlockwise. Therefore, it is impossible for a helicopter without a tail motor to stay stable because of the anticlockwise moment of force exerted on it. A spinning tail motor provides a clockwise moment of force to counterbalance the moment of force produced by the main motors and stabilizes the fuselage.

Thus without a tail motor, one of the gadgets of Doraemon - the bamboo dragonfly - could never enable Nobita to fly steadily. He would just spin!