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I would like to understand how exactly a rotating helicopter propeller creates equal but opposite torque on the helicopter body. I learned that if there is no contra propeller (tail rotor), helicopters spin due to unbalanced torque created by the rotating propeller.

So, there’s an electric motor and gas turbine engine to rotate the propeller, doesn’t matter how the propeller is powered. While it rotates, how can it apply equal but opposite torque on helicopter body? Through what contact point? This is the first one I need to understand.

The second one is if we need contact point for this or not? If I had a flying plate on which an electric motor with a propeller shaft upwards like a helicopter propeller is mounted with bolts, would it be spin in opposite direction of the propeller’s rotation direction?

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"If I had a flying plate on which an electric motor with a propeller shaft upwards like a helicopter propeller is mounted with bolts, would it be spin in opposite direction of the propeller’s rotation direction?"

Yes it would. The essential reason is the nature of the force interaction between bodies, that is Newton's third law... If body A exerts a force on body B, then B exerts an equal and opposite force on A. In the case of your plate and propeller, the equal and opposite forces are applied tangentially to the propellor shaft and off-centre to the plate, resulting in equal and opposite torques on propellor and plate.

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Obviously a contact point is required: how, otherwise, would the propellor be made to spin? A typical set-up is that the propellor is on a shaft which is spun by torque from the motor via a gearbox. The torque applied to spin the propellor creates a reaction which causes the helicopter body to want to spin in the opposite sense.

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