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On the majority of modern aircraft, the rudder is used for two purposes:

1) The rudder is used to keep the nose of the plane pointing in the direction it is going. Planes are turned by banking the plane (lowering one wing and raising the other) and then using some of the lift to curve the plane's flight path. When this happens, something called "adverse yaw" causes the plane's nose to turn in the opposite direction. The rudder is used to counter adverse yaw and keep the plane pointing in the direction it is going (called 'coordinated flight').

2) During some flight operations such as landing in a crosswind, the rudder is used to intentionally cause the plane's nose to point in a different direction from the direction the plane is going (called a 'slip'). This is used most commonly to keep the plane aligned with a runway during landing.

Contrary to popular belief, the rudder is not used to turn the plane. If you used the rudder to turn the plane, you'd wind doing a dutch roll, which would make the passengers very uncomfortable (they would feel like they were sliding), waste fuel (the plane would experience enormous drag as some of the side of the plane faced into the wind), and at low speeds or high turn rates, it would be unsafe and could cause one wing to suddenly stop producing lift if the airflow detaches from the surface of the wing.

On some aircraft, the rudder is also used to counter various twisting forces that the plane experiences. For example, on single-engine, propeller-driven aircraft, there is a leftward twisting force that is applied to the plane during climb caused by the way air flows off the turning propeller. The rudder is used to correct for this force and prevent the plane from entering uncoordinated flight and (if you didn't apply any rudder at all) entering a power on stall.

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