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when an aircraft banks to left or right, why don't passengers cups fall off the table in front of them?
For the same reason that a person on a bicycle or motorcycle doesn't fall over when he leans into a turn—which is also the same reason why an ice skater can lean to one side while turning without falling.
It's all about acceleration and physics.
Normally, when an airplane is in straight and level flight, the only force acting on passengers is gravity, which tries to pull them straight down. When the airplane turns, however, centrifugal force created by inertia as the airplane accelerates to one side in the turn tends to force passengers in the opposite direction, away from the turn. The tighter the turn, the greater this sideways force becomes.
However, when an airplane turns, it also banks. The tighter the turn, the greater tha bank. In a turn the passengers are not being pulled straight down, but instead are being pulled down at an angle. The bank of the aircraft matches the angle at which the passengers (and the rest of the plane) are being pulled. Thus, in a turn, passengers and their coffee cups are still being pulled directly towards the floor of the aircraft, and not to one side.
In addition to airplanes, skaters, and motorcycle and bike riders, one sees this same phenomenon when trains enter a curve (the trains are often tilted slightly to allow the train to bank into the turn) and when racing cars enter a turn (the track is often banked).
Instead of being pushed to one side, passengers in a turning airplane feel as if they are moving upwards, because the force acting upon them increases (as it would if they rose upwards), but it remains normal to the floor of the aircraft (that is, straight towards the floor). When the plane comes out of the turn, passengers will feel as if they are going down.
Mathematically, the force acting on the passengers is a vector. In level flight, the vector is composed only of gravity, and points straight towards the ground and towards the floor of the cabin. In a turn, the vector increases in magnitude as the lateral acceleration of the turn is added to it, and it shifts to one side from a purely vertical direction because the added acceleration is lateral. The banking of the aircraft keeps this tilted vector perpendicular to the floor of the cabin, so passengers feel a bit heavier (giving them the impression that they are rising), but things still are pulled directly towards the cabin floor.
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