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Planes fly upside_down for the same reason that they fly right-side up, due to the lifting force, know as lift, generated by the wings.
A wing generates lift when flying forward due to two factors - the cross-sectional shape of the wing and the angle_of_attack. The angle_of_attack is the angle of the wing relative to the direction of the flight. If the nose of the wing, called the leading edge, is tilted upwards relative to the direction of flight, the angle_of_attack is positive and vice versa.
The cross-sectional shape of the wing is designed such that it efficiently produces lift at a very small angle_of_attack and with low drag. Drag is the force that opposes the forward motion of the wing. Hence the need for engines on the plane to push the wing forward against the drag. The higher the drag, the bigger(and heavier) the engines and hence higher the amount of fuel required to fly the plane. This is why we strive to design a cross-sectional shape that will produce high lift with low drag.
As mentioned earlier, the second way to generate lift is through the angle_of_attack. The more the leading edge is tipped upwards the larger the lift. For example, you could build a wing that is simply a flat sheet and generate a lift by angling it upwards (positive angle_of_attack) with respect to the direction of flight. This is how most kites fly. However at a large angle_of_attack, you would generate a lot of drag. That is why a flat sheet does not make a usable wing.
Now let us come back to planes flying upside_down. To keep flying the plane's wings must generate an upward force, otherwise the plane would come crashing down. The required lift cannot come from the wing cross-sectional shape since it is generally shaped to generate lift when flying right-side up. In fact, the cross-sectional shape would generate a downward lift when flying upside_down. So the upward lift must come from tilting the plane up (skywards) to create a sufficiently large positive angle_of_attack to generate the needed lift force.
But as mentioned above, a high angle_of_attack also generates high drag and so the plane's engines must be powerful enough to keep pushing the plane forward if it is to continue flying up-side down. That is why planes designed for normal level flight cannot fly upside_down for very long. In fact, unless the wing is strong enough to withstand the forces when flying upside_down, it will fall apart. Hence, few planes except stunt planes and fighter aircraft specifically designed for such maneuvers can fly upside_down.
To summarize, planes fly upside_down by tilting their nose skywards to create sufficient lift through increased angle_of_attack, and revving up the engines to counteract the large drag created. Provided of course that the wings are strong enough to take it!
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