It’s intuitive that the horizontal extrusions (like the wings) would help with stability, but what about the vertical one, extruding upwards from the back?
How does it interact with the wind and help keep the plane steady? If anything, wouldn’t that surface be “pushed” over by the wind, like a sail? And does it serve any other purposes?
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Horizontal extrusions don’t actually help with stability, they simply provide lift.
*Two* horizontal extrusions similarly don’t help with stability in and of themselves – they stabilize a plane because each one separately provides roughly equal amounts of force. If the two wings are spaced equally from the plane’s center – which *does* happen because plane fuselages are symmetric – then the two forces balance and contribute to preventing unwanted roll.
The problem is that if you look at a plane from the side, the two wings essentially provide lift at one location along the length of the fuselage. This is not good – if this location is offset from the center of mass, then you will create unwanted pitch. It’s like trying to balance a pencil on one finger – you have to get that pencil *perfectly* balanced, or it’s gonna roll off your finger.
~~You’ll notice that many planes have smaller tail wings as well. These serve to help with unwanted pitch, because the smaller tail wings provide lift while offset from the center of mass. With the above pencil example, it’s like trying to balance a pencil on *two* fingers – MUCH easier, and there are multiple places where you can put your two fingers for it to work.~~ See below explanation
A vertical stabilizer (which is what you are describing) stabilizes the plane’s yaw direction. Without it, there is nothing preventing the plane from going into a flat spin. [Here](https://gfycat.com/pettypoliticalarthropods) are intentional flat spins; you can imagine how scary it would be to get into them *unintentionally*. Strictly speaking, two vertical stabilizers would be better – but the plane has to land eventually, and a vertical stabilizer pointing into the ground would make the plane difficult to land without damaging the tail.
Lastly, remember that plane design is all about understanding where the plane operates. Some planes don’t have a vertical stabilizer and two rear horizontal stabilizers, but will instead have [V-tails](https://www.planeandpilotmag.com/images/stories/2005/jan/bonanza_lead2.jpg) that accomplish the same as a combination of horizontal and vertical stabilizers. Some planes lack rear stabilizers at all – these tend to be highly mobile craft like fighter jets, because in highly mobile craft the lack of control corresponds to high maneuverability, or stealth aircraft, where more surfaces corresponds to less stealth. To make up for the lack of stabilizer, these aircraft also often have different wing shapes, like delta wings or flying wings. Flying at high altitude vs. low altitude and/or supersonic (and even hypersonic) vs. subsonic also affects wing configuration.
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