Thanks for taking the time to write all that out, AirRabbit. You have indeed made it easier for others to understand the aerodynamics involved. Especially the vector component details - my professors would be proud! So let's put it all together:
Viewed from behind:
1) Plane banks left (with enough elevator to keep it from losing altitude).
2) Lift vector now points diagonally up to the left. (Plane starts to turn.)
3) Gravity vector points down.
4) Centrifugal force vector points to the right.
5) If you sum up 3) and 4) the resulting vector points diagonally down to the right.
6) Now 2) and 5) are opposite to each other and the plane is coordinated without using the rudder (assuming that the bank was hard enough to cause enough of a loss of lift to give gravity an edge which is enough to keep the turn coordinated).
Hey Corsair –
I admire your thinking process – at least you’re not afraid to venture out there looking for other ways of explaining something; and I would encourage you to keep that kind of “spirit.” However, let me point out a couple of things that may have crept into the mix.
In your summation, points 1, 2, and 3 are correct. But let’s consider point 4, and the two forces at work: centrifugal and centripetal. According to Mr. Newton, a moving body travels along a straight path with constant velocity unless an outside force acts on that body. For circular motion to occur, there must be a constant force acting on a body, pushing it toward the center of the circular path. This force is the centripetal (or “center-seeking”) force. You are correct; there is a centrifugal force at work, exactly opposite in direction and equal in magnitude as compared to the centripetal force. But, in our case, which one acts on the airplane? For a planet orbiting the sun, the force is gravitational; for an object twirled on a string, the force is mechanical; for an electron orbiting an atom, it is electrical. Remember, the force we’re seeking to identify is pushing the airplane toward the center of the circular path. What force do you suppose is constantly acting on the airplane; pushing it toward the center of the circular path? Here, it is the aerodynamic force, Lift, acting on the airplane. What, you say? I thought Lift acted to get and keep the airplane in the air. Yes, true. But, as you correctly pointed out in your point 2, the “Lift vector now points diagonally up to the left.” If you break that down into its component vectors, I think you’ll see my point. Certainly one component is vertical (straight up – acting against gravity and when equal to the weight of the airplane, will allow the airplane to maintain altitude) and horizontal (in this case, to the left, toward the center of the circular path). So here, it is the centripetal force that is constantly acting on the airplane, constantly pushing it toward the center of the circular flight path. It is important to know that the centrifugal force does not act on the airplane; the only force acting on the airplane is the centripetal force. However, I think it’s equally important to recognize that it is the opposite, but equal, centrifugal force that, when added to the weight vector, is what makes things “feel” heavier. And by “things” I mean the airplane and everything in the airplane, including you. You probably already know this, but if you calculate what additional “Lift” is necessary to keep the airplane at a constant altitude for the varying amount of bank angles from zero through 90 degrees (where 90 degrees is with the wings exactly vertical) and calculate what would be necessary at 60 degrees of bank - calculating the relevant vectors and dividing them into their component parts, you’ll eventually get to the point that the amount of centrifugal force generated, combined with the weight, will equal 2 times the weight. This is commonly known as a “2-g turn,” and it’s applicable for any airplane at any gross weight.
The rudder DOES play an important part here, but it can’t be replaced by gravity. The rudder is what keeps the relative wind flowing perpendicular to the lifting surfaces, or keeps the airplane in “coordinated flight.” Remember that the rudder affects the airplane “around the vertical axis.” That is the only axis about which it can affect the airplane. So if you rotate this vertical axis to the left, looking at it from behind the airplane as in your example, the rudder will act on the airplane to move the nose of the airplane either up and to the right (where too much will put the airplane in a slip) or down and to the left (where too much will put the airplane in a skid).
Your point #5 is, as you say, a summation of the centrifugal and gravity vectors and will point down and to the right. However, this is the force equal to and opposite of the Lift being generated (as you say in your point #6), and is the summed value that, at 60 degrees of bank, will give you twice the force of gravity (the “2-g turn”) but that is not what puts the airplane in coordinated flight. The coordination function remains with the rudder. Of course, when you do get over to 90 degrees of bank, and you stay there, there is no amount of increased Lift that will be able to counter-act the force of gravity. If you stay there more than just a couple of seconds, you’re going to have to find a way to counter the force of gravity or you will descend like the proverbial rock. Most acrobatic pilots will tell you they use “top rudder” and a lot of power – “top rudder” (or “right rudder”) will provide a nose movement to the right (again, assuming our left bank example), but there isn’t enough surface to generate enough Lift to keep the nose at that position and the pilot will have to rely on an increased amount of thrust (which will then be pointed above the horizon) and, if there is enough of it, will hold the altitude for a while. That’s why you don’t see a lot of “knife edge” passes in airplanes, except those with the relative thrust of an F-15 or F-16 for the airplane involved – and even then, the pilot doesn’t overstay his welcome at 90 degrees of bank. Mother nature isn’t very tolerant of rude guests!