If you have not been told already, you will someday be told something like "an airplane flies because its wings are curved on top." Unfortunately, this is only partly true and it's definitely misleading. Think about it. Some, actually most, airplanes can fly upside down as well as right side up. When it is upside down the curved surface is on the "bottom" of the wing. And consider the $2 balsa wood plane with the wind-up propellor. Its wings are flat on both sides and yet it flies. So what's going on?

Let's look at a little basic airfoil theory. A wing is an example of an airfoil. When an airfoil is in motion through the air the reaction of the air produces forces on it. The component of force perpendicular to the direction of motion is called "lift". Lift is thought of as an upward force since the direction of motion of an airplane is usually horizontal.

Anyhow the amount of lift produced by the airfoil in motion depends on several variables. In the case of the wing the shape is just one and probably not even the most important one. The equation below is a pretty good way to calculate lift.

The × means multiplication and the letters are abbreviations for variable quantities:
L is for lift, ρ is for air density (e.g. pounds per cubic foot), V is for velocity (actually speed through the air, feet per second), A is for area of the wing (length x width), and C_L is for "lift coefficient". So lift depends on a lot of things.

Notice that if any of the quantities are zero, we get zero lift. To get non-zero lift we need some air with density, not empty space. We need to be moving though it with some velocity, and our wing has to have some area (or it doesn't exist)! Notice that the velocity is especially important since it is squared. If we double our velocity, we quadruple our lift, if everything else stays constant.

So what is this "lift coefficient"? This is finally where we get to the shape of the wing. Where we can see its effect most easily is in the Figure 4 below. (When we talk about "shape" of the wing" here we are talking about the "profile". That is, what we see if we look at the wing from the tip end toward the fuselage. If we look down on the wing from above we see what is called the "planform".) But first we must emphasize that lift coefficient is affected not only by the airfoil profile, but just as much by a quantity called "angle of attack". So let's see what that is.

Let's look at a "symmetrical" profile with 0° angle of attack in Figure 1 below.

Now look at the same profile at +10° angle of attack in Figure 2.

A negative angle of attack would have the leaading edge of the airfoil point downward relative to the direction of motion.

A little imagination should make you think that a positive angle of attack would deflect the relative airstream downward. And so it does, and that is why lift is produced. The airfoil is forced upward as it deflects the air mass downward.

Now let's look at an airfoil which has the familiar curvaceous top, and therefore is called "cambered", in Figure 3.

This profile can produce lift at 0° angle of attack because it deflects the airstream downward by a different mechanism than the symmetrical airfoil. In the cambered airfoil the deflection occurs because the air is "sticky". It adheres to the airfoil surfaces and since the top surface of the cambered profile curves down more than the bottom curves up, the net airflow has a downward component.

So let's get back to lift coefficient. The graph below, Figure 4, shows curves which tell us how the lift coefficient is affected by the angle of attack, often abbreviated by the symbol α (pronounced AL-fuh) for a given profile. The two curves are for the two profiles discussed above

As you can see, angle of attack is an extremely important variable affecting lift coefficient. First look at the curve for the symmetrical profile. Notice that when α = 0, the lift coefficient C_L = 0. That means no lift, no matter what the speed or size of the wing or air density is. Now look at the curve for the cambered profile. It has positive C_L for α = 0, but that is not terribly important. It is prefered for most aircraft because its drag coefficient is lower for the important values of lift coefficient. I will not go any further with drag coefficient except to say that it is similar to lift coefficient but determines the air resistance to motion of the airfoil instead of its lift. Therefore it determines how much power it takes to produce the velocity required to get a high enough lift for the aircraft to fly.

Another interesting concept can be inferred from figure 4. Even though varying C_L in flight by varying α is an an important means of control, there is another way. What if the pilot could vary C_L by changing the shape of the wing? This would move the whole curve up or down, so C_L would vary even though α stayed the same. In the early days of flying some planes were built so that the pilot could bend the wing to produce this effect, but later it was seen to be better structurally to build in a hinge running along the span of the airfoil. This produces a profile like that of Figure 5 below.

Moving the aft part of the airfoil up and down by means of this hinge varies C_L without changing α and is easier than bending the airfoil.

This mechanism is commonly used in a number of places on an airplane. On the wing there are commonly two sets of these hinges. One set (a pair actually) is located nearer the fuselage, and the trailing edges aft of the hinges are called "flaps". The other pair is located nearer the wing tips and the trailing edges are called "ailerons". The two flaps move in the same direction and increase the lift of the wings when they are lowered, as they are shown in Figure 5. The two ailerons are moved in opposite directions from one another. This gives one wing a higher lift coefficient than the other one and so makes the airplane tip (actually called "roll") as one wing goes higher than the other one.

When you are a passenger in an airplane you will see the flaps deflected downward when the airplane is in its landing phase. This gives the wing a lot higher lift coefficient so the airplane can fly slower when it lands. The maximally lowered flaps also produce a high drag coefficient but this is OK since the landing phase of the flight is short and fuel economy is not much effected. Besides that the plane is going "downhill" which reduces the engine power required to overcome the the high drag. During takeoff you the passenger might see that the flaps are lowered just a little. The gives a higher lift coefficient with only a small increase in drag. The drag reduces the acceleration capability of the airplane and so reduces the benefit of the higher lift coefficient, so lowering flaps for takeoff might not be done for all airplanes.

There are other interesting things about flying which can be inferred from Figure 4. First notice that the C_L does not keep increasing when α is increased more and more, but decreases abruptly at about α=15°. This phenomenon is known as "stalling". A pilot wants to avoid a sudden loss of lift, so he is careful not to let his angle of attack get too high.

A careful look at Figure 4 also shows how an airplane can fly upside down. First realize that if the airplane is upside down then negative C_L means that the lift generated will be in the up direction whereas normally positive C_L is upwards lift. So now see that if we have enough negative α we can get a sizeable negative lift coefficient even with the cambered wing profile. That implies that if we are flying fast enough we can get enough lift to fly upside down. It's even easier to fly upside down with a symmetrical wing profile and many acrobatic airplanes are built that way.

The control of angle of attack is a very important part of piloting. So how is it done, you might well ask? Well, airplanes have another airfoil which looks a lot like a wing but is much smaller and located (usually) at the tail end. It is called the "horizontal stabilizer" or sometimes "tail plane". These are usually, if not always, of symmetrical profile and usually have their own hinged flaps which are called "elevators". (Some designs rotate the whole tail plane.) Changes in lift coefficient of the tail plane by whatever means is used to rotate ("pitch") the airplane nose up or nose down. Guess what - since the wing is rigidly attached to the body ("fuselage") of the airplane the wing rotates with it and thus the angle of attack of the wing is controlled by the pilot using the elevators.

There is another symmetrical airfoil at the rear end of the airplane called the "vertical stabilizer". It also has a hinged flap called the "rudder". The rudder is used in flight mostly to counteract undesirable sideways forces on the airplane that come from various situations. Normally it is desirable to keep the airplane flying without relative wind hitting the airplane from the side, which can produce high drag. So the rudder is used to keep the airplane "flying straight into the wind". In takeoff and landing operations "flying straight into the wind" is not the important goal. Flying straight along the runway is. In this case, the rudder is the most important control used to acheive this.

Although I have tried to be general in my descriptions, there are many variations in airplane designs. It seems almost everything has been tried at one time or another, so it would not be hard to find exceptions to many of my statements. But on the other hand there is no shortage of educational material on aerodynamics and airplane design, so if something I said seems doubtful, then by all means look further.

Before you decide to jump off you house roof with a pair of wings strapped to your back, it would be a good idea to calculate your lift first. Or rather use a spreadsheet approach to see how your wing measures up.

First note that the density of air at comfortable temperatures and low elevations is about 0.0024 pounds-mass per cubic foot. Let's say your wing is well designed and can get a lift coefficient of 1.50, and the weight of you and your wing together is 200 pounds. Now let's manipulate the formula above to get it in a form more suitable for this problem. Say,

where L (lift) has been replaced by W (weight) since we want our lift equal to our weight. Evaluating the right hand side gets 112,000 when units of pounds, feet and seconds are used.

Now for a given size wing we can calculate how fast we must fly for it to lift pilot and plane.

or for a given speed we can calculate how big the wing has to be to lift us

Fun, huh?

The area of a wing 16 feet from tip-to-tip and 3 feet wide is 48 ft². So we see that we have to fly at 48 ft/sec to lift our weight. No one can run anywhere near that fast, even in track shoes and shorts and carrying no extra weight. So it is wise to do such a calculation before you run and jump.