The fly isn't being supported by a magically maintained column of air that follows him around. He flies because his wings are providing lift by moving air. The air he moves simply dissapates its energy buy moving the air around it. The fly in flight inside an aircraft does not add weight to the aircraft, any more than an aircraft in flight adds weight to the ground beneath it. If you are under the flight path of a commercial jet, the jet's weight will not crush you.
Lets make this a lot more scalable since a single fly vs an entire aircraft is just orders of magnitude different. Any device that could measure a plane's weight would have an error range of many flies.
Lets scale this to a fly inside 1ft cube made of plastic wrap and tooth picks (with some small holes to allow oxygen for the poor fly). Under this cube is a scale so the system we are weighing is that of the cube + fly
There are a few states the fly could be in:
- Resting on a side of the cube
- Pushing off the bottom of the cube (about to take flight)
- Flapping its wings and maintaining position
- Flapping its wings and ascending
- Flapping its wings and descending
Case 1:
Weight = F
total = m
total * a (in this case we'll assume g).
m
total = m
fly + m
cube
This case should be self evident. To simply the other cases we are going to assume that m
cube is constant (thus F
cube is constant) and only deal with F
fly.
Case 2:
As the fly pushes off the bottom it will cause the scale to read an increase. If you don't believe me grab a bathroom scale and try it. You've basically increased a so F increases.
Case 3:
This is the tricky one. First we need to realize that for the fly to maintain position the sum of all the forces acting on that fly has to be 0. Otherwise you'd have acceleration which is movement. So we the force of gravity has to be equally matched with an upward force. This force is generated by the flapping of the wings. The force caused by the wings gets transfer through the air and will eventually hit the sides of the cube. Now this is the important part: It does get transfer in a straight line, it gets dissipated. So what might have a great force in a small area (high pressure) becomes a great force over an larger area (lower pressure). Now lets say all the pressure hits the bottom, it will then hit the scale and register that force. I honestly don't know what the scale would show if the pressure hits the sides. So this means that the total weight is still F
cube + F
fly and F
fly can be closely approximated as m
fly * g.
Case 4:
Now if the fly to ascend the upward force has to be greater than the downward force. Given the same assumptions about the pressure hitting only the bottom then the scale would show an increase.
Case 5:
Basically the reverse of case 4 so the scale would show a decrease.
So, yes a fly flapping its wings in an enclosed aircraft does contribute to the total weight of the aircraft.
Back to the OP's question it really depends on what system you are measuring.
If you put a scale on my stomach while I was hovering in mid-water then the scale will read 0. Since I'm not ascending nor descending the total forces on my equal 0. Since weight is a force and F = m * a it'll think I weight 0. This is because I'm the system.
Now lets put me in a giant aquarium (such that I won't displace water out of it) with a scale under the aquarium. Since the system is now the aquarium + me then what matters is the total forces acting on that that system. We'll you've got the mass of the aquarium, my mass, and the mass of the water all times gravity. The force the water pushes up on me with does get transfer down to the bottom where the scale can register it. (Fly scenario)
