A plane has 10,000 canaries on board and has exceeded its take-off weight by 5000 canaries, can the plane still take off if you keep half of the canaries flying?
No. The force required to keep the canaries up is the same as their weight. I think… 
If the plane is airtight, no. If there is open exchange of air between the airplane’s interior and the surrounding air, yes.
BTW, why does the plane have 10,000 canaries aboard?
I think the open exchange of air thing is correct, but it’s hard to imagine how it would work. For the canaries to fly they have to support their weight by transfering downwards momentum to the air (downwash). Somehow you’d have to get that air out without it transferring downwards momentum back to the the aeroplane’s structure.
The canaries were on an Air Force jet, on their way to Alaska to participate in a project called HAARP.
Of course, It’s not going to be perfect. 
The open exchange of air thing might work, like a screen window. However, wouldn’t that mean that you would not be able to fly at an altitude that requires pressurization? We need the canaries alive, for now.
Picture a canary in a corked bottle on a scale. When the canary begins flying, the scale doesn’t read less***!***
Have you tried this?
I saw a demo answering this question on that goofy kid’s show Newton’s Apple. A guy had a remote control helicopter sitting on a scale. He showed that when it hovered above it the scale registered the same weight (from the force of the air blowing down) as when it was sitting on it.
This is a simple application of Newton’s Third Law, “If object A exerts a force on object B, then object B exerts an oppositely directed force of equal magnitude on object A.” You have to first consider the force that’s holding the canaries up in the air and then find what the downward force is acting against. It has to be the airplane; there’s nothing else to contain it.
Hail Ants’s experiment from Newton’s Apple is an excellent one.
My uncle told me that if you had a tub full of water with a catfish in it, that it weighs less if the catfish is swimming that if it were just sitting on the bottom.
But he also told me that if you cut off dogs’ tails, their offspring will similarly be born without tails.
Your uncle has a knack for being wrong. Does it run in the family?
The canaries would have to be flying forward at 500 miles per hour. otherwise, the plane is carrying them and overcoming the inertia of their bodies by transmitting force through the air. I think.
Actually the closest was the comment about the 747 flying over your house. The plane is not supported on a column of air because lift is provided by the pressure differential between the air moving (relatively speaking) faster over the curved upper wing surface than the air under the flat lower surface. Ie the lift comes from the fact that there is more pressure under the wing than above it. Only in ground effect aircraft (Ekranoplans) does the downwash contribute. To get lift you have to move the plane forward and this force is provided by the plane’s fuel or the bird’s energy stores. Ie the plane is, for the purposes of this discussion weightless. It’s weight is cancelled by the energy used to propel it. Before you ask gliders use the potential energy from the launcher or thermals in which case the energy comes from the sun. It is the energy from the fuel which is traded against the mass of the plane or bird. Thus, temporarily at least the weight is removed from the measurement with making the atmosphere any heavier.
Maybe…
Not true. There are three (equivalent) ways of making a lift calculation that I can remember:
- Calculate the pressure distribution
- Calculate the ‘circulation’ (related to equivalent vorticity)
- Calculate the downwash.
Which ever method you use, downwash is always present. The ‘Ground effect’ that you mention is increased lift due to the effective strength of the image vortex (negligibly small at 66000 feet). By the way, downwash is just the verticle velocity component imparted to the air. It’s not a verticle column of air that’s somehow moving along directly underneath the aeroplane, that’s why your house doesn’t fall down when a 747 flies over. Now if a giant hovercraft flew over your house…
But what if you’re not at 66000 feet. Suppose I’m standing on a runway as a 747 is taking off. The airplane passes just a few feet over my head. Now, I’m sure I would be knocked over by the rush of air displaced by the moving vehicle, but surely I wouldn’t be crushed by the airplane’s weight.
(Do not try this at the local airport. They won’t think it’s funny.)
The 66,000 feet thing is 33,000 times 2. The image vortex doesn’t actually exist, it’s just a convenient way of calculating the effective flow field for ground effect, but that’s all a bit technical. As far as planes going over your head on a run way are concerned…
Maybe you would be knocked to the ground, I don’t know, you’d certainly find yourself in one hell of a wind. The force that you would experience would be:
Force=Velocity of wind squaredDrag coefficient0.5density of airprojected area of your body
Think about this equation and it explains why oak trees can blow down in a gale that a human can stand up in. The drag coefficient is a measure of how streamlined you are and 0.5 is just a convention. Obviously this isn’t the same as being crushed by the weight of a 747.
However the momentum of the air in the downwash will have to be transferred at some point (momentum is a vector quantity and the air can’t pass through the ground). This may take place over a wide area and at low velocity depending on the flow field (something much too complicated to be calculated). Not all the molecules of air from the 747’s down wash are going to reduce their vertical velocity to zero on top of your head whatever the proximity of the plane, so you’re in no danger of being crushed. 
Oh, the “column of air” reference comes from this wacky thread:
http://boards.straightdope.com/sdmb/showthread.php?threadid=13796&pagenumber=1