OBTW, when the pilot stands on his brakes and revs the engines, I don’t THINK he’s generating a force equal to the total thrust of the engines, just enough force to stop the wheels from turning. Maybe it has something to do with inertia, et al, but it is beyond my aging Physics education.
Enough force to stop the wheels from turning is exactly the same thing as the total thrust of the engine (assuming the engine is horizontal). Most planes can produce more thrust when at higher speeds though.
Could you please give us the equation for frictional force as a function of relative speed? Because all of the models for dry friction with which I’m familiar have the force being approximately constant with speed. In other words, it’s possible to construct a system such that the frictional force would be enough to prevent the plane from taking off, but such an airplane wouldn’t be able to take off from a normal runway, either.
I’m not actually convinced of that, since the wheels are rotating, and thus a non-inertial frame of reference. General relativity would come into play at some point. For that matter, proper acceleration is only applicable for someone attached to the conveyor belt; passengers in the plane would see Lorentz effects as the belt accelerated relative to their frame of reference. The belt would be able to accelerate as long as it wanted without bumping into c, but the motionless plane would see the speed of the belt top out at c, long before which it would contract until it was too small for the plane to stand on. 
(I don’t want to think about what would happen to the wheels, rotating up to c.)
Well, I’m trying not to get too much into this without taking away from my present work (let’s hear it for PowerPoint!). But O.K., without having to drag out my first year Engineering Statics and Dynamics text book, here’s a little something to chew on, using a 747. Data is taken from the Boeing website, as well as Goodyear Aviation.
Using one common 747 configuration, the aircraft at takeoff is 130 tons. ~80% of that weight is distributed on the aircraft’s 16 main gear tires, which are sizable, 49" tall and 19" wide - the designation is H49x19.0-22, for aircraft tires the second number is the nominal width, not aspect ratio as in passenger cars. (oh, each tire costs $3,600).
Length of the tire’s ground footprint under load is 20% of it’s circumference.
49" x PI x 20% = 31"
footprint is 31" x 19.0" = 589in²
With 65 tons of downward force on each of the main tires, there is
65/589 = 220lbs/in²
on the contact surfaces. Now, at this point I haven’t gotten the time to look up the coefficient of friction (µ) between the tires and the tarmac, but it’s pretty damn high… µ is the tangential force required to create slippage, divided by 220lbs/in². It’s gonna be pretty damned high, high enough that there will be no slipping of the tires on the tarmac. So we know that the tires and the tarmac will roll together, with a resultant significant backwards force on the tires… This means that a sufficiently powerful treadmill can provide enough backwards force to keep the plane on the ground, because the rolling resistance plus to friction between the tarmac and the treadmill is too significant to overcome.
Now, as far as the landing gear assembly is concerned, guess what? The frictional resistance of the tire assemblies (ball bearings, etc) is pretty considerable until the plane starts going down the runaway and some lift is provided by the wings. As the plane’s wind speed increases, the plane’s relative weight on the wheel assemblies decreases, and frictional resistance decreases, etc.
What does this all mean?
If the plane, from the very get-go, does not generate any air speed, the frictional resistance of the landing gear, added to the backwards force generated by the treadmill - it’s possible that, on a treadmill, with the plane at full thrust, the plane will not only be stationary, but the wheels may be spinning at significantly less than they would at takeoff speed - maybe even a speed relative to the treadmill of less than 50mph, far less than the average takeoff speed of 155mph.
According to Goodyear aviation: “Heavy loads and high speeds cause the heat generation in tires to exceed that of all other tires and can have a very detrimental effect…rubber dissipates heat slowly…for this reason, aircraft tires and only be used intermittently.”. So, rotate the tires too fast and they burn up. Think about that.
So, a 747 cannot take off, nor can any other commercial ariliner. But guess what? Start plugging in these figures for a light and powerful plane, and the problem appears differently… a commerical airliner can’t take off, but maybe a jet fighter can?
The moral of this story is that people that like non-engineering thought experiments shouldn’t build airplanes.
I think it’s finally getting through to me that you think BR#2 is a neat problem.
Yeah, pretty much. Exactly what happens isn’t intuitively obvious, I don’t think. There’s also a lot of effects in play, so you can get into a whole string of, “what happens?” “Well, this does.” “But ignoring that, what happens?” “Well, this does.” “But ignoring that, what happens?”
I think it’s sort of an interesting exploration of forces and motions, examining them by pushing the thought experiment to its boundaries. I imagine most other people think of it as a tiresome exercise in semantics, but my enthusiasm is undampened. 
Very impressive, Paradoxic. But I didn’t see any speed dependance anywhere in there. We are agreed, of course, that there is some amount of friction in the landing gears. I think we’re also agreed that a plane on a normal runway can overcome that friction, start moving, accelerate forward, and eventually take off. I assert that the friction in the treadmill case will be no greater than the friction in the runway case, and that the plane will therefore still accelerate forward and eventually take off. You assert, however, that the friction will increase with increasing relative speed. You still haven’t given any support to that assertation, however.
First post (Yay, me.)
K, on this topic: so many wasted words for such a simple question. When conducting mind experiments, one doesn’t worry the mechanics. Thus, Einstein could hop on a train that could approach the speed of light, etc.
The simple and obvious answer to the OP is: Given that a treadmill can be constructed to keep the plane stationary, the plane will not take off. No lift is being generated, lift coming from airflow over the wings. QED.
First, welcome to the boards. Second, reading the whole thread would help to avoid problems like this. As has been stated by multiple level headed people you are answering a different question than the people who think the plane will take off. You are defining the speed of the plane relative to the moving surface of the treadmill. The others are defining it relative to an immobile object. Think on this young one, and return with wisdom.
Airplanes don’t stand on the runway, they sit on the runway. Try overcoming that friction.
Sorry about being late to this forum. I was going over the archives and found that Cecil’s conclusion on the ‘plane on a tredmill’ question is in error. The plane if it remains stationary to the air surrounding it, will not not fly no matter how fast the prop spins and the wheels turn. Lift is provided by airspeed. The problem described only mentions groundspeed. Without sufficient air flow over the wings there is no lift.
Sorry folks, this turkey won’t fly.
You fail entirely to comprehend the ‘problem.’ Perhaps reading the whole thread would help you.
You don’t comprehend that the issue is what happens to the force created by the turning of the propellers or the igniting of the jet fuel. That force acts upon the plane. If you can explain how the plane can stay motionless relative to the air under those circumstances, then we will accept your answer.
No! :eek: Cecil in error?? :dubious: Say it ain’t so, Joe! 
I hate arguments semantics. Here’s what is really going on here. The original question was badly phrased. This created two factions, one that took the question literally as is, and one that instead tried to assess the intention of the original question. Arguments over semantics are stupid, it’s a lot of wasted energy to prove nothing except that you interpret something differently. Instead lets focus on what we do agree on.
You need relative motion of a plane to air to create lift. That’s it, we’re done, lets move on.
And you have relative motion of the plane to air. The plane will move through the air at the same speed as it would without the treadmill. What happens to the treadmill and to the wheels depends on your interpretation, but in either interpretation, the plane will move forward through the air.
That was the whole point of the original question (not the question as answered by Cecil, but the one somebody originally contacted him about): It’s a thought problem, an exercise in examining how people interpret badly-worded questions. The original questioner could not care less whether or not the plane takes off; he wanted to find out how people would interpret the question.
What about this column’s question makes you think that? It was a straight forward question, asked by a guy who understood the correct answer, but couldn’t get his buddies to agree.
As for the version asked on the Internet, I’ve yet to see it asked in a way that allows the friction component to actually overcome the force of the engines. I suppose it might be buried in one of the many threads on the subject here on the board, but I’ve not bothered to find it. 
From the column you link to:
As you point out, one problem here is the wording of the question. Your version straightforwardly states that the conveyor moves backward at the same rate that the plane moves forward. If the plane’s forward speed is 100 miles per hour, the conveyor rolls 100 MPH backward, and the wheels rotate at 200 MPH. Assuming you’ve got Indy-car-quality tires and wheel bearings, no problem. However, some versions put matters this way: “The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation.” This language leads to a paradox…
I’ve truncated it there because the rest of Cecil’s argument is, to be charitable, not well thought-out.
Luckily, though, he did a bit better job of it in the second column on this subject. The point here is that you needn’t rely on friction to balance thrust from the engine, but rather the inertia of the spinning wheel.
We agree that the problem was in some forms given worded in a way that offered multiple interpretation. However, that does not mean that the person who thought the problem up did so to test your ability (or lack thereof) to spot a paradoxical wording.
Further, and this is VITAL, there simply is no way to reach the type of result in what Cecil calls BR2 without completely changing the wording of the problem. As long as the problem is worded to deal with the velocity of the aircraft, or the wheels, you cannot achieve the solution advocated and discussed in the second column.
All of which means: As originally formulated, the problem was intended (CLEARLY) to make people stop and think that airplanes are not cars. Any other approach to the original question is simply an attempt to be a contrarian.