View Single Post
Old 02-03-2006, 04:10 PM
zut zut is offline
Charter Member
Join Date: Apr 2000
Location: Detroit, MI
Posts: 3,725
It's all about the interpretation of the question. Unfortunately, Cecil commingled two different interpretations in his column. Let me repost something:

There are plenty of answers to this question, because the key to the question is what you assume from the beginning. The assumptions are the key. Let's start off at the top:

A. Suppose we actually built a treadmill and put a 747 on it, and had the tradmill match the wspeed of the wheels. Would the 747 take off? If "exactly matching the speed of the wheels" means that the treadmill matches the *hub speed* of the wheels (the speed of the wheel center), then yes. The treadmill simply accelerates in the opposite direction that the plane does. The wheels wind up *rotating* twice as fast as they normally would, but the plane will take off, leaving a treadmill behind that's rotating in the opposite direction.

B. But that problem is trivial. Let's assume that "exactly matching the speed of the wheels" means "matching the outer diameter surface velocity"--the velocity with respect to the hub. Would the 747 take off? Almost certainly it would, but only because we can't build a treadmill capable of keeping up with the thrust transmitted to the plane by the engines--in other words, we violate the spirit of the question, because the treadmill isn't matching the wheel velocity.

C. OK, that's stupid. It's a thought experiment. Posit a magic treadmill that can accelerate as fast as desired. And it doesn't break. I imagine the wheels will skid on the treadmill, because the friction won't be able to transmit the necessary force. In that case, we again violate the spirit of the question, and--

D. It's a thought experiment, smart guy. Assume there's enough friction to rotate the tires. All right. When the engine lights off, the treadmill will accelerate until the force transmitted through the wheel hub to the plane exactly balances the thrust. The plane would stay stationary as the thrust power was dissipated in the wheel bearings (as friction), tires (hysteresis), and in accelerating the wheel to ever-increasing speeds (All of these effects wind up transmitting force between the plane and the treadmill). Since all the power is dissipated in the wheels, eventually either the bearings would overheat, the tires would blow, or the wheel would rip itself apart due to inertial forces. After that, the plane crashes and burns. Then you've destroyed a rather expensive magic treadmill.

E. Thought experiment, I said! Let's posit ultra-strong and heat resistant tires. All right. It turns out the real world is rather complicated. If the treadmill is a long, runway-sized treadmill, it will eventually, running thousands of miles an hour, pull in air at high enough velocity that the plane will lift off at zero ground speed (but substantial air speed). However, now you're running into trans-sonic compressibility effects...

F. No speed of sound effects! And assume magic air that doesn't become entrained with the treadmill motion. And don't throw in any other crazy stuff, either. In that case, the treadmill speeds up (still balancing the plane's thrust force) and the plane stays in place until the engines run out of fuel. I imagine the treadmill goes pretty fast at that point. The plane stays put until the fuel's gone, at which point the magic treadmill whips it backwards.

G. Backwards, shmackwards. Now we're getting somewhere. What if we had infinite fuel? Then the wheels keep going until they're running near light speed, and relativistic effects take over. The wheels get smaller, I suppose...

H. None of that! No relativity-- Hey, wait a minute. Back up. Suppose we have *zero friction* bearings and tires. That doesn't seem so unreasonable for a thought experiment. Well, zero friction tires would mean they just skid on the runway, since nothing turns them. So the plane will take off, tires motionless, and the treadmill won't move.

I. Hey! Quit it! I already said the tires don't skid! Sorry. Just friction on the tire/treadmill interface, then, but none in the bearing or sidewall. With zero friction in the bearing, you lose the friction coupling between the treadmill and the jet. But you still have inertial coupling. The jet power goes into *accelerating* the wheels, and you have the same case as you do *with* friction. The jet stays stationary as the wheel accelerates; the wheel just accelerates faster.

J. Well, how about the other way around? *Massless* wheels, but you still have friction? Here it starts to get complex. As you accelerate the wheels, the bearings will change shape and heat up and so forth, so it's reasonable to guess that the "friction coefficient" goes *up* with increasing speed. If that's the case, then when the engines start, the treadmill accelerates up to whatever speed will give *enough* friction to balance the thrust. The plane stays stationary, wheels rotating at some reasonably constant (but large) velocity, dissapating the engine power through friction.

K. But I want massless wheels and a *constant* coefficient of friction. Indestructable wheels, remember? None of this hand-waving "it's gonna get bigger" crap. OK. It *is* a thought experiment. With a limited "friction coefficient," only a limited amount of energy can be absorbed by the friction. When the engine lights off, the treadmill instantly accelerates to infinite speed. It's never able to counteract the thrust force, and thus plane takes off, leaving the infinite-speed treadmill behind.

L. Ah. OK, one last step. What if we had no bearing friction *and* massless tires? What happens then? Pretty much the same thing. There's now *no* energy losses in the wheels and tires, *no* coupling between the treadmill and the plane--no bearing friction, no inertial effects, no air resistance, and no way for the treadmill to affect the plane's motion. The same thing would happen as above, with the plane taking off, leaving the infinite-speed treadmill behind. However, there's one added interesting thing: This is now an unstable runaway system. There's no resistance to treadmill motion, and a positive feedback circuit. Imagine the poor mechanic who bumps a wheel, setting it in motion. A very slight roll by the tire is sensed, and the treadmill luches forward. The tire goes faster, the treadmill goes faster, the tire goes faster.... Since we've posited an instantly-accelerating treadmill and no relativity and no air resistance and no wheel inertia, the treadmill goes from zero to infinity in no time flat. Try to keep your balance on that.

Pick your scenario--they're all correct.