Airplane On A Treadmill

A forum for good logic/math puzzles.

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phlip: I saw your wording above which was why I was sure to specify my own - to reduce miscommunication, and can see what you're getting at.

In fact, by my wording, there's a logical incosistency if you use a car. Car moves -> Belt moves other way -> Car's stationary -> ERROR. Which is fine, because it's a plane - not a car. The question phrased as I have it still tends to lead people into the dyno/car comparison and they make the same mistake.

My biggest trouble with your wording is that, although I feel what "at the speed the vehicle would be doing if the belt weren't there" means, I'm not sure it can be precisely defined. I haven't followed it through, but worry it might be open to contradictory legitimate interpretations.

The problem as worded in the original post of this thread is appalling though.

...designed to MATCH the speed of the wheels of an airplane...

Firstly, define "speed of the wheel". Is that "the rate at which the axle of the wheel gets from one place to another", or "the reading on a speedometer connected to the wheel". They are quite different. A wheel not in contact with the ground that has been fired from a cannon will have a high speed BUT as it isn't rotating a speedometer connected to it will display zero.

Conversely a motionless wheel on a moving treadmill isn't going anywhere (so has no speed), but a speedometer connected to it will show a non-zero value.

If "speed of the wheels" in the question DOES mean "how quickly the wheels get from one place to another" then, because they're connected to the plane, it means EXACTLY the same thing my phrasing of the question does.

If instead "speed of the wheels" means "reading shown on a speedometer connected to the wheel" then the question BY DEFINITION (literally) says that the plane cannot move.

ie. A speedometer on a wheel touching a conveyor belt will show the "speed at which the wheel is getting somewhere else" PLUS "the speed at which the belt is moving under the wheel". So if the question defines the belt as one that moves at the speed shown on the wheel's speedometer, the "speed at which the wheel is getting somewhere else" is DEFINED as zero. As the aeroplane's attached to the wheel then its speed is ALSO DEFINED as zero. There's no mechanical process holding the aircraft there, just the phrasing of the question has effectively become: "There is a belt with a motionless aircraft that will always remain motionless on it. Can the aircraft move?" Obviously a trivial question, and so I reject the possibility that the question could mean that.

EDIT: Written as the last four posts were made. Everything since Rey's.
EDIT2: Added philip to start of post.
EDIT3: Added "that will always remain motionless" to last paragraph.
EDIT4: Added EDIT2, EDIT3, & EDIT4 notes.
EDIT5: Changed "the rate at which the axle of the wheel moves" to "the rate at which the axle of the wheel gets from one place to another" and EDIT5 note.
Last edited by Birdman on Sun Feb 25, 2007 6:57 am UTC, edited 5 times in total.

Birdman

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Which is why the way phlip and Birdman described the question makes a lot more sense. (Go Aussies!)

Hawknc
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@Rat

Sure, in reality, the plane wouldn't take off. But under the scenario described AND using ideal conditions, it would.
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Mathmagic
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Rey wrote:
phlip wrote:Why would his legs stay behind?

Again think of water skiing

In water skiing your legs go forward. Otherwise you end up sinking.

If you lean back on the treadmill with rollerskates and pull, you'd go forwards alright. It's like standing on an ice-skating rink and pulling yourself around the edges. Sure it's not trivial to balance, but you can do it...

SpitValve
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Uh, actually, I think the non-physics-educated people are right.

I've completed three semesters of Honors physics, and my roommate is an aero-space major.

At first I thought that yes, the plane could clearly take off, because there was a forward force that isn't matched by the frictional force of the wheels.

However, the wheels don't exert a frictional force on the plane; they exert a rolling resistance, which is dependent on velocity. In normal physics classes this doesn't matter because the velocity-dependent component is an order of magnitude smaller than the constant component at reasonable physics-lab speeds, but for aerodynamic work, it matters. I got the equations for rolling resistance from my roommate's aerodynamics textbook. (F=Crr*N and Crr=Crr1+Crr2*V)

Here's the equations, with hypothetical numbers for a Boeing 747-100:

: Corrected numbers (Thanks, Birdman.)

Mass of the plane:
m = 735000 lb=333000kg
Normal force:
N = g*m = 3.3*10^6N

Coefficients of rolling resistance (from aerodynamics textbook):
Crr1 = 0.0025
Crr2 = 0.00015 s/m
Crr = Crr1 + Crr2 * V
The force of rolling resistance is the coefficient of rolling resistance times the normal force.
F = Crr * N

Engine thrust:
F = 4*50000lbf=890000 N

Fnet = Frr - Fthrust = 3.3*10^6N*(0.0025+0.00015s/m*V)-890000N

For the plane to not move, the net force must be 0. Solving the equation, this occurs at v=1800m/s=4000mph

(I originally got 165 mph)

With the treadmill rolling at 165mph, the rolling resistance on the plane cancels the thrust of the plane's engines. The plane thus isn't accelerating. Assuming the treadmill can accelerate as quickly as the engines can ramp up from unpowered to full thrust, the plane has no airspeed, so it won't take off.

The Boeing's takeoff speed is 180mph, though, so how is this possible? With a real Boeing, when it is traveling at 165mph, it is generating considerable lift, which reduces the normal force, which reduces the force of rolling resistance by a large amount.

As a check of my numbers, I looked at the ad of a VW Touareg V10 towing a 747. The Touareg has a max torque of 855Nm; with a .5 m wheel radius, that's a maximum of 1700 N. The rolling resistance must be smaller than that, so the value of Crr1 must be lower than about 0.0002, and Crr2 is probably also smaller. Using the new value of Crr1 and keeping the same Crr2, I get a speed of v=200 mph. Even if we make Crr2 10 times smaller, that gives an equilibrium velocity of 2000mph, which, while unreasonable, isn't impossible.

(With corrected numbers, Crr1 doesn't matter, only Crr2, and I haven't found any reliable values for this.)
Last edited by nagromo on Sun Feb 25, 2007 7:58 am UTC, edited 1 time in total.
nagromo

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nagromo:

Uh, just so you know a 747 (747-400ER, to be precise) has a maximum take-off weight of just under 415 000 kg. And each of its four CF-6 engines provides a little over 270 000 N thrust to give a total of 1 080 000 N available thrust.

So your thrust's 10% of actual available thrust and the mass is 80% too high so your maximum speed is too low (don't the masses cancel? Which would put max speed at 1660 mph. Well above both take-off speed and VNE)

That said, though, it's good to have a quantised measurement of how rolling resistance can retard an aircraft's motion. Thanks (there should be a thumbs-up smilie).

Birdman

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Oops, yeah. My bad on the numbers. I'm tired and I forgot to use unit conversions. I was using numbers for the 747-100, weighing 3.3 MN with 890 kN thrust, and I got 1800 m/s (4000 mph).

Either way, Crr2 is the main unknown. My value is about right for the ultra-low rolling resistance wheels in a solar car, but it's probably higher for a 747, giving a lower equilibrium speed.

The main point is that there is an equilibrium speed, and it's much, much lower than the speed of light, and not too many times higher than the speed of sound.
nagromo

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nagromo wrote:The main point is that there is an equilibrium speed, and it's much, much lower than the speed of light, and not too many times higher than the speed of sound.

Most speeds are

And just to conclude for the fans following at home, the plane will take off well before it reaches the equilibrium speed, as the equilibrium speed is faster than most passenger jets can go. The aeroplane takes off!

Physics was the winner on the day.

SpitValve
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Wow, I spend a day or two playing wesnoth and this is what I come back to. I'm with genewitch, rat, the "BY DEFINITION" bit of birdman's post and nagromo's actually worked out real numbers. Hooray for real numbers.

I can understand what birdman is saying when he says that the problem is reduced to one of definition and that is therefore a little weird, but isn't that the way that logic puzzles are? This is analogous to saying that all theorems in mathematics are tautologies, and therefore incredibly dull. It misses the point of how difficult it is to form interesting tautologies, or see the implications of definitions. The puzzle becomes interesting here because it doesn't matter that the thrust isn't being provided by the wheels and that the friction can oppose the thrust and create an equilibrium velocity. Both problems seem interesting to me.

I guess I'm with the retards on this, as someone will no doubt point out soon enough.

PS. Mathemagic, you don't say that the wheels are frictionless in the OP. If they are frictionless then the only way the wheels and treadmill can match speeds in the speedometer sense outlined by birdman is if nothing is moving, which would be a strange puzzle indeed.

jestingrabbit

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This argument all comes down to how individuals parse the question to decide which speed they believe the treadmill is moving at.

Some people assume that the treadmill must always move fast enough that the friction in the wheels will negate the thrust. At that kind of speed the wheels melt off and it becomes a question of whether a plane can take off while lying on un-geared belly. Also, with speeds this high we're going to get some significant wind currents generated above the treadmill, which may give the plane a big enough airspeed to get in the air. We're talking stupidly high speeds here; speeds that are so stupidly high that this option becomes a stupid option to take.

The other options are all that the treadmill will move with a velocity equal in magnitude and opposite in direction to some point on the wheel. Because the treadmill is confined to horizontal movement, there are only three points on the wheel that we can really consider.

The first is the point at the exact top of the wheel, which will be moving forward at twice the speed of the plane relative to the surface of the treadmill. This would mean that the treadmill has to move backwards at, erm, twice the speed of the plane relative to the treadmill. This requires the treadmill to be moving faster than its own speed and you get a feedback loop that immediately flares out to infinity. This is one of those options that we can declare to be stupid.

The second available point is the point at the exact bottom of the wheel, which doesn't move at all relative to the treadmill. It has a speed of zero, so the treadmill's speed will be zero and the plane takes off on a normal stationary runway. I like this point, but I don't think it matches with the spirit of the question. This brings us to...

The third part of the wheel!

The third available point is the centre of the wheel. The velocity of the centre of the wheel is also the average velocity of all other points on the wheel. If the centre of the wheel (and by extension the entire plane) is moving forward at 10m/s, the treadmill will be moving at 10m/s backwards. This means the bottom point of the wheel will be moving backwards at 10m/s and the top will be moving forwards at 30m/s. The plane will have an airspeed of 10m/s and a speed relative to the surface of the treadmill of 20m/s with the wheels spinning twice as fast as they would normally need to at that airspeed. The friction of the faster-spinning wheels would slightly increase the thrust necessary to achieve take-off speed, but aeroplanes tend to be built pretty well and I don't think they'd have too much trouble.

The last two options are the only reasonable (or even POSSIBLE) interpretations of the question and both of them allow for the plane to accelerate to a speed where its wings will provide enough lift for it to take off.

If any of you continue to claim that the plane can't move "because the treadmill is matching speed with the wheels", I will send an army of demonic robot squid-raptor hybrids to destroy you and everyone you've ever cared about.
Do unto others as you would have them do unto you. Unless you're a masochist. Most people just don't find that stuff cool, okay?
Teaspoon

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SpitValve wrote:
nagromo wrote:The main point is that there is an equilibrium speed, and it's much, much lower than the speed of light, and not too many times higher than the speed of sound.

Most speeds are

And just to conclude for the fans following at home, the plane will take off well before it reaches the equilibrium speed, as the equilibrium speed is faster than most passenger jets can go. The aeroplane takes off!

Physics was the winner on the day.

No, the plane doesn't take off.

The equilibrium speed is the speed at which the forward thrust matches the backward rolling resistance of the wheels. The plane has 4000mph ground speed and 0mph airspeed, and there's 0 net force on the plane (and also no lift). The plane is in equilibrium: it will never take off on a treadmill going 4000 mph.

Teaspoon wrote:This argument all comes down to how individuals parse the question to decide which speed they believe the treadmill is moving at.

Some people assume that the treadmill must always move fast enough that the friction in the wheels will negate the thrust. At that kind of speed the wheels melt off and it becomes a question of whether a plane can take off while lying on un-geared belly. Also, with speeds this high we're going to get some significant wind currents generated above the treadmill, which may give the plane a big enough airspeed to get in the air. We're talking stupidly high speeds here; speeds that are so stupidly high that this option becomes a stupid option to take.

Not too stupidly high. 4000mph (mach 5) is an approximate upper cap. If a plane has 10 times more rolling resistance than a solar car (which is optimized for extreme efficiency, and cutting each pound of drag matters), then the equilibrium speed is about 400mph, just over half the speed of sound. While that's still very fast and we won't actually build such a treadmill, a plane's wheels might be able to survive such speeds.

With your second and third options, it seems very clear that the plane takes off. The first one is the only one that produces an interesting discussion.

If any of you continue to claim that the plane can't move "because the treadmill is matching speed with the wheels", I will send an army of demonic robot squid-raptor hybrids to destroy you and everyone you've ever cared about.

I claim that the plane can't move because the rolling resistance of the wheels is a force that is equal and opposite of the force of thrust, producing 0 net force and 0 acceleration. If the engines ramp up from 0 output to full 890 kN thrust over time and the treadmill increases speed to match, the plane will never overcome this rolling resistance, so it will never start moving.

Of course, a set of wheels with real, physical bearings probably couldn't survive providing 890 kN of resistance, so in real live the plane would probably crash and burn, but either way it won't take off.
nagromo

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Nagromo, I also think your definitions of groundspeed and airspeed are slightly skewed...unless you're saying that the aircraft is facing a 4000mph headwind. Indicated groundspeed, perhaps you mean.

Hawknc
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Yes.

If it's a Harrier.
Class: 12th level Epiphenomenalist Alignment: Rational

Aoeniac

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Some people assume that the treadmill must always move fast enough that the friction in the wheels will negate the thrust. At that kind of speed the wheels melt off and it becomes a question of whether a plane can take off while lying on un-geared belly. Also, with speeds this high we're going to get some significant wind currents generated above the treadmill, which may give the plane a big enough airspeed to get in the air. We're talking stupidly high speeds here; speeds that are so stupidly high that this option becomes a stupid option to take.

The other options are all that the treadmill will move with a velocity equal in magnitude and opposite in direction to some point on the wheel. Because the treadmill is confined to horizontal movement, there are only three points on the wheel that we can really consider.

The first is the point at the exact top of the wheel, which will be moving forward at twice the speed of the plane relative to the surface of the treadmill. This would mean that the treadmill has to move backwards at, erm, twice the speed of the plane relative to the treadmill. This requires the treadmill to be moving faster than its own speed and you get a feedback loop that immediately flares out to infinity. This is one of those options that we can declare to be stupid.

if this option is stupid then so is the question in the OP because it is the answer... impossible and incomprehensible and stupid..

maybe the wheels/treadmill do have a maximum speeds though, i mean maybe it doesnt have to be infinitely fast... im leaving the possibility open
Last edited by Rat on Sun Feb 25, 2007 12:01 pm UTC, edited 2 times in total.

Rat
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Yeah, it's fairly common sense that "the speed of the wheel" can be taken as the linear speed of the surface of the wheel, i.e. what the wheel would register if it were attached to a speedometer. So you can take it that way, which is causing most of the stress here, or you can take it as written poorly and actually meaning the speed of the aircraft, which is another common way of phrasing the question.

Hawknc
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since we realize now that it is the various wordings of the question that screwed people i hope everyone can agree with my picture here...

the answer to the question in the first post of this thread is that the plane does not take off because it states that the treadmill matches the speed of the WHEELS

Rat
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Rat wrote:the answer to the question in the first post of this thread is that the plane does not take off because it states that the treadmill matches the speed of the WHEELS

Yeah, the speed that hawknc and teaspoon are saying the question refers to would be referred to, unambiguously, as the speed of plane (relative to the ground or air, they are the same in a windless situation).

jestingrabbit

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Rat, what it seems you are not getting is how the wheels on an airplane works, which my example was trying to clarify. You could have the treadmill going at ANY SPEED (within limits or ratoinality, im sure if they are going at light speed some crazy shit will occr ) AND THE PLANE WILL NOT BE EFFECTED SEVERELY

Why is this?
BECAUSE THE WHEELS ON AN AIRPLANE DO NOT WORK THE WAY THEY DO ON A CAR, THEY ARE NOT DIRECTLY LINKED TO THE AIRPLANE AND OR ITS PROPULSION SYSTEM!
THEY ARE DESIGNED TO BE A LOW FRICTION SURFACE TO THE GROUND!

Again, have the airplane standing still, start moving the treadmill at any desired speed, any whichsoever (within reason) lets say TWICE the speed the airplane will have when it takes off. The airplane will once again slowly start moving backwards?
HOwever, when the jet engines start pushing A DIRECT FORCE MUCH GREATER THAN THE FRICTOINAL FORCE BETWEEN THE WHEELS AND THE BALL BEARING will start pushing the plane FORWARD!!
You can really change the wheels for a hovering mechanism, NOTHING CHANGES DRASTICALLY!
If there was a greatly effecient way to build such a hovering mechanism, you could have that on airplanes, they would still be able to takeoff the problem would to some extent be landing, since you could not break, but TAKE OFF WOULD NOT BE AFFECTED
imagine that one?
your average airplane on a NORMAL airstrip, with a hover mechanism, basially htere are no wheels but the airplane is hovering a few feet above the gronud. The engines start pushing and it will get into the air (as in, above those few feet ) prety much just as quickly as a normal plane with wheels. This is because THE WHEELS TO NOT PROPOLL THE PLANE!
Now place this hover plane, which is in functionality pretty much identical to a normal plane, place it on this treadmill, AND NOMATTER AT WHAT SPEED THE TREADMILL IS AT, IT WILL TAKE OFF!

I hope this makes as much sence as it should. The speed of the treadmill DOESN'T MATTER! It would also be TWICE that of the airplane, because it DOES NOT PUSH THE AIRPLANE BACKWARDS! WHILE THE JET ENGINES PUSH THE AIRPLANE VERY EFECTIVELY FORWARD!
//Yepp, THE fjafjan (who's THE fjafjan?)
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fjafjan, read the post by nagromo earlier on this page. It has a calculation that the treadmill speed required for a 747 to be incapable of takeoff is at most 4000 mph, and probably significantly less. Nowhere near the speed of light.

The fact friction increases with speed allows the possibility that the treadmill can go fast enough such that the thrust is equally opposed by the friction of the wheels. That the wheels don't do the thrusting is irrelevant. The forces cancel and there is no net acceleration.

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nagromo:

By the time the (horizontal) forces are in equilibrium (ie net force = 0) the aircraft is moving very quickly indeed. So the plane can certainly, by YOUR maths, take off.

It's like a skydiver's terminal velocity. They're in equilibrium; their weight is equal to their drag and they aren't accelerating (Fy = 0; ay = 0) but they're definitely moving (vy ~= -60ms^-1). By the time the aeroplane reaches the point where rolling resistance of the wheels causes it to stop accelerating it will have met and exceeded its takeoff velocity (so it could take off) and its maximum speed anyway (so it's going faster than any 747 in history).

Aoeniac:
Oh hush.

Rat:
Please do us a favour and define "speed of the wheels".

Do you mean:
a) The measured speed according to a speedometer attached to the wheels; or
b) The rate at which the entire wheel goes from one place to another place. That is, if the speed is not zero and you go away - the wheel will not be where you left it when you come back.

If you mean a above, then I totally agree with you. The plane can never take off, because it can never move (because the problem is defined as a case where the aircraft is forever stationary - not because of any physics or other mechanistic means).

If you mean b, then you're totally wrong. There is nothing stopping the wheel from moving (neither by definition, nor physically). It will simply move rotating at twice the rate expected for that speed.

Hawknc:
4000mph groundspeed and 0mph airspeed would show a 4000mph tailwind, but yeah

jestingrabbit:
I'd say you see the problem pretty well (mostly because you say you agree with that bit of my post ). To me, if you take the problem as one where it's the speed of the plane in question (not wheels) and it's measured relative to the stationary earth then the problem's no longer one of straight definition and more interesting than one which just gets untangled to "A plane can't move. Can it move?". That would be an interesting linguistic question, perhaps - but not so satisfying as a logic puzzle.

Actually I think this puzzle is just a really fun way to start light-hearted (mostly) arguments on forums and at work. Stays away from the taboo subjects (religion, politics etc) and especially good if you work at an airport for putting the know-alls in their place.

As for the frictionless wheels, it's another definition problem. Did mathemagic mean "no part of the wheel feels the effect of friction, including the tread and hence the motion or otherwise of the belt/treadmill is of no consequence", or "the wheel bearing is frictionless", or even "the wheel bearing is frictionless and the wheel has zero rotational inertia"? I certainly don't know.

edited to add: The whole "a treadmill going 4000mph could stop a 747 accelerating" thing is fine, but for the treadmill to be moving, the 747 has to be moving. What nagromo's calculations show is that a 747 could reach a speed of 2000mph (with the treadmill going 2000mph the other way that makes a relative motion of 4000mph) before the treadmill could stop it from accelerating. Or, another way of saying it is 'a treadmill CANNOT stop a 747 going at any speed below 2000mph - including 190mph or so required for takeoff'. Alternatively 'A 747 can continue to accelerate to 2000mph on the treadmill described before it reaches its limiting speed'. Which means: the treadmill can't stop the plane taking off (only a poorly defined problem can).

ALL:

I would appreciate it if everyone henceforth would state how they interpret the phrase
speed of the wheels

My guess is that it'd sort a whole lot of this out straight away. Many people are assuming it means one thing and that others take it to mean the same thing, all with different meanings.

edited to add:
fjafjan:
By my reading of the question, you are spot on.

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Duh, yes, I did indeed mean tailwind. It's okay, I'm only going to be designing aircraft in a year, it doesn't matter if I make mistakes like that...

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Hawknc:

Hey, half your range estimates will be waaay too optimistic, and half will be waaay to pessimistic. That means it'll average out and be OK, right?

Also I can't wait to see you get the crosswind and headwind components mixed up in the documentation: "in crosswinds of 40kts - 60kts, land straight with flaps up and power at idle".

Finally (hooray multi-edits) and speaking of 4000mph groundspeeds, I went down to an AIAA conference in Canberra late last year on hypersonic aircraft and scramjets. It was pretty neat. UQ has a great scramjet programme going, and have built a scramjet with the most awesome 3D inlet ever. If you've got much time left in your degree it'd be worth joining, even if just for the cheap textbooks. Even after postage from the US I was always \$40 - \$100 better off than getting them at the UNSW bookshop. The monthly magazine is pretty good too. [/thread hijack]

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Been meaning to join for years, it's my last year now but I might join anyway. One of my friends gave a presentation to the Melbourne AIAA conference last year on his final year project - didn't win the prize sadly, but even getting accepted is a hard ask.

Okay, thread goes back on topic...now.

Hawknc
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Birdman wrote:ALL:

I would appreciate it if everyone henceforth would state how they interpret the phrase
speed of the wheels

My guess is that it'd sort a whole lot of this out straight away. Many people are assuming it means one thing and that others take it to mean the same thing, all with different meanings.

My initial reading of the question is that "the speed of the wheels" refers to the speedometer value that you'd get if you strapped a speedo onto it. I maintain that that is the correct reading of the phrase too.

If you mean "the rate at which the axle of the wheel gets from one place to another" you could as easily refer to the speed of the nose of the airplane, or just the ground speed (or perhaps earth speed, I mean the speed relative to the earth, not the speed relative to the treadmill) of the airplane.

The only reason that you'd specify that your talking about the speed of the wheels is if you are getting a speed from the wheels that is different from a speed you could get elsewhere. The speedometer speed is the only speed like this that you can get from the wheels.

I agree that this is where the argument is. I think the trap that the puzzle has in store when you interpret it my way is the fact that although its not the wheels providing the thrust, they can still stop you. I don't think that's a linguistic trick, I think its an interesting physics factoid.

ps I would expect that at the conveyor speed required you'd have a turbulent flow near the belt, not the sort of thing that you could use to bootstrap into causing lift.

jestingrabbit

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The turbulent boundary layer coming off a moving treadmill would be too low to reach the wings of any airliner anyway.

Hawknc
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Ok, I read the first two pages or so of the thread, and I'm not necessarily disagreeing with it. I think that there's something that I'm missing, however, and I'd like to have it clarified.

A) The wheels are moving at 100 m.p.h.
B) The treadmill is moving at 100 m.p.h.

By reasoning, we can infer that the plane is sitting still (this is WITHOUT the engine on at all. Right now we're just focusing on the wheels).

So let's say we turn the engine on, and it uses thrust to generate lift. To generate lift, the plane must move forward in space (and again, correct me if I'm wrong) to encounter the wind to lift the plane.

I accept that the plane would lift. However, wouldn't the plane moving forward increase the speed of the wheels by ancillary effect? If so, the wheels would be going faster than the treadmill.

In summary: I understand that the plane would take off, but not how the wheels would not increase in speed. Thanks in advance for the explanation.
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First, when I said 4000mph groundspeed, I meant that a speedometer attached to the wheels would measure 4000mph. The plane actually has 0mph airspeed, 0mph groundspeed, and 4000mph treadmillspeed.

ALL:

I would appreciate it if everyone henceforth would state how they interpret the phrase
speed of the wheels

My guess is that it'd sort a whole lot of this out straight away. Many people are assuming it means one thing and that others take it to mean the same thing, all with different meanings.

I interpret this as the speed as measured by a spedometer attached to the wheels. All other interpretations make it clear that the plane can take off; this is the only interesting interpretation.

Birdman:
For the treadmill to be moving, the wheels have to be moving relative to the treadmill; the plane doesn't have to move relative to the ground.

My math shows that when the engines are putting out full thrust, the treadmill can cancel out forces by turning at 4000 mph.

The jet turbines can't turn on instantly; they take some time to ramp up. Because of the statement of the problem, the treadmill will attempt to match the speed of the wheels.

Say that the turbines go from powered down to full thrust linearly over the course of 30 seconds. Over that time, the thrust is modeled by this equation:

Ft=29667*t

with 0 thrust at t=0, 890 kN thrust at t=30, and linear interpolation between.

We can now solve my previous equation for this thrust to get the treadmill velocity as a function of t.

29667*t=3300000*(0.0025+0.00015*v)

Solving this, we get v=59.93*t-16.67 (in m/s).

This means that until the entines put out 8250N of thrust, the constant factor of rolling resistance is enough to stop the plane; the treadmill doesn't even have to move. After that, if the treadmill accelerates at 59.93 m/s^2 (about 6 g's) for 30 seconds while the engine ramps up to full thrust, the net force on the plane will be 0 the entire time, so the plane will never start moving.

This is the only way for the problem's formulation to be satisfied; if the plane accelerates relative to the ground, then the treadmill isn't turning at the speed of the speedometer attached to the wheels. This way is possible (theoretically), and it satisfies both physics and the question's formulation.
nagromo

Posts: 10
Joined: Sun Feb 25, 2007 6:24 am UTC

fjafjan.. ive said over and over again that i GET how a plane works... get your head out of your ass and just THINK about what is going on here...

it depends on what the treadmill is matching.. if it is matching the speed at which the airplane goes from point a to point b then sure it can take off...

if it matches the speed at which the wheels rotate then the plane will not take off.. it doesnt matter what causes the wheel to rotate.. if the treadmill is matching the speed at which the wheel rotates but in the opposite direction then the plane will not move.. the question states that the treadmill matches the speed of the WHEEL... so it is all of YOU who are not getting it...

OBVIOUSLY if the plane is moving forward then the treadmill is not matching the speed of the wheels rotation, the wheels are obviously rotating faster than the treadmill...

mathmagic wrote:designed to MATCH the speed of the wheels of an airplane

it is more reasonable to believe that he meant the speed at which the wheels rotate rather than the speed of the plane moving from point A to point B...

before you post and try to argue with me think about what i am saying for a second, realize there was confusion, and realize that the answer is: no. the plane will not take off.

Rat
Rattus Trolleri

Posts: 929
Joined: Tue Nov 07, 2006 4:40 pm UTC

Rat wrote:fjafjan.. ive said over and over again that i GET how a plane works... get your head out of your ass and just THINK about what is going on here...

it depends on what the treadmill is matching.. if it is matching the speed at which the airplane goes from point a to point b then sure it can take off...

if it matches the speed at which the wheels rotate then the plane will not take off.. it doesnt matter what causes the wheel to rotate.. if the treadmill is matching the speed at which the wheel rotates but in the opposite direction then the plane will not move.. the question states that the treadmill matches the speed of the WHEEL... so it is all of YOU who are not getting it...

OBVIOUSLY if the plane is moving forward then the treadmill is not matching the speed of the wheels rotation, the wheels are obviously rotating faster than the treadmill...

mathmagic wrote:designed to MATCH the speed of the wheels of an airplane

it is more reasonable to believe that he meant the speed at which the wheels rotate rather than the speed of the plane moving from point A to point B...

before you post and try to argue with me think about what i am saying for a second, realize there was confusion, and realize that the answer is: no. the plane will not take off.
I agree with you in that if the plane is moving forward, the treadmill cannot match the speed of the wheels. However, due to the way the plane works, the plane must take off. Therefore, I am beginning to think that it is more likely that they meant the initial speed of the wheel in relation to the treadmill, for this is the only way this is possible.
Zach

Posts: 81
Joined: Thu Feb 15, 2007 1:18 am UTC
Location: Lexington, Kentucky.

Rat, you're continuing to ignore our request that you explain how you interpret "the speed of the wheel".

The fact is, your interpretation of "matching speed" seems invalid because it describes a physically impossible situation.

If the treadmill surface moves at the same speed as the linear velocity of the uppermost point of the wheel:

We start with the plane stationary, so wheel speed and treadmill speed are zero.

The engines begin to thrust the plane forward. An instant passes and it begins to roll at speed x (which may be incredibly tiny, but is an actual speed). The top of the wheel is now moving at 2x and the bottom is stationary.

The conveyor belt sets its speed to that of the top of the wheel (2x). This makes the bottom of the wheel move backwards at the speed of the treadmill (2x). The plane is moving at x, so the top of the wheel has to be moving forward at 4x to allow that speed of movement, which means the treadmill is moving at 4x, which means the bottom of the wheel is moving at 4x, which means the top of the wheel is moving at 6x, which means...

Yeah, infinite speeds coming right up.

Or do you mean that the treadmill moves at the speed that a speedometer measuring the rotation of the wheel would calculate? I think that gets us into another infinite-growth loop anyway.

Lemme see, an instant of thrust to get us out of the all-zeroes situation... Plane is now moving forward at x, speedometer calculates wheel to be moving at x, treadmill moves backwards at x, plane is still moving forward at x so speedometer now calculates 2x, which means the treadmill goes at 2x so the speedometer calculates 3x.

Yeah, there it is.

Both of those interpretations are broken because they immediately loop to physical impossibilities as soon as the plane gets nudged, and there's no movement of the wheels or conveyor belt if the plane doesn't get any speed up in the first place.
Do unto others as you would have them do unto you. Unless you're a masochist. Most people just don't find that stuff cool, okay?
Teaspoon

Posts: 348
Joined: Tue Sep 12, 2006 11:37 pm UTC
Location: Where you least expect me

Zach wrote:
Rat wrote:fjafjan.. ive said over and over again that i GET how a plane works... get your head out of your ass and just THINK about what is going on here...

it depends on what the treadmill is matching.. if it is matching the speed at which the airplane goes from point a to point b then sure it can take off...

if it matches the speed at which the wheels rotate then the plane will not take off.. it doesnt matter what causes the wheel to rotate.. if the treadmill is matching the speed at which the wheel rotates but in the opposite direction then the plane will not move.. the question states that the treadmill matches the speed of the WHEEL... so it is all of YOU who are not getting it...

OBVIOUSLY if the plane is moving forward then the treadmill is not matching the speed of the wheels rotation, the wheels are obviously rotating faster than the treadmill...

mathmagic wrote:designed to MATCH the speed of the wheels of an airplane

it is more reasonable to believe that he meant the speed at which the wheels rotate rather than the speed of the plane moving from point A to point B...

before you post and try to argue with me think about what i am saying for a second, realize there was confusion, and realize that the answer is: no. the plane will not take off.
I agree with you in that if the plane is moving forward, the treadmill cannot match the speed of the wheels. However, due to the way the plane works, the plane must take off. Therefore, I am beginning to think that it is more likely that they meant the initial speed of the wheel in relation to the treadmill, for this is the only way this is possible.

what? just because a plane CAN fly doesnt mean it can just float away.. if it's not moving it cant take off.. and if the conveyor is matching the speed of the wheels it does not move...

the initial speed of the wheel would be nothing? so then the treadmill isnt moving at all? so its a plane taking off on a runway? i guess that's one of the ways possible.....

edit: teaspoon: i am now ignoring you because you continue to ignore my explanations of what i mean by "speed of the wheel"

Rat
Rattus Trolleri

Posts: 929
Joined: Tue Nov 07, 2006 4:40 pm UTC

I had on-topic things to say and then I saw that: http://www.unitedmedia.com/comics/dilbert/archive/images/dilbert2007026109124.gif

Oh well, sorry. Carry on.

tendays

Posts: 945
Joined: Sat Feb 17, 2007 6:21 pm UTC
Location: INDIA

Okay, this is your homework.

1. Go to Toys-R-Us and get a toy plane. Notice how the wheels spin freely.

2. Go to the gym and turn on a tread mill at any speed.

3. Place the toy airplane on the treadmill and move it forward with your finger. (Your finger represents the thrust of the airplane)

4. Now, is there any speed you can turn the treadmill to that would prevent you from moving the airplane forward? No.

Rat wrote:
Zach wrote:
Rat wrote:fjafjan.. ive said over and over again that i GET how a plane works... get your head out of your ass and just THINK about what is going on here...

it depends on what the treadmill is matching.. if it is matching the speed at which the airplane goes from point a to point b then sure it can take off...

if it matches the speed at which the wheels rotate then the plane will not take off.. it doesnt matter what causes the wheel to rotate.. if the treadmill is matching the speed at which the wheel rotates but in the opposite direction then the plane will not move.. the question states that the treadmill matches the speed of the WHEEL... so it is all of YOU who are not getting it...

OBVIOUSLY if the plane is moving forward then the treadmill is not matching the speed of the wheels rotation, the wheels are obviously rotating faster than the treadmill...

mathmagic wrote:designed to MATCH the speed of the wheels of an airplane

it is more reasonable to believe that he meant the speed at which the wheels rotate rather than the speed of the plane moving from point A to point B...

before you post and try to argue with me think about what i am saying for a second, realize there was confusion, and realize that the answer is: no. the plane will not take off.
I agree with you in that if the plane is moving forward, the treadmill cannot match the speed of the wheels. However, due to the way the plane works, the plane must take off. Therefore, I am beginning to think that it is more likely that they meant the initial speed of the wheel in relation to the treadmill, for this is the only way this is possible.

what? just because a plane CAN fly doesnt mean it can just float away.. if it's not moving it cant take off.. and if the conveyor is matching the speed of the wheels it does not move...

the initial speed of the wheel would be nothing? so then the treadmill isnt moving at all? so its a plane taking off on a runway? i guess that's one of the ways possible.....

edit: teaspoon: i am now ignoring you because you continue to ignore my explanations of what i mean by "speed of the wheel"
26.2 (If you have to ask why, you'll never understand)

rpoulin79

Posts: 84
Joined: Tue Feb 13, 2007 1:41 am UTC
Location: Michigan

Okay, this is your homework.

1. Go to Toys-R-Us and get a toy plane. Notice how the wheels spin freely.

2. Go to the gym and turn on a tread mill at any speed.

3. Place the toy airplane on the treadmill and move it forward with your finger. (Your finger represents the thrust of the airplane)

4. Now, is there any speed you can turn the treadmill to that would prevent you from moving the airplane forward? No.

Rat wrote:
Zach wrote:
Rat wrote:fjafjan.. ive said over and over again that i GET how a plane works... get your head out of your ass and just THINK about what is going on here...

it depends on what the treadmill is matching.. if it is matching the speed at which the airplane goes from point a to point b then sure it can take off...

if it matches the speed at which the wheels rotate then the plane will not take off.. it doesnt matter what causes the wheel to rotate.. if the treadmill is matching the speed at which the wheel rotates but in the opposite direction then the plane will not move.. the question states that the treadmill matches the speed of the WHEEL... so it is all of YOU who are not getting it...

OBVIOUSLY if the plane is moving forward then the treadmill is not matching the speed of the wheels rotation, the wheels are obviously rotating faster than the treadmill...

mathmagic wrote:designed to MATCH the speed of the wheels of an airplane

it is more reasonable to believe that he meant the speed at which the wheels rotate rather than the speed of the plane moving from point A to point B...

before you post and try to argue with me think about what i am saying for a second, realize there was confusion, and realize that the answer is: no. the plane will not take off.
I agree with you in that if the plane is moving forward, the treadmill cannot match the speed of the wheels. However, due to the way the plane works, the plane must take off. Therefore, I am beginning to think that it is more likely that they meant the initial speed of the wheel in relation to the treadmill, for this is the only way this is possible.

what? just because a plane CAN fly doesnt mean it can just float away.. if it's not moving it cant take off.. and if the conveyor is matching the speed of the wheels it does not move...

the initial speed of the wheel would be nothing? so then the treadmill isnt moving at all? so its a plane taking off on a runway? i guess that's one of the ways possible.....

edit: teaspoon: i am now ignoring you because you continue to ignore my explanations of what i mean by "speed of the wheel"
26.2 (If you have to ask why, you'll never understand)

rpoulin79

Posts: 84
Joined: Tue Feb 13, 2007 1:41 am UTC
Location: Michigan

Okay, this is your homework.

1. Go to Toys-R-Us and get a toy plane. Notice how the wheels spin freely.

2. Go to the gym and turn on a tread mill at any speed.

3. Place the toy airplane on the treadmill and move it forward with your finger. (Your finger represents the thrust of the airplane)

4. Now, is there any speed you can turn the treadmill to that would prevent you from moving the airplane forward? No.

Rat wrote:
Zach wrote:
Rat wrote:fjafjan.. ive said over and over again that i GET how a plane works... get your head out of your ass and just THINK about what is going on here...

it depends on what the treadmill is matching.. if it is matching the speed at which the airplane goes from point a to point b then sure it can take off...

if it matches the speed at which the wheels rotate then the plane will not take off.. it doesnt matter what causes the wheel to rotate.. if the treadmill is matching the speed at which the wheel rotates but in the opposite direction then the plane will not move.. the question states that the treadmill matches the speed of the WHEEL... so it is all of YOU who are not getting it...

OBVIOUSLY if the plane is moving forward then the treadmill is not matching the speed of the wheels rotation, the wheels are obviously rotating faster than the treadmill...

mathmagic wrote:designed to MATCH the speed of the wheels of an airplane

it is more reasonable to believe that he meant the speed at which the wheels rotate rather than the speed of the plane moving from point A to point B...

before you post and try to argue with me think about what i am saying for a second, realize there was confusion, and realize that the answer is: no. the plane will not take off.
I agree with you in that if the plane is moving forward, the treadmill cannot match the speed of the wheels. However, due to the way the plane works, the plane must take off. Therefore, I am beginning to think that it is more likely that they meant the initial speed of the wheel in relation to the treadmill, for this is the only way this is possible.

what? just because a plane CAN fly doesnt mean it can just float away.. if it's not moving it cant take off.. and if the conveyor is matching the speed of the wheels it does not move...

the initial speed of the wheel would be nothing? so then the treadmill isnt moving at all? so its a plane taking off on a runway? i guess that's one of the ways possible.....

edit: teaspoon: i am now ignoring you because you continue to ignore my explanations of what i mean by "speed of the wheel"
26.2 (If you have to ask why, you'll never understand)

rpoulin79

Posts: 84
Joined: Tue Feb 13, 2007 1:41 am UTC
Location: Michigan

Nice triple-post rpoulin

Teaspoon:

According to his interpretation of "speed of the wheel", Rat is perfectly correct. The plane is defined in the problem as one which never moves. There is no mechanical interaction which stops it, it's just defined as a "forever motionless aircraft". Therefore, it can't take off.

Also, Rat acknowledges that if the speed in question is the rate at which the plane moves, then the plane can take off.

The argument you two should be having is NOT "can the plane take off" but "precisely how badly was the question phased initially, and how should it be interpreted?".

Birdman

Posts: 208
Joined: Mon Nov 27, 2006 4:28 am UTC
Location: Sydney, Australia

Rat wrote:
Zach wrote:
Rat wrote:fjafjan.. ive said over and over again that i GET how a plane works... get your head out of your ass and just THINK about what is going on here...

it depends on what the treadmill is matching.. if it is matching the speed at which the airplane goes from point a to point b then sure it can take off...

if it matches the speed at which the wheels rotate then the plane will not take off.. it doesnt matter what causes the wheel to rotate.. if the treadmill is matching the speed at which the wheel rotates but in the opposite direction then the plane will not move.. the question states that the treadmill matches the speed of the WHEEL... so it is all of YOU who are not getting it...

OBVIOUSLY if the plane is moving forward then the treadmill is not matching the speed of the wheels rotation, the wheels are obviously rotating faster than the treadmill...

mathmagic wrote:designed to MATCH the speed of the wheels of an airplane

it is more reasonable to believe that he meant the speed at which the wheels rotate rather than the speed of the plane moving from point A to point B...

before you post and try to argue with me think about what i am saying for a second, realize there was confusion, and realize that the answer is: no. the plane will not take off.
I agree with you in that if the plane is moving forward, the treadmill cannot match the speed of the wheels. However, due to the way the plane works, the plane must take off. Therefore, I am beginning to think that it is more likely that they meant the initial speed of the wheel in relation to the treadmill, for this is the only way this is possible.

what? just because a plane CAN fly doesnt mean it can just float away.. if it's not moving it cant take off.. and if the conveyor is matching the speed of the wheels it does not move...

the initial speed of the wheel would be nothing? so then the treadmill isnt moving at all? so its a plane taking off on a runway? i guess that's one of the ways possible.....

edit: teaspoon: i am now ignoring you because you continue to ignore my explanations of what i mean by "speed of the wheel"

I think that you're sort of looking at it right, but not quite.

This is a response to you and rpoulin.

Imagine a plane. The plane is on a treadmill, and the wheels are spinning at 100 m.p.h. while the treadmill runs backwards at 100 m.p.h. Therefore, the plane is stationary. Turn on the engines. The thrust pushes the plane forward. Since the plane is moving forward, the wheels must logically be moving faster than the treadmill.

Therefore, while it is true that initially the wheels and treadmill are going the same speed, and the plane will take off, this cannot remain true. The wheels will spin faster than the treadmill can move, and there's simply no way around it.

rpoulin:

Alright, you're pushing the plane forward with your finger. No, no matter how fast you turn up the treadmill, it cannot stop you from moving that plane. However, if the treadmill is going 10 m.p.h. and the wheels are going 10 m.p.h., the plane is not moving. That "thrust" (your finger) is causing the wheels to rotate slightly faster.

That's my logic, anyway.
Zach

Posts: 81
Joined: Thu Feb 15, 2007 1:18 am UTC
Location: Lexington, Kentucky.

Re: Airplane On A Treadmill

mathmagic wrote:Alright, here's a physics problem for you.

If there was a treadmill of infinite length, and the width of a runway, and it was designed to MATCH the speed of the wheels of an airplane, would the airplane be able to take off the ground?

Assume the wheels are fully contacted with the ground until the moment of takeoff (completely off the ground).

This was being discussed on another message board, but it will probably be easier for all YOU smart folk.

No where does is say the the plane cannot move.

And, (I realize that this has all been covered before) speed is scalar quantity describing motion in a linear direction, therefore the linear speed of the centroid of the wheel. Rat can reinterpret the question all he wants, it doesn't make the question what he wants it to be. I don't think the question is stated unclearly at all. Speed means what it means. Open up any Dynamics text book, turn the section covering Kinematics of Rigid Bodies and you will verify that. I can call the plane a pencil, but that doesn't make it one.

If the speed of the treadmill matches the speed of the wheels in the direction against that which the plane is moving, then the rotational velocity of the wheels is twice that of what they would normally be. If the speed of the treadmill matches the speed of the wheels in the direction the plane is moving, then the treadmill is moving at the same speed as the plane and the wheels don't turn. Either way, they plane is moving forward and will take off.
26.2 (If you have to ask why, you'll never understand)

rpoulin79

Posts: 84
Joined: Tue Feb 13, 2007 1:41 am UTC
Location: Michigan

Zach wrote:This is a response to you and rpoulin.

Imagine a plane. The plane is on a treadmill, and the wheels are spinning at 100 m.p.h. while the treadmill runs backwards at 100 m.p.h. Therefore, the plane is stationary. Turn on the engines. The thrust pushes the plane forward. Since the plane is moving forward, the wheels must logically be moving faster than the treadmill.

Therefore, while it is true that initially the wheels and treadmill are going the same speed, and the plane will take off, this cannot remain true. The wheels will spin faster than the treadmill can move, and there's simply no way around it.

rpoulin:

Alright, you're pushing the plane forward with your finger. No, no matter how fast you turn up the treadmill, it cannot stop you from moving that plane. However, if the treadmill is going 10 m.p.h. and the wheels are going 10 m.p.h., the plane is not moving. That "thrust" (your finger) is causing the wheels to rotate slightly faster.

That's my logic, anyway.

1. Wheels can't spin at 100 mph. Wheels can only spin at RPM, rad/s, etc... I'm not trying to be a know it all, I'm just stressing that your confusing rotation velocity and speed. NOT THE SAME THING.

2. The plane is moving at 10mph, isn't it? Put it on a treadmill going 180 mph or greater with no wheels and its going to take off isn't it?
26.2 (If you have to ask why, you'll never understand)

rpoulin79

Posts: 84
Joined: Tue Feb 13, 2007 1:41 am UTC
Location: Michigan

rpoulin79 wrote:
Zach wrote:This is a response to you and rpoulin.

Imagine a plane. The plane is on a treadmill, and the wheels are spinning at 100 m.p.h. while the treadmill runs backwards at 100 m.p.h. Therefore, the plane is stationary. Turn on the engines. The thrust pushes the plane forward. Since the plane is moving forward, the wheels must logically be moving faster than the treadmill.

Therefore, while it is true that initially the wheels and treadmill are going the same speed, and the plane will take off, this cannot remain true. The wheels will spin faster than the treadmill can move, and there's simply no way around it.

rpoulin:

Alright, you're pushing the plane forward with your finger. No, no matter how fast you turn up the treadmill, it cannot stop you from moving that plane. However, if the treadmill is going 10 m.p.h. and the wheels are going 10 m.p.h., the plane is not moving. That "thrust" (your finger) is causing the wheels to rotate slightly faster.

That's my logic, anyway.

1. Wheels can't spin at 100 mph. Wheels can only spin at RPM, rad/s, etc... I'm not trying to be a know it all, I'm just stressing that your confusing rotation velocity and speed. NOT THE SAME THING.

2. The plane is moving at 10mph, isn't it? Put it on a treadmill going 180 mph or greater with no wheels and its going to take off isn't it?

the word speed can go both ways.. in my opinion it makes more sense to read the question as "matches the speed at which the wheels rotate"... as in, the treadmill is spinning at the speed that the SPEEDometer attached to the wheels reads...

Rat
Rattus Trolleri

Posts: 929
Joined: Tue Nov 07, 2006 4:40 pm UTC

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