If you accept the version of the problem that, “the conveyor belt is (somehow) designed to exactly match the speed of the wheels,” the problem seems to be more a matter of logic than physics. The plane can’t “roll” forward faster than the conveyor belt if for no other reason than the problem doesn’t allow it. Therefore, the rules of the problem prevent the wheels from being used to reduce the friction of the plane against the conveyor until it takes off. The plane could however skid down the runway (until it takes off). This doesn’t violate the rules of the problem because the wheels aren’t rolling nor is the conveyor belt moving. To the extent the wheels do roll, the plane still has to complete its takeoff using a full skid (and the pilot would seem wise to apply the brakes in order to shut off the conveyor belt!). The question then is can the plane take off, in effect, with the wheels locked. The answer is, “I don’t know because the problem doesn’t specify anything about the plane (and it’s engines).” But I would guess that most jet airliners couldn’t.
Again, it depends on what interpretation you put on “speed of the wheels”.
If (as many have) you choose to interpret this as the speed of the wheel hub (which is of course the same as the speed of the airplane - fuselage, wings etc.) then the problem does allow the plane to roll forward. It could, for example, be headed north at 80 knots while the treadmill trundles south at 80 knots and the wheels spin as they would at 160 knots on normal pavement.
did I just wake up in yesterday again?
Don’t you think the horizontal force (treadmill->wheel) will be converted into rotational force? With no wheel bearing friction, that would be a pretty good conversion. I confess I don’t know how to calculate how much this would be.
If the conversion efficiency is 100%, the plane won’t move. If less, it would, at least initially and slightly.
ok, I just had a thought about this. Please excuse me if it has been brought up already.
I read the famous post with all the scenarios and understood it and found it excellent. I am just not sure how this idea pertains to those scenarios.
The problem states that the belt matches the speed of the wheels. The problem arises when we assume it to mean the speedometer speed of the wheel.
Now, in its most basic form, all this statements means is that there is no slippage between the rubber of the wheels and the tarmac.
The treadmill could be spinning at any speed and in any direction and the statement holds up and the plane just remains stationary. But then it is not the belt matching the speed of the wheels but the other way around, the wheels are matching the speed of the belt. Does this difference mean anything?
If the belt senses and matches the speed of the wheels, the only way the wheel can have any (rotational) speed (considering there is no engine transmission) is by moving forward. But we already established that cannot happen as the belt will not allow it. What is happening then to the thrust of the engine?
Please be patient with me and explain this to me without pointing out that it has been covered a thousand times. I really have no agenda here for either solution. Just trying to figure this out.
Sorry I should have been more precise. Having read the log, I thought your interpretation of the problem was solved long ago. I was OBVIOUSLY refering to “the outer diameter surface velocity” which I thought ZUT and the rest of you all agreed was “the more interesting problem.” I just thought all the analysis about hub wheel inertia and such, while brilliant, overlooked what the problem is really all about, which is to see whether you can discover what the real issue is (i.e. whether a jet has enough power to skid down the run way and take off with it’s wheels locked).
The “insteresting” part of this problem is not really about physics. Who really cares whether a passanger jet can or can’t so skid? Someone can just look that up in a book.
There’s no “conversion” of force in the sense that you’re thinking. There’s no requirement that a certain proportion of a force be allocated to translational motion and another proportion be allocated to rotational motion. A single force can cause both translational and rotational motion, depending on how it’s aligned with the center of mass. (As a side note, this is apparently a relatively difficult concept to grasp. At least one other regular poster has been militantly and snidely pro-ignorance on the subject, comparing the acceleration to a perpetual motion machine. Not everyone likes to pay attention, it seems.)
Anyway, let’s look at the wheel. There’s a couple ways of looking at this. First, look at the wheel alone, unattached from the airplane. If the treadmill moves beneath it, and there is no slip, there will be a force between the treadmill and wheel, right? That force will accelerate the wheel translationally, because the force is unopposed, and F = ma. The force also creates a torque around the wheel center of mass, so the force will also accelerate the wheel rotationally (because T = I [symbol]a[/symbol]).
There’s a more involved explanation in this post. Also, you can prove it to yourself experimentally. Take a small roll of tape (or something else small and round) and sit it on a piece of paper. If you pull the paper very slowly, the tape will just sit on the paper because the surfaces will compress slightly. If you pull the paper very quickly, the paper will slip out from underneath the tape because the coefficient of friction is too low. But if you pull it at medium speed, the tape will move forwards and roll backwards at the same time. Try it.
Now, when you attach the wheel back to the plane, clearly it can’t translate without moving the plane, also. If you assume the plane stays in the same place (i.e., there is no translation), then the force between the treadmill and wheel must be transmitted to the plane–otherwise, there would be an unbalanced force, and the wheel would accelerate in translation, because F = ma.
So that force–the one between the treadmill and the wheel–is transmitted to the plane, and if the force is large enough it will cancel out the engine thrust.
ZUT, you are a giant… But you are also the devil! The problem is designed to see who does or doesn’t get diverted down the very tangent of this log. And you are clearly being paid to facilitate that! You solved the physics behind this problem long ago, now you are just giving free physics lesson (which is the reason I came here and read the log in the first place). Thank you. But I can’t help but wonder if it’s not time for you to crack the next problem.
Zut, I appreciate your physics lesson, I really do, and I’m aware of the action in your experimental tape roll, but the inertial values there are quite a bit different.
If the plane were locked to the ground, all force initiated by the treadmill would have to be translated into rotary motion (or heat, I guess). Compared to the inertia of a wheel, the much greater mass of the plane would have an effect similar to the locked-down scenario, and the plane wouldn’t move a wholeheckuva lot.
Aye, but again, the relative value of engine thrust is large compared to the friction/wheel/inertial forces, unless you are proposing the treadmill accelerate to something like a million miles an hour in one second. I guess it is necessary to define reasonable ranges to really answer this thought experiment.
Heh. I’m pretty sure Princhester wasn’t serious. Still, thanks for the compliment.
If i’m understanding you correctly, you’re thinking about this similar to the way Hairyman was a few days ago: for the treadmill to respond, the wheels must start moving. FOr the wheels to start moving, the plane must lurch forward. For the plate to lurch forward, the treadmill must not be matching the speed…and that violates the problem. Yes?
I think, in this case, you’re reading the problem too strictly. In a real engineering setting, if you wanted to experimentally test this scenario, you’d build a feedback circuit that would control the treadmill with some small dither. In other words, the speeds would be matched to within some small amount. What’s a “small amount”? I dunno. You tell me. How close do you need to match speeds to be “close enough”?
Also, remember that the tires are made out of rubber. Rubber flexes. The tire distorts where it contacts the ground, and it shears under torque loading, and it slips and skids a bit at the edges of the contact patch. That means that the meaning of “the speed of the wheels” isn’t exactly defined, particularly when the engines just begin to light off.
Actually, this problem has been surprisingly educational for me. What happens when you explore every single possible aspect of a thought experiment? In detail? In the…half dozen…extensive threads on this problem, a number of people have pointed out things I didn’t know.
Yup.
Yup. Your statement is a bit misleading, in that the force isn’t converted into rotary motion and heat; the energy associated with the force is converted into rotary motion and heat. An important distinction, because energy is apportioned exactly, but force is not. But in any case, the conclusion is sound: the plane wouldn’t move a wholeheckuva lot.
Bingo. For a real plane, the required treadmill acceleration is reasonably large, although nowhere near your guess. As I recall, someone–David Simmons, I think–calculated the required acceleration in a previous thread, and it was large, but not unmanageably so.
This sort of defines the line between between people who treat this problem as a practical experiment and a thought experiment. Conceptually, you can think of a treadmill that will accelerate at whatever rate is desired. Practically, there are limits on what you can build*. There’s nothing wrong with either approach to the problem, so long as you state the assumptions up front–most all of the disagreements about the problem are between people who don’t understand that other people are making different assumptions.
[sub]*Although I should point out that most versions of this problem don’t specify a particular airpllane–it’s just “a plane”. So you could make this a much more manageable problem by constructing a small-scale plane with, say, enourmous jead wheels.[/sub]
zut, thanks for the response. I just keep thinking of “ideal” setups.
Oh I was, I was.
ZUT,
A final post to summarize my understanding of the/your solution so you can (hopefully) verify it for me.
In the interesting version of the problem* (i.e. HYPOTHETICAL conveyor’s velocity matches the wheels’ circumference velocity), if you assume that a coefficient of friction high enough to ensure the wheels will roll is necessary to fulfill the sprit of the question (you know from my post 141/146 that I believe that assumption is very mistaken, but nevertheless) then the problem is one of basic momentum. The energy of the conveyor will apportion its resulting (changes in) velocities (i.e. the rotation of the wheels vs. the (lack of) movement of the plane) based on the relative masses of the plane and the wheels (and their hub friction). Granted, a lot of the velocity obviously ends up in rotation but that’s not the issue. And yes, if you assume massless wheels and frictionless hubs, the plane won’t move backward, but that would seems to violate the spirit of the problem.
Those who (probably got the answer wrong but ought to know better and therefore) want to defeat the hypothetical conveyor without obvious arguments like the conveyor will melt before it stops the plane, which are easily defeated on the basis of semantics (i.e. “It’s a HYPOTHETICAL conveyor, stupid!”) wonder whether the speed of light might limit the speed of the conveyor. But, accelerating a mass like the wheels to the speed of light involves so much energy that nothing remotely like that is required to hold back a jet.
Is that sufficient, or do I need more?
*In the pedestrian version of the problem, where the conveyor matches the velocity of the fuselage, the conveyor merely has the power to double the rotation speed of the wheels, which clearly won’t stop a jet.
PS Thank you. I appreciate the generosity of your time, understanding and interest in this cool problem.
As a side note, I think the coefficient of friction actually is high enough on most planes. The thrust-to-weight ratio of commercial planes is low enough, and the tire coefficient of friction is high enough, that the tires can stand full thrust, at least as a first approximation. I’m not really sure, though.
Right.
That’s pretty much sufficient. If you really want to get into it, I suppose you could explore what happens to the tires if the treadmill does approach the speed of light…but I don’t really know, and it’s certainly OK to assume we’ll stay well below that limit.
I’m nearly certain that this is correct. Evidence that it is would be the fact that from a given speed a jet can, without skidding its tires, brake to a stop in less distance than it took to accelerate.
TO break it down.
- For the plane to take off, it must move relative to the air. (We’re all agreed on this.)
2… A conveyor belt cannot possibly apply enough force through the wheels of a plane to counter the thrust from its engines - at most it can go so fast that it makes the wheels fall off. - If the plane moves relative to the air (unless the air is moving towards the plane courtesy of a headwind or a giant bloody fan) then its wheels move.
- If its wheels move, then the conveyor belt tries to push back on them at the same speed, which doubles the wheel speed, which doubles the conveyor belt speed, etc until both are moving at the speed of light and we have a paradox.
If the wheels move, the universe collapses in on itself. For the question to make sense, the wheels mustn’t move.
To avoid the paradox, we just have to resolve the assumptions in #3 above. Two ways:
1. Assume a headwind or a giant fan: the plane can take off without moving its wheels, and all criteria of the problem are met. And the plane takes off.
2. The wheels are nailed so they can’t move and the plane has sufficient thrust to drag itself down the runway despite its non-moving wheels. Again, the wheels don’t move so the conveyor belt doesn’t leap into a destructive feedback loop. (This is easier if it’s on snow, or ice, or some low-friction surface, so it doesn’t catch on fire. But with sufficient thrust, it could drag itself across the stickiest of sticky substances and still take off.)
I suppose there’s a solution where the plane doesn’t take off: someone’s attached the body of the plane to the conveyor belt with a very strong rope or something. Then the force of the engines is being directly countered by the force of the conveyor belt and the universe doesn’t have to explode because, once again, the wheels aren’t going to move.
So the answer is “Yes”. And “no.”
Actually, can I rephrase that?
The question can only be answered by requesting more information:
- Is there a headwind or a giant fan blowing the air towards the plane? If so, the plane can take off because its wheels don’t have to move relative to the ground for the wings to move relative to the air. If not;
- Are we allowed to hold the wheels in place so that if the plane moves forward the wheels don’t turn? If so, the plane can take off because its wheels don’t have to move relative to the ground for the wings to move relative to the air. If not;
- Does the pilot care about destroying the universe or not? If so, the pilot is restrained by his or her care for fellow humans into not turning on the engines, a paradox/freaky physics disaster is avoided and the plane doesn’t take off. If not:
The plane engines start. The wheels begin to move. The conveyor belt pushes back, trying desperately to slow the plane. It’s not enough. It can never be enough. The wheels are now turning twice as fast as before. The conveyor belt doubles its speed. The wheels are now spinning twice as fast again. The conveyor belt tries to match, fails, and pretty much instantaneously both the wheels and the conveyor belt are moving at the speed of light (or they collapse/explode).
Then Einstein is called in because nobody has any idea how sufficient energy was found to move the conveyor belt at the speed of light, and also because no-one’s entirely sure what happens now that the conveyor belt and plane wheels are stuck in time. (The plane, now a confusing mix of time zones, now finds existing as much of a problem as taking off.)
So the answer is: Depends.
(can’t lift this)
My my my my airplane will not… lift off
It will not get aloft
What is this treadmill here?
Spinning my wheels and spillin’ my beer
It’s as fast… as I can fly
And I can’t accelerate although I try
It keeps my plane down … I’m so miffed
And that’s because I got (uh) no lift
(can’t lift this)
Please report to the nearest punishment center.