Ah, you are correct, sir, thanks for pointing out that one flaw to the explanation. Cecil may wish to consider rewording his reply to that question in the databank, owing the brilliance of the members of the peanut gallery. BTW, if you’re not old enough to remember the peanut gallery, please look it up!
CPL
Well, duh, all that whistling is going to generate wind, plus, their sound is just going to get cancelled out by the sound the jet engines make (those suckers are loud).
I guess the question is what keeps the plane from moving forward. Actually the magic conveyor belt could move at 10 times the forward progress of the plane and it wouldn’t matter as long as the forward movement of the plane reached it’s take off airspeed. I have reconsidered and am switching over to the “will take off” camp. I’m not sure what i had in mind anchoring the plane over the conveyor belt. With no attachments, the wheels can spin at any speed. The speed of the conveyor belt would be similar to changing the size of the tires affecting only the rotation speed of the hubs. What if the plane pulled onto the the magic conveyor belt spinning forward at 60 knots? With the same magic frictionless wheels no advantage would be imparted to the take-off speed of the plane. The wheel could even be spinning backward at take-off.
D’oh, it might help for me to check the question first …
So, I do withdraw that part, but it makes no difference, for the reasons cited by Sullivan above. I hereby adopt Sullivan’s answer and reasoning. More spinning doesn’t add proportionally more backward force.
Clay
I don’t see how such an interpretation makes sense.
If the control system is measuring the airplane speed relative to the belt, then realtive to what is it measuring the belt’s speed?
There are just two options: The airplane, and the ground.
If we assume the belt measures its speed wrt the airplane, the belt simply never moves.
If we assume “the ground”, this makes no sense. You can’t match one speed to another unless both are measured using the same reference frame.
Anyway, the obvious intent of the problem is the notion that the belt prevents the airplane from achieving airspeed. That is impossible regardless of how the belt’s control system works. The only way the belt can apply force to the airplane is through the rolling friction of the tires + wheel bearings. This does not vary with wheel rotational speed, so no matter how fast the belt moves, the airplane still moves wrt the actual ground, and therefore wrt the air.
Hah.
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.
Fine. You take the winch, I’ll take the wench.
Actually, Telemark, if the conveyer belt is accelerating in the reverse direction, it will impart a force in the opposite direction as the engine thrust. This is true even if the wheels have frictionless bearings.
(I also missed this point in the previous thread, and was called out on it by treis.)
That being said, however, the plane will still take off, as the force imparted would be miniscule in any real-world situation.
But it’s incorrect to state that a treadmill cannot exert an opposing force on the plane.
Because of the friction between the tires and the conveyer belt (even with frictionless bearings), the situation is not the same as a plane taking off from a frictionless surface.
Although I agree with Cecil’s answer that the plane will take off under the stated problem conditions, I would like to point out an interesting technicality due to the wheels. Consider the following problem:
A plane with massive wheels (assume uniform disks, for example) but frictionless axels rests on a conveyor belt, and the wheels roll without slipping (i.e. the point of contact between the wheels and the belt matches the speed of the belt). Assume that the plane produces a finite thrust. Is it possible to move the conveyor belt in such a way that the plane will not take off?
The answer to this problem is, in fact, that the belt CAN prevent the plane from taking off. The reason is the following.
In order to accelare the wheels (rotationally) it is necessary to exert a torque on the wheels. This torque is provided by a FORCE from the belt. The force will accelerate the center of mass of the wheel, but the center of the wheel is coupled to the aircraft, and so the aircraft feels this force! (The force is backwards) Thus, by constantly accelerating the conveyor belt, the force of the engines can be cancelled for ANY finite thrust. The actual speed of the belt isn’t important, it is the acceleration that is correlated to the reaction force. Note that for a reasonable aircraft, the belt has to accelerate REALLY fast because the rotational inertia of the wheels is so small compared to the mass of the plane. This also applies to the rollerblades on a treadmill–you can make it impossible to pull yourself forward if the treadmill is accelerating sufficiently rapidly. It is relatively straightforward to calculate the ratio of belt acceleration to thrust given: the moment of inertia of the wheels and the mass of the plane+wheels… simply model the system as a single wheel of the combined moment of inertia with the mass of the total system.
An important note is that this IS NOT DISSIPATIVE. No energy is lost in the process as the wheel does not move relative to the belt. We really don’t need friction at all as we could imagine the wheel and coveyor being gears.
Finally, I’d like to emphasize the point that I’ve made that it is the ACCELERATION of the wheels that is important, other comments have said that it is the speed of the belt that matters, but it doesn’t at all, it is only the acceleration.
Hope people find this interesting!
Tony
I see that CurtC, Zut, and now EntangleMint have all made the same point as I just did in post #108.
You don’t get a much more comprehensive summary of the problem than Zut’s post, which was originally posted in the previous thread on this question. Nice job! Cecil should have used your answer.
This is the post from the first page that lit the lightbulb for me to “get it” and I’ve had a simple thought experiment to illustrate it ever since which I’ve been eager to post as I waded through the whole thread.
Soooo, though I think the answer to the question has been more than adequately explained, I’m posting my thought experiment anyway.
Dangle the jet engine on a string from a large ceiling. Does it move forward? Of course. (yes, assuming it’s been ignited smart-ass).
Now attach a wheel to the bottom of it at just the exact length from the jet engine so that it touches the ground. Assuming the wheel is frictionless enough to be a negligible force, the jet moves forward in the exact same way.
Make the ground a treadmill. Does it make a difference? No. It would be no different if, instead of there being any ground, a giant hand reached out and spun the wheel with its giant index finger. No matter how fast you could spin that wheel with your finger, the jet would move forward all the same.
Of course this ignores many of the wonderfully illustrated (though not practically relevant) scenarios in zut’s really cool post.
I suppose if the treadmill was travelling at the speed of light, the physics of the situation would indeed be affected.
Let’s imagine the problem again with the interpretation that most people seem to like. The airplane is rolling down the conveyor belt WITH forward velocity relative to a stational observer. Now let’s imagine that the pilot switches the engines off. This is in the real world, with all associated drag and friction effects.
What happens? As the thrust from the engines decreases, the plane will eventually stop moving forward, and START MOVING BACKWARDS due to friction from contact between the wheels and the conveyor belt. To the people who think that the conveyor belt cannot exert any force on the aircraft - why then does the plane move backward after the thrust from the engines goes to zero?
Since the conveyor belt can exert a backwards force on the plane, it all boils down to whether or not the conveyor belt can exert a backwards force large enough to counteract the forward thrust produced from the engines. Practically? I really don’t think so, but there’s always those “carbon nano-fibers” . . …
This scenario can be compared to a jogger running on a treadmill. Imagine, for the instant, that the jogger gets off the treadmill, moves it outside, puts on some rollerskates, and straps on a jetpack. The jogger then gets on the treadmill, starts the treadmill, activates a sensor that determines how much thrust is produced from the jetpack and relays it to the treadmill computer which adjusts the speed of the treadmill to compensate by spinning quickly backwards. Now, since the treadmill can exert a backwards force on the jogger despite the rollerskates, the jogger simply stays in place with respect to a stational observer. Does he feel a strong wind blowing on him from the front? No, because he is not accelerating with respect to the ground. He will probably get blisters on his back and feet from the jetpack and the rollerskates with wheels turning at many, many orders of magnitude higher than intended (the cloth/rubber part of the skates will melt off within the first 20-30 seconds or so), but he will not feel any more wind passing over him than if he got off the treadmill and took off the jetpack and the skates.
At least that’s what I think.
Nobody thinks that. The point though is that this force is tiny compared to the engine’s thrust.
Remember – drag from rolling wheels (or from simple sliding friction) does not increase with wheel speed! No matter how fast the treadmill moves, it exerts the SAME drag force against the jogger.
Therefore the treadmill cannot hope to counteract the jetpack’s thrust by moving faster… unless you have either an exceptionally wimpy jetpack, or exceptionally friction-y wheels.
When the engines are stopped, and the conveyor is running, the plane moves backwards. Force is being transmitted through the wheels. Once the engines are on, the wheels are insignificant relative to the thrust of the engines.
OK, some of you are confused as to what I’m saying, and I probably worded it very badly as well. Let me start over.
“A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?”
That’s the question.
The plane (not the tires) and the treadmill are going the same speed. The plane’s speed is 250mph and the treadmill’s speed is -250mph. Therefore, the plane will be stationary (250 + -250 = 0). The plane will be moving in respect to the treadmill, but not to the Earth or anyone/anything on Earth.
The treadmill doesn’t make the air move with it. The plane will be stationary in respect to everything except the treadmill, and that includes the air surrounding it. Because there is no air flowing over the wings, there will be no lift.
The plane will NOT lift off of the treadmill.
There is no reason to add the “well, if the wheels can handle the speed” or “the friction between the blibbity bla” to this problem because it’s not asked in the question.
Um, the original version of this question seems almost certain to have been an attempt at a thought experiment designed to confuse people by sucking them into making a false assumption, namely, that the plane does not move relative to the air.
The only way the plane cannot move relative to the air is if the conveyor belt is able to impart force onto the plane itself. Clearly, friction will allow the belt to turn the wheels (unless the wheels themselves or the belt are made of a frictionless substance). But the fact the wheels turn is meaningless. Unless they are able to in turn convey some of that force to the plane itself, the plane is unaffected.
This question is most easily answered by drawing a force diagram. When doing so, you see that only five forces affect the plane:
The last force is minimal at best; it consists of the limited amount of friction caused by the axle or wheel hub rubbing against whatever housing it turns in. If we postulate a frictionless axle, then 5 goes away entirely, except for the very limited amount of static friction involved before the wheels begin to roll relative to the pavement. And it really doesn’t matter how fast the treadmill goes; there still isn’t any friction force acting on the plane in that situation (frictionless axle). The engines thrust the housing forward; the wheels and the axle come along for the ride.
The only way to keep the plane from moving forward is to impart a force upon it in the opposite direction. That means increasing the friction factor to equal the thrust. But there simply isn’t any good way to make that happen within the scenario postulated by the question.
The question is a classic example of diverting your attention from the real issue. It assumes that you will make the mistake of analogizing to the car or the human on the treadmill, where a completely different force diagram is involved. Once you fall into that trap, you are doomed. As a teacher, I see this sort of question all the time these days on tests that measure achievement in mathematics adn science, though, of course, not quite this sophisticated. 
You still fail to understand the point to the question. The treadmill can move as fast as it wants. The plane’s motion relative to the rest of the world outside the treadmill is totally unaffected by the treadmill, except to the very limited extent that there is some friction in the system that allows the wheels to turn in the socket provided. All you have to do to understand this is to think about holding a plane in your hand, in contact with a moving belt, and ask yourself if you would or would not be able to move the plane relative to the rest of the world.
Or think about it this way: Assume the plane is moving at the speed of 100 feet per second. The belt is moving at the same velocity, in the opposite direction. At time 0, mark the point on the belt where the tires are in contact with it. Then take a picture one second later. In that time, the belt has moved 100 feet “backward”, the plane has moved 100 feet “forward” relative to the rest of the world, and in the process, the wheels are in contact with a point on the belt 200 feet from where they were a second ago.
Overcome if you can the limitation in your mind that persists in assuming that the fact the belt is moving at any speed forces the plane to move at that speed in the same direction, absent its thrust in the opposite direction. All the belt can do is move the wheels; the plane itself is only very barely affected by the force imparted to the wheels. With a frictionless axle, it wouldn’t even move at all, no matter how fast you made the belt go.
This subject is the internet’s version of “Who’s on first?”
Why do planes need long runways? The answer is to gain speed, creating sufficient airflow over the wings to take off. I’m a flight instructor and have studied aerodynamics in detail.
Cecil says, “what gets a plane moving are its propellers or jet turbines.” True, but the engines do not create LIFT. The issue is NOT the thrust of the engines, speed of the wheels, or speed of the plane relative to the ground. The only thing that matters is the relative wind and the wings. Say for example, there is no treadmill, and the engines are off. If there is sufficient wind going straight down the runway, then the plane can theortical take off. It will just hover there, but the wings will think it’s speeding through the air. This is why, at 10,000 feet, if all engines fail, the pilot’s first instinct is to maintain sufficient airflow over the wings. He may have to pitch the nose of the plane down slightly in order to do this, but the plane will glide very nicely. The point: a plane does not take off because of the power of the engines: it takes off due to the aerodynamic effect of wind over the wings (creating lift).
Scott
This is where your logic breaks down. This only makes sense if the directional motion of the conveyor belt is communicated to the plane. It would make sense if you were running at 250 mph on a belt moving 250 mph in the opposite direction–then you would not make any progress relative to the ground. But freely spinning wheels prevent the motion of the belt from being communicated to the plane, and the plane is free to move forward.