Plane vs. Conveyor belt.......old but interesting

[Atomsk]

My favorite answer to this question is in a post by the user rich in reply to this blog post: http://ddhr.org/2006/06/26/plane-on-a-conveyor-belt/
It states:

I'm loving these posts recently... I feel the need to weigh in on this once again.
I could be really annoying and give the quick useless answer, or I could be long winded and give a more complete answer. I think I'll be even more long winded and give both!
1. Useless answer: You did not define what type of plane this is. I therefore define the plane to be a harrier. It can take off because a harrier can take off vertically. Nyea
2. And here we go...
The original prompt did not specify frictionless wheels, and this matters a great deal, so I will address both the case of frictionless wheels and wheels with friction. I will assume that the plane's engine can exhaust air at an arbitrary speed and ignore transonic effects if such speeds are necessary. I will also assume that the conveyor belt can move at an arbitrary speed, is of infinite length, and that its control system can adjust its speed instantly. (And if you want the quick answer, yes, the plane takes off.)
2a: With frictionless wheels:
This premise is almost untenable. Such a control system could not exist as to keep the belt moving at the same speed as the wheels. All of the linear momentum that the belt transfers to the wheel is used up by the wheel as angular momentum. Nothing is transferred to the plane. In this case, the only force on the plane in the x direction is forward, and is from the exhaust air pushing on the back of the fan blades (I'm assuming a standard turbofan, but this works for whatever). Thus, the plane will begin to move forward (relative to a stationary observer not on the belt), and smoothly accelerates to a takeoff. Before takeoff, though, interesting things are happening to the wheels and belt: This process spins the wheels faster, forcing the belt to move faster (though at any point in time it simply CANNOT actually move fast enough to match the speed of the wheels rotation). The belt will then continually accelerate trying to compensate for the fact that the wheels will always be spinning a little faster due to the plane's forward motion. However, the acceleration is not without an upper bound. What will happen before too long is that the coefficient of static (if you think this should be kinetic, you should read up on how a wheel actually works) friction will be overcome, and the wheels will skid. I originally thought that this would slow the plane down, but, again, the frictionless nature of the wheels means that the dragging from the belt only serves to slow the rotation of the wheels. What happens next depends on how this perfect control system works. If it is measuring the rotational velocity of the wheels, then it will slow down since the wheels slowed during the skid. As the belt and wheels both slow, they will reestablish static friction, and the wheels will roll again, and the belt will speed up again. This will continue to modulate, not affecting the plane's forward motion, but putting a ceiling on the speed of the belt. Then, the plane takes off, and the belt and wheels can finally slow down to rest. (Yes, I know I overanalyzed what was happening to the wheels and belt, but I thought it mattered for a while, and it comes into play below.)
2b: The wheels have some friction:
The plane takes off, but not for the reasons you most likely think. The forward force from the exhaust air causes the plane to start to roll forward, which causes the belt to speed up, spinning the wheels faster. Since there is friction in the wheels, this does create a backward force on the plane. Since the belt is controlled to match the speed of the wheels, then this backward force will always be equal and opposite to the force applied by the air. By definition, the controller will not allow the wheels to outpace the belt, so they cannot roll forward. The plane will stay stationary with respect to an observer not on the belt. If the observer is anywhere nearby, they are likely to be deafened, as the belt's motor and plane's engines scream as they continually put out more power seemingly without bound. However, this actually does not go on forever. At some point, the wheels will skid for much the same reason as in 2a. The coefficient of static friction will be overcome by the force applied by the accelerating belt. Since the coefficient of kinetic friction will be lower, the plane begins to move forward, dragging its wheels the whole way. (I'm assuming the tires won't be worn out, and that the coefficients of static and kinetic friction remain constant despite the drastic heating that is likely occurring at this point.) At this point, the rotational speed of the wheels determines what happens. The forward motion of the plane, and the rearward motion of the belt are both contributing to the rotation of the wheels. The efficiency with which their linear momenta are translated into rotational momentum is dictated by the coefficient of kinetic friction. This spawns 3 cases: 2b1, 2b2, and 2b3. (In each case, the plane takes off, but I'm trying to be thorough)
2b1: The coefficient of kinetic friction is so low that the wheels slow down. This causes the belt to slow, until the point that static friction is reestablished, and the wheels roll again. However, the belt and wheels are going far too slow to balance the thrust that the plane is putting out at this point, so the plane continues forward, causing the wheels to roll faster, which once again overcomes static friction, and we are sliding again. All this serves to do is possibly momentarily hinder the plane's acceleration, so it still takes off.
2b2: The coefficient of kinetic friction is at exactly the right value that the wheels spin with constant speed, so the belt moves with constant speed as well, maintaining the regime where the wheels are skidding. The plane continues to accelerate smoothly and takes off.
2b3: The coefficient of kinetic friction is high enough that the combined effect of the plane's forward motion and the belt's rearward motion cause the wheels and belt to accelerate without bound until the plane takes off. During this time, we are still skidding, so the plane will accelerate smoothly and take off (putting the wheels and belt out of their misery).
And with that, I put you out of your misery as well. Sorry it was so long, but I think I've covered all the main sticking points.

[john970]

A commercial airliner would probably not take off. The tires are rated for takeoff/landing speeds, which would be doubled in this case and they would likely blow a tire(s) before rotation speed was reached.

I'm a commercial pilot.

--

edit: Speeds wouldn't necessarily be doubled - didn't specify conveyor details.

[1ster]

I can't believe we're actually debating this. There's no lift, people!

If this actually worked, don't we think someone (the military, NASA, whoever) would have done it already?

[Torqueless]

Originally Posted by 1ster

I can't believe we're actually debating this. There's no lift, people!

If this actually worked, don't we think someone (the military, NASA, whoever) would have done it already?

read atomsk's post, it covers several possibilities that all result in the plane taking flight. As long as the plane has enough foward motion, assuming a mechanically sound plane, there will be lift.

[plexiglass]

my brain hurts



power tread mill matching the rotational speed of the wheels I could see where there would be no forward movement for the plane.

tread mill is free moving then the plane would be able to move forward therefore air moving over the wings and lift occurs when the faster moving air over the wings creates a lower pressure above the wing than below....yada yada yada......



if a bear craps in the woods is there a tree nearby to smell it?

and if I could develop a special breed of rabbit that has fur that crap does not stick to....would there be a bear market?


did I mention my brain hurts....

[Atomsk]

Like Rick said in my post. It does not matter, power matching treadmill, friction or no friction, eventually the coefficient of static friction will not be high enough to keep the wheels rolling and they will begin to slide and the plane will drag them across the treadmill and take off.

[Torqueless]

Alright let's assume a plane accelerates in one direction and the treadmill accelerates proportional to the plane in the opposite direction. The plane's engine, propeller whatever, acts on the air not the ground. Plane's wheels are designed to spin freely and reduce friction between the ground, this means the wheels will spin twice as fast due to the two opposing accelerations but it will not affect the foward motion of the plane.

[Sherifftruman]

Originally Posted by Torqueless

Alright let's assume a plane accelerates in one direction and the treadmill accelerates proportional to the plane in the opposite direction. The plane's engine, propeller whatever, acts on the air not the ground. Plane's wheels are designed to spin freely and reduce friction between the ground, this means the wheels will spin twice as fast due to the two opposing accelerations but it will not affect the foward motion of the plane.

Exactly.

[1ster]

I would like to retract my previous answer. I find this argument very convincing. I originally made the same mistake as the author below, but his final analysis is persuasive.

Ask the pilot

What does it take to make a plane fly? Can it take off from a conveyor belt? The pilot weighs in on an old brainteaser.

By Patrick Smith

Jan. 5, 2007 | Last month, New York Times technology columnist David Pogue set the Web abuzz by rekindling an old brainteaser about whether a theoretical airplane would be able to take off from a theoretical treadmill. The puzzle was "ripping around the Internet" (Pogue's words), and appeals for clarification quickly reached Ask the Pilot's in box. Will it or won't it fly, people wanted to know, imploring me to weigh in.

Belatedly, and grudgingly, I will now do so. Such topics tend to induce the rapid closure of my eyelids, and while I'd like to tell you this is the kind of shop talk that keeps aviators engaged and alert in those quiet midnight hours high above the ocean, nothing could be further from the truth. (Mostly they're just bemoaning the loss of their pensions and talking about movies.) Nevertheless, here goes ...

"Imagine a plane is sitting on a massive conveyor belt," poses Pogue's Dec. 11 Times blog, "as wide and as long as a runway. The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?"

When I last checked, more than 860 people had posted their opinions, split about 50/50 between those who say the airplane will fly and those who insist it can't. If you look carefully you can locate my own contribution, flatly declaring that no, absolutely not, the aircraft will not get off the ground. "If this is truly 'ripping around the Internet,'" I snarked, "then heaven help us. The plane will not fly. Of course it won't fly."

And why should it? How can it fly if it's not moving? For an aircraft to get and stay aloft, it needs lift; it needs air passing above and below its wings. And for that it needs to move. For a cursory lesson on how this works, simply shove your arm out the window of a speeding car. Shape your hand into an approximation of a wing, angle it slightly into the oncoming wind, and voilà, it's flying.

Now, imagine you are in that same car, on a treadmill. The car's wheels are spinning -- be it at 60 mph or 600 mph -- but when you put your hand out the window, does it rise up? Of course not. Your hand won't fly, and the plane won't fly either for exactly the same reason: because for all its efforts, the vehicle isn't moving. You have zero relative speed and zero lift.

This seemed so obvious that it needed a caveat: "On the other hand, if you were able to generate a tremendous enough amount of thrust," I noted in a follow-up post, "and redirect the vector of that thrust downward, you could, conceivably, lift the plane off like a rocket. Heck, you can make anything 'fly' if you stick enough power under it. But that isn't fair to the spirit of the premise."
Except, wait a minute, what is the premise?

Go back and read it. "Imagine a plane is sitting on a massive conveyor belt," it says, "as wide and as long as a runway." I'd glanced right over those key words, "as long as a runway." I was so caught up in the image of a motionless plane on a regular old treadmill -- like the kind you might see at the gym -- that I missed the whole question. Looking back, that does seem a dull and senseless riddle: Can a plane fly if it can't move? Obviously not. There has to be more to it.

And there is. At heart, this has nothing to do with the principles of lift but, rather, with those of friction and acceleration. The gist of the question is better understood as follows: Will an airplane, under its own power, remain motionless on a 10,000-foot-long treadmill, or will it roll forward? Will it accelerate and fly?

Turns out the answer is yes. A distinctly theoretical yes, for reasons we'll get to shortly, but for all intents and purposes of the puzzle, that's yes enough.

A car won't accelerate on a treadmill; the belt will always match the rotation of the tires. You will not accelerate on a treadmill; the belt always runs in sync with your footfalls. But an airplane is different. An airplane's wheels are not powered by gears or a drive train. They hang inertly below, and are free-spinning. The thrust force that moves the plane along couldn't care less about the ground. The engines are not fighting against the surface, they are fighting against the air.

Confusing, I know, and in the interest of full disclosure, physics was the one class in high school that I outright failed and had to take twice (you try ciphering out equations while listening to Minor Threat on a pair of clandestinely strung ear buds). And some of you might remember what happened the last time I combined things aeronautical and mathematical. So let's get somebody else to explain.

According to Paul J. Camp, a professor in the department of physics at Spelman College, it's all pretty simple. "At first, the conveyor will hold the plane still. But only to a certain point, after which, driven by thrust from its engines, the craft will accelerate."

But the problem clearly states: The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction.

"The key is in the behavior of friction," Camp says. "Friction is a peculiar force in that it has an upper limit. For instance, push an object on your desk, but not hard enough to move it. Why doesn't it move? Because the friction force exactly balances the force of your push. At some point you push hard enough to set the object in motion. This is the point where friction has topped out and is not capable of growing any larger."

With the airplane and treadmill, there is, at the outset, friction force capable of rotating the tires at the proper speed to keep the plane stationary. However, as the thrust is increased, that force eventually maxes out. (Two separate frictions are at play here, actually, one between the tires and belt, the other between the plane's axles/bearings and its wheels. The first will max out before the second.)

"And at that point the wheels no longer roll, they slide," says Camp. "Or rather, they roll and slide at the same time. Tire motion is now decoupled from the belt motion. No matter how much you whiz up the treadmill, you won't add any more rotational velocity to the wheels because friction is already doing everything it is capable of. The plane skids toward takeoff -- likely accompanied by much smoke and a powerful rubbery stink."

And there you have it, at least on paper. Bear in mind that for a plane to reach that point of decoupling would require two things above and beyond the pale of normal engineering. First, a remarkable amount of power -- far more than any jetliner, and probably any military plane, is capable of developing. The illustration on Pogue's blog is of an Airbus A320; some sort of rocket plane would be more appropriate. Second, no existing aircraft tires could take such abuse. The rotational velocity required before reaching the friction limit would have them bursting within seconds, causing the plane to be flung backward. Believe it or not, landing gear isn't engineered with giant treadmills in mind, and pilots need to adhere to maximum groundspeed limits, lest their tires wind up like this. These limits occasionally present problems during tailwind operations or in the case of flap and slat malfunctions -- scenarios dictating the need for unusually high takeoff or landing speeds.

For good measure, the treadmill itself, as described, could never be built. It can't "exactly match the speed of the wheels," because the wheels will turn at the speed of the treadmill plus the speed of the plane relative to the ground. When the speed of the plane is greater than zero (which it is the moment its wheels start to spin; otherwise they would never move), then the problem becomes impossible. By definition, the wheels have to be turning faster than the treadmill.

Whose idea was this crazy problem?

Meanwhile I can't decide if this is good or bad news for the conveyor belt industry and treadmill enthusiasts worldwide. Though, as they say, any publicity is good publicity.

[Atomsk]

:w00t: yep basically what good ol' rick said in that blog post that i quoted above.

[Papethova]

Looks like Mythbusters is doing this on tonights episode

[BrownBoy]

At first, I thought it wouldn't take off due to lack of air passing over the wings but after some research my mind changed and I didn't know what to believe.

P.S. I just saw the episode...it takes off guys.

[RobLS]

It will not take off, in the original question, and the following correction by the poster, it said nothing about the wheels being the driving force, only that the conveyor matched the plane's movement in the opposite direction. Therefore, no matter where the force is being applied, the force is still matched in the opposite direction of the conveyor. Now if you wanted the plan to take off, assume its a harrier and it will take off vertically.

[chemhalo]

Originally Posted by Papethova

Looks like Mythbusters is doing this on tonights episode

Was it just me or did it look like mythbusters botched it?

The plane was moving forward. The tarp they pulled under the plane did not match the speed that the plane was traveling. When I used to fly with my friend who had a small Cessna, he told me the plan would take off at a MUCH LOWER speed than we always did and that he would keep the plan from taking off with the flaps(?) until we got to a much safer speed to take off.

I think this is what happened on mythbusters. It was an incredibly small plane so it may have needed even less airspeed than the Cessna. The propeller will provide forward thrust but not lift. Unfortunately the plane was moving forward wish did create lift and got the plane up.

Edit: actually reading that long post above explains why

[B25 Jim]

From the article above:
For good measure, the treadmill itself, as described, could never be built. It can't "exactly match the speed of the wheels," because the wheels will turn at the speed of the treadmill plus the speed of the plane relative to the ground. When the speed of the plane is greater than zero (which it is the moment its wheels start to spin; otherwise they would never move), then the problem becomes impossible. By definition, the wheels have to be turning faster than the treadmill.

You have to keep in mind, the wheels freely turn unless brakes are applied. There is nothing holding the wheels back besides the friction of the bearings. And for John970... the speed on the tires will greatly exceed the takeoff/landing speed... not double necessarily, but close...

Jim
Airline Mechanic and Engineer.

[john970]

Originally Posted by B25 Jim

You have to keep in mind, the wheels freely turn unless brakes are applied. There is nothing holding the wheels back besides the friction of the bearings. And for John970... the speed on the tires will greatly exceed the takeoff/landing speed... not double necessarily, but close...

Read my edit right after I posted it...

The fact that such a treadmill could never be built is irrelevant (albeit true), the tires would fail before that happened.

--edit:
After a bit more thought, if you strictly interpret the question you're right - the question is invalid as no such treadmill could possibly exist. Assuming there is a reaction time allowed before the treadmill reacts, I believe my above statement stands and the real solution would depend on said reaction time, wheel speed rating, and the Vs0 of the aircraft in question.

[B25 Jim]

Originally Posted by john970

Read my edit right after I posted it...

The fact that such a treadmill could never be built is irrelevant (albeit true), the tires would fail before that happened.

--edit:
After a bit more thought, if you strictly interpret the question you're right - the question is invalid as no such treadmill could possibly exist. Assuming there is a reaction time allowed before the treadmill reacts, I believe my above statement stands and the real solution would depend on said reaction time, wheel speed rating, and the Vs0 of the aircraft in question.

Yes, you are correct, if the speed of the treadmill was over a certain speed, it would destroy the tires. We are both interpreting for different factors. Reguardless, it WOULD take off, if the tires were not destroyed.

[cracka]

Wow, do we have some pragmatists around here or what? :wink:

Hypotheticals like this aren't based on what can be built practically. Who cares if the treadmill can be built or the tires will shred? The question only exists to dispel the common misconception about how the propulsion of an airplane is applied when it's on the ground, so it's not invalid on that basis.

The Mythbusters demo, while certainly not perfect, does address the fundamental question nicely. Not bad on a shoestring budget.

[B25 Jim]

Originally Posted by cracka

Wow, do we have some pragmatists around here or what? :wink:

Hypotheticals like this aren't based on what can be built practically. Who cares if the treadmill can be built or the tires will shred? The question only exists to dispel the common misconception about how the propulsion of an airplane is applied when it's on the ground, so it's not invalid on that basis.

The Mythbusters demo, while certainly not perfect, does address the fundamental question nicely. Not bad on a shoestring budget.

You ask questions about aircraft to aircraft people and we are likely to be a little on the serious side. These types of conversations are what I spend most of my day on, though usually a little more on the practical side in relation to flight. And John, being a pilot, would spend time worrying about what speed the tires are at, which is justified. Just remember, the guys who are flying and fixing your planes are accounting for these factors :biggrin:

[cracka]

Originally Posted by B25 Jim

Just remember, the guys who are flying and fixing your planes are accounting for these factors :biggrin:

Fair enough. You go right on being pragmatic. :biggrin:

[RobLS]

this is a theoretical question, you have to take it for that.

[B25 Jim]

Originally Posted by RobLS

this is a theoretical question, you have to take it for that.

I understand it is theoretical. And the question was answered on the first page. Yes it will take off. But, just as everyone kept discussing the possibilities tied to that, so did we. We just took it a different direction. :wink: