Plane on a Treadmill - SOLUTION

Cecil's Columns/Staff Reports
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MODERATOR EDIT: Please note that this thread was started in 2006, has been revived several times since. Please be careful about what you’re responding to, since it could be way (way!) old. This thread is in response to this Straight Dope column and this followup. – CKDH

Cecil is wrong. The plane does not take off. This is why:

Original Statement:
“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?”

Keep in mind that no system is frictionless. We are assuming real systems, and the conveyer as described is well within present capabilities. It can be built.

Fact 1: The force generated by an engine is limited.
Fact 2: Friction is a force
Fact 3: The mass of the airplane is an “inertial force”
Fact 3: The frictional force of the wheel assemblies against the ground, whenever referred to below, contains the friction of the tires against the ground, as well as the wheel assembly (bearings, axle, etc). This also includes the downward force of the mass of the airplane upon the entire assembly. This entire frictional force is not inconsequential.

Fact 4: The conveyer, as it moves faster, exerts more and more backwards force upon the wheels, and this force is not inconsequential. Cecil dismissed this, but this is considerable and is a significant part of the frictional force of the wheel assemblies. Unfortunately, this is Cecil’s big physics flaw. The backwards force of the conveyer is equal to the force generated by the conveyor system. THIS is what everyone has ignored, and this data is required in order for the system to be closed. You cannot ignore the force generated by the conveyor system itself. It is equal to the backwards frictional force against the wheels. You cannot ignore the force of the conveyor system itself. The conveyor is moving, a force is being generated to create that motion, and that force, the force to move the conveyor upon which a big hulking mass of airplane is sitting, is large and considerable.

As the below shows, it is the force generated by the conveyor system that prevents the plane from achieving lift.

Fact: The plane has no horizontal motion.
Item: For the plane to have horizontal motion, the force of the engines must be greater than:

[The big hulking inertial mass of the plane] + [frictional force of the wheel assemblies with the ground (including conveyor system)]

If the control system tracks the rotation of the plane’s wheels exactly, then it follows that the full, force of the engines at that moment have force exactly equal to the frictional force of the wheel assemblies against the ground (bearings, ground frictional force, etc). Therefore, as long as the treadmill tracks as in the original statement, the full force of the engines are exactly equal to the frictional force of the wheel assemblies against the ground. (it may take some time to realize this). As the constant force of the engines are equal to the frictional force of the wheel assembly and the ground friction, no force is available to counteract the inertial force of the mass of the airplane.

Fact: In the original statement, the velocity of the plane relative to the conveyer, is not the same as the velocity of the plane relative to the air.
Item: No matter how fast the conveyer moves, with or without anything on it, it won’t make the air move any faster six feet above it. Therefore, velocity of the air over the wings is independent of the conveyor speed.

Fact: A plane takes off not by it’s speed relative to the ground, but by the velocity of the air relative to it’s wings.
Item: We can agree that, in the original statement, a plane without wings will not take off. See the next Fact for a further explanation.

FACT: THE AIR FLOW OF THE ENGINES ALONE ARE NOT SUFFICIENT TO PROVIDE LIFT!
Proof: An airplane at the end of a runway as it’s about to take off, regardless of whether the airplane is propeller driven, or has engines mounted on it’s wings, is at full throttle/full thrust. At the beginning of the runway, this full throttle/thrust is not sufficient to pull enough air over the wings for the plane to take off - if it did, planes would take off immediately and would not require a runway.
This exercise can be compared to an airplane sitting at the beginning of a runway with it’s engines at full throttle.
A plane’s engines provide forward motion to increase the flow of air relative to the wings. The air flow provided by the engines alone are not sufficient to provide lift. Forward motion is required. If the force of the engines is equal to the frictional force of the wheel assemblies and the conveyer assemblies, then there is no forward motion. Without forward motion, the airflow generated by the engines alone are insufficient to create enough airflow across the wings to provide lift.

(If we can talk about frictionless surfaces, infinite thrust engines, etc. we change the experiment.)

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Please change:
Proof: An airplane at the end of a runway as it’s about to take off, regardless of whether the airplane is propeller driven, or has engines mounted on it’s wings, is at full throttle/full thrust. At the beginning of the runway, this full throttle/thrust is not sufficient to pull enough air over the wings for the plane to take off - if it did, planes would take off immediately and would not require a runway.

To:
Proof: An airplane at the start of a runway as it’s about to take off, regardless of whether the airplane is propeller driven, or has engines mounted on it’s wings, is at full throttle/full thrust. At the start of the runway, this full throttle/thrust is not sufficient to pull enough air over the wings for the plane to take off - if it did, planes would take off immediately and would not require a runway.

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Welcome to the Straight Dope, Paradoxic. Your conclusion is certainly reasonable, but it depends quite a bit on the assumptions that you make about the situation. In fact, I listed a lot of those possible assumptions in this post relating to Cecil’s earlier thread.

In particular, you are correct that, according to the original problem (at least, some versions of the original problem) the plane will be held stationary with respect to the ground, and thus the conveyor belt will apply the necessary force to counteract the thrust. And this force is, by definition, not inconsiderable. The real question that a lot of people asked was if it was even possible for the treadmill to apply such a force. Again, it depends on your assumptions.

However, a few things: First, mass is not an “inertial force.” It’s a component of the inertial force, but the force only occurs when the mass is accelerating (F=ma). Since we’re assuming no motion of the plane at all, there’s no inertial force associated with the bulk mass of the plane.

Second, assuming that the frictional force associated with the wheels will balance the engine thrust is problematical. Clearly, this frictional force is nowhere near the engine thrust when the plane takes off from a concrete runway. It’s possible that the frictional force increases with increased velocity, but it would have to increase substantially, and this is by no means assured. A more significant force, I would argue, is the force resulting from the rotational acceleration of the wheels (as Cecil mentions).

Third, you are clearly correct that the airflow from the engines won’t lift the plane, and that bulk flow over the wings is necessary. However, it is certainly possible that if the conveyor were long enough and ran fast enough for a long enough time, air could become entrained with the moving belt, moving fast enough to lift the plane off the belt.

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No.

The force generated by the conveyor system itself is what everyone has ignored, and you are still ignoring it.

Clearly, the physics show that the conveyor system itself has to be responsible for a force equal to the thrust of the engines. That’s all it takes. If the force of the conveyor system equals the thrust of the engines, the aircraft can go nowhere.

Suppose, for example, you created a conveyor system powered by multiple turbine engines similar to the ones used to power the aircraft?

And the frictional force HAS to be considerable because the physics dictates that the force generated by this turbine-driven conveyor system can not go anywhere else. You must look at this as a closed system and consider the force of the conveyor.

You are making assumptions, and using words like “clearly” and “it’s possible”, because you are still considering this to be a thought experiment. You have to look at this as a closed system, and consider the force generated by the conveyor system itself.

You are making unwarranted assumptions. Please, don’t make me have to take time out to demonstrate the math on this…!

You need to stop treating this as a thought experiment and start looking at this as a closed-system physics equation. You cannot ignore the force generated by the conveyor. The conveyor system must be generating a lot of force to move the belt, and that force must go somewhere, right?

But I reject your assumption. I cannot see any fluid dymanics equation where air can become “entrained” with the moving belt. Even if you look at a speeding car, even at 160mph, you can’t “entrain” enough air six feet away from the car to be of sufficient velocity to lift an airplane. Again, please don’t make me do the ball-busting math on this…

I know that, as a matter of business, Cecil is supposed to be always right, but in this case, the excuses must stop because the plane cannot take off.

You need to look at this as a closed system, and stop making assumptions about friction and airflow.

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Congratulations, you’ve jumped straight to BR #2: If the conveyor belt exerts backward force via friction et al. on the axles of the jet plane equal to the thrust generated by the jet engines, the plane will not move, and thus not take off.

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And the friction can evidence itself in terms of HEAT. It’s to be expected that you’d have some really hot wheels by doing this, heat dissipated into the atmosphere.

To put it simply: In this closed system, if we’ve got a turbine-driven conveyor system, such as the turbines used to drive the aircraft, and if this conveyor system uses the same force as the aircraft engines, then this force has to go somewhere.

Friction is not as trivial as most people make it out to be.

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No I’m not, actually. I even said, “according to the original problem, the plane will be held stationary with respect to the ground, and thus the conveyor belt will apply the necessary force to counteract the thrust.” As far as I can tell, I’m agreeing with you.

Let me be pedantic far a moment: If the force of the conveyor system on the plane equals the thrust of the engines, the aircraft can go nowhere. Agreed? Now, the size of that force has to do with the construction of the plane wheels and the motion of the treadmill. According to the original problem, the treadmill will do whatever it takes to keep the plane stationary. And just what that is depends on your assumptions about the setup of the problem.

For example, if the frictional drag in the wheel hub is constant with velocity, the treadmill will continuously accelerate the wheels in order to apply the appropriate force.

What assumptions? I didn’t think I really made any at all. And sure, go ahead and do the math. Please do state your assumptions up front, though, because the conclusion is heavily dependent on initial assumptions.

Sure, but again, it depends on your assumptions. It takes a lot of force to accelerate a huge conveyor belt with no plane on it at all. That force goes somewhere also, and it’s clearly not into the plane.

Like I said, it depends on the size and speed of the belt. There must be a boundary layer associated with the belt, and I would think that at some ridiculously high belt speed, the air over the wings would move fast enough to produce adequate lift. Granted, we’re talking about awfully high speeds, but awfully high belt speeds are required to keep the plane in place.

I honestly can’t tell if I agree with you or not. Can you expand on this? For example, in this turbine system, is the belt accelerating, or staying at a constant speed? If we look at the forces and torques applied to the wheel, what would they be?

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O.K., then, I guess we agree. Actually, what I am objecting to is your use of the word “clearly”. This problem is difficult as a thought problem, and is more obvious when viewed as a closed-system physics equation.

If the plane does not move horizintally, then according to the rest of my analysis, the plane does not take off because the airflow generated by the engines alone is not sufficient to create enough lift with the wings.

Sorry, Cecil.

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Let’s say the airplane needs 100mph ground speed to take off.

Let’s also assume that the landing gear can support the plane at 200 mph (forward speed of the plane, plus the backwards speed of the conveyor).

If the engines can overcome the additional forces against the landing gear of 200mph over 100 mph (not a big deal) and the landing gear can withstand twice the limit of ground speed (again, not a big deal) The plane will take off.

The question states that the conveyor matches the speed of the plane. The plane moves 100mph forward (ground speed) The conveyor moves at 100mph backwards (ground speed). The wheels turn as if the plane was traveling 200mph.

What’s to understand? The plane is moving 100mph down a moving runway moving in the opposite direction. So what if the runway is moving? The plane has 100mph air over the wings. It takes off.

To put it another way ---- What is the frictional force against the airplanes wheel at 100mph?

What is it a 200mph? Three times the resistance? Four?

Now, we also have to deal with the resistance created by the rotational velocity of the tire spinning up to 200mph, instead of 100mph. Race cars overcome that routinely.

Consider the wind resistance as a plane approaches take off speed. It must be 100 times the resistance from the landing gear. The landing gear resistance is nothing.

It’s gonna be there, for sure, but hardly something that couldn’t be overcome.

How about this? Would it be possible for a plane to take off with a 100mph tail wind? Same thing. Not something anyone would dare to do, but something that would be about as easy as building the hypothetical treadmill.

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You have the same problem that everyone else who thinks the plane does not take off has: you are reading the problem differently from the rest of us. To quote an earlier post of mine:

So I ask you, if the speed of the plane is measured relative to a stationary object, and it is that speed that the treadmill mirrors, will the plane take off?

That is the question that Cecil (correctly) answered.

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I am keen to see someone build one. The total thrust from a 747’s engines is about 250,000 pounds. That’s 250,000 pounds of horizontal force applied to the center of mass of the airplane (more or less). In order to stop the plane from moving, the conveyor belt needs to apply an equal force in the opposite direction—and it can do it only by rubbing itself against the free-wheeling wheels. The conveyor belt would need to move at ungodly speeds; the tires would melt, the bearings would seize (which would, granted, help the belt out), the landing gear would snap off, and eventually the plane itself would be reduced to scrap.

This is something like trying to keep a freight train from moving forward by applying a belt sander to one side of the engine. It’s theoretically possible, sure, with a magic belt sander and an impervious train, but there’s no way you could build such a critter.

I think it’s much more likely that the question is asking about the plane’s speed relative to the ground. It doesn’t make sense to me to say the speed of the belt is “exactly the same” as the speed of the plane when the speed of the belt would have to be much much faster than the plane’s top speed could ever be (~600 mph for a 747). The belt is no longer matching the plane’s speed, in this thought experiment; it’s moving fast enough that friction prevents the plane from moving, which is not the same thing.

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Yeah. And I want my Air Car. :smiley:

Building it is not the issue.

So is the force against the landing gear by the moving runway.

No argument there. Learned that falling down and skinning my knee. .
[/quote]

The airplane only needs to double it’s wheel speed to get to proper ground speed to take off. No big deal when we talk about moving conveyer belts.

In what possible way is this a closed system? The airplanes engines interact with the air that has absolutely nothing what so ever to do with the conveyor. The reason the airplane takes off is because it’s an open system. The power to the airplane and the power to the conveyor are so far removed I can’t begin to understand how anyone would consider this a closed system.

Let me ask everyone this.

Put a toy car on your home treadmill. Tie a string to it. Start the tread mill at 1foot per second. Could your pull the car to you at 1fps? Having the toy cars wheels turn at 2fps Could you pull it off the tread mill? If you believe that you could pull the toy car off the tread mill, then the plane can take off.

It stunned me that some people thought this could happen. “Ummm…. what are the wings for? Hold the engines on I guess”. Wow.

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Agreed, with a quibble. This gets into turbulence, which obviously requires a lot more information than we’re given. A large enough belt, once it settles into a steady state, will indeed move the air along with it, but speed isn’t really what you want. As the belt increases speed, the air will split into various laminar flow regions, with fast-moving ones near the belt and slow-moving ones near the outside world, until a certain threshold is crossed and all the layers dissolve into turbulence. If this belt were miles wide and circled the earth, you could probably create enough of a wind to launch a 747, if you increased the speed steadily; if it were about the size of the plane, you’d probably just create a lot of vortices. (That’s just an off-the-cuff guess, though.)

Note, Paradoxic, that we’re not subject to 1,000 mph winds here on the surface of the earth. This isn’t because the atmosphere has rotational inertia; it’s because the atmosphere is pulled along with the big spherical “conveyor belt” we call the earth.

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Isnt that more to do with gravity than the friction generated by the surface of the earth?

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Upon review, I would say this is not an open system, but two closed systems.

Heat and work are exchanged, but not nearly enough to prevent the plane from taking off.

UNLESS you go to BF#2 wherein you dive into theoretical physics and have a constantly accelerating treadmill. I don’t think that that was the point of the original question.

And, I also think that some folks are thinking about a tread mill that is only big enough to support the plane. Especially those folks that make air-craft carrier references.

I’m talking about a tread mill as long or a bit longer than a runway.

A runway that can move.

Hmmmm…… A runway that can move.

“The plane moves in one direction” Right. There’s the answer to the poorly worded question. If the plane moves, it can generate lift. There’s the so called Plane on a Treadmill – SOLUTION

The plane takes off. More work, certainly. Nearly impossible to control on the ground as the wheels are spinning at twice the rate they normally would. And the transition would be tricky at best. But if this critter could be built, a plane could take off from it.

Cecil nailed it. Twice.

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Bad analogy, because the treadmill in this case isn’t reacting to the plane’s motion. That changes the problem considerably.

In any case, there are multiple interpretations of the problem, and the “answer” depends on your interpretation and your initial assumptions. flight lays out the two basic camps: The first is where the belt matches the plane speed with respect to the ground (that’s what you’re talking about). The second is where the belt matches the wheel rotation speed (or the belt matches the plane speed with respect to the belt). This requires the plane to remain stationary; otherwise the condition of the problem is violated. This second interpretation is (I think) what Paradoxic is talking about.

I’ve seen different wordings of the question (Cecil’s original column had multiple versions, for example) that more or less strongly imply one interpretation or another. So I don’t think it’s necessarily conclusive to tie the “answer” to one wording or another, because other people have likely made conclusions based on alternate problem statements.

Agreed, also. I was objecting to Paradoxic’s statement that "no matter how fast the conveyer moves, with or without anything on it, it won’t make the air move any faster six feet above it. " My sense is that a runway-sized conveyor belt running at a few hundred miles per hour would move the air six feet above with some measurable velocity. Enough to create lift? I doubt it. Unless it’s a huge belt going at ridiculous speeds for a long time, in which case you run into compressibility effects, and you get a supersonic shock wave around the wheels… That’s getting pretty far afield from the original question, though.

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The treadmill is irrelevant because it doesn’t matter whether the plane is on it’s wheels, or suspended from above by a set of steel chains hanging from a curtain rod that runs the length of the runway.

The people who refuse to believe the plane takes off are confusing how engines push an aeroplane forward with the idea of how motorized wheels would pull the plane forward.

Planes are not pushed to takeoff by motorized wheels.

If you imagine a curtain rod running the length of the runway, from which the plane is suspended by a chain (which can move along the length of the rod) you see that the treadmill below is irrelevant to how the engines push the plane forward.

Regards,

Vic

P.S. An ordinary car with wings would not take off because the car is driven by wheels, which push against the runway for speed.

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heh. a car with wings at a negative angle of attack could accelerate to say 400 knots. at least in this thought experiment. when the pilot pulls back on the stick the car with wings would take off. but it would not fly. because as you point out the wheels are no longer generating thrust.

as for why a plane can or cannot take off on a treadmill… the math is here http://boards.straightdope.com/sdmb/showpost.php?p=7162071&postcount=45
enjoy.

the non-math summary:
if you assume the treadmill matches the plane’s airspeed then the plane clearly takes off.
if you assume the treadmill matches the planes’s tire’s speed then the plane clearly does not take off. but the reasons why not are usually confused. the treadmill does not apply any force to the plane. it can’t. not significantly anyway. the plane is physically able to move. but if it does move then the belt speed is no longer equal to the tire speed. and a premise of the problem is violated. therefore the correct conclusion is that the premise is wrong. ie the belt speed cannot be set to the tire speed when the plane is moving relative to the air. which would be required for takeoff. the incorrect conclusion is that the treadmill holds the plane still. presumably by magic. or equivalently, closed physics models. or two. ;->

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No. It’s nearly a perfect analogy.

It’s two closed systems working against each other. With the landing gear as the connection.

Sheeessss. Put a Cessna 172 on a runway.

What type of force is exerted on it’s wheels and landing gear at a take of speed?

Compared to the force that the engine needs to pull the plane forward, and get air over its wings it’s nothing.

Nothing at all.

Now. The force on our 172 landing gear certainly counts for something. Double the speed on the gear and it’s going to be more. But not much.

Does any one believe that ANY airplane could not take off because of the effect that the landing gear is spinning twice as fast as normal?

IT WON’T MATTER. It would be a ***** to steer before transition, but the plane would take off. It has ground speed. Air over the wings. That’s all that matters.

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Ah, now I see that each of us sees here what we wish to see. I hang my plane from shower curtain rods and others believe wheel drag shall pin the plane down. Some do math and others draw pictures. I think there is much more utility (and truth) in inebriation and celebration. If any of you are in NY, you can join my krewe at the Alt.Oscar Party Sunday in chelsea at http://www.costumeculture.org/

It is very politically incorrect. That is easy for me because I was not intelligently designed. I definitely evolved from a monkey.

I’m done geeking out now. Time to go rock star.

Regards,

Squid Vicious