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Paradoxic 03-03-2006 03:26 PM

Plane on a Treadmill - SOLUTION
 
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. -- 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.)

Paradoxic 03-03-2006 04:27 PM

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.

zut 03-03-2006 04:38 PM

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.

Paradoxic 03-03-2006 05:25 PM

The Plane Will Not Take Off
 
Quote:

Originally Posted by zut
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.

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.

Quote:

Originally Posted by zut
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).

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?

Quote:

Originally Posted by zut
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.

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.

Ethilrist 03-03-2006 05:29 PM

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.

Paradoxic 03-03-2006 05:39 PM

Friction is Our Friend
 
Quote:

Originally Posted by Ethilrist
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.

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.

zut 03-03-2006 06:21 PM

Quote:

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

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.

Quote:

Originally Posted by Paradoxic
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.

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.

Quote:

Originally Posted by Paradoxic
You are making unwarranted assumptions. Please, don't make me have to take time out to demonstrate the math on this...!

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.

Quote:

Originally Posted by Paradoxic
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?

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.

Quote:

Originally Posted by Paradoxic
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...

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.

Quote:

Originally Posted by Paradoxic
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.

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?

Paradoxic 03-03-2006 08:31 PM

The Plane Does Not Take Off
 
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.

enipla 03-03-2006 09:28 PM

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.

flight 03-03-2006 09:32 PM

Quote:

Originally Posted by Paradoxic
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.

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:
Quote:

Originally Posted by me with a little tweaking
Most people are coming to correct solution, they just interpret speed differently and diverge from there.

Group 1 (Cecil's flying group) say that when you say speed you are talking relative to a nearby building or a section of ground that is not the treadmill.

Group 2 (the stuck to the groud group) define speed relative to the ground immediately beneath the plane, aka, the moving surface of the treadmill.

Thus, for group 1 there is an increasing backward force on the wheels of the plane as it moves faster relative to a stationary object, but nowhere near enough to stop it from taking off. The wheels are creating friction as if the speed of the plane relative to the ground were 2V.

For group 2 however, the plane has a speed V relative to the surface of the treadmill. The surface of the treadmill has a speed -V relative to a stationary object. Thus (neglecting relativity) the plane has a speed 0 relative to a stationary object. An observer of the plane that has his view of the treadmill obstructed will not notice any motion in the plane. With no motion there is no lift and therefore the plane stays on the ground, whatever V you are talking about.

So, depending on what you define speed relative to, you have two very different problems.

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.

jere7my 03-03-2006 11:41 PM

Quote:

Originally Posted by Paradoxic
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.

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.

enipla 03-03-2006 11:52 PM

Quote:

Originally Posted by Paradoxic
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. .

Yeah. And I want my Air Car. :D

Building it is not the issue.

Quote:

Originally Posted by Paradoxic
Fact 1: The force generated by an engine is limited.

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

Quote:

Originally Posted by Paradoxic
Fact 2: Friction is a force

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

Quote:

Originally Posted by Paradoxic
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)

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.

Quote:

Originally Posted by Paradoxic
THIS is what everyone has ignored, and this data is required in order for the system to be closed.

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.

Quote:

Originally Posted by Paradoxic
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.

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

jere7my 03-03-2006 11:57 PM

Quote:

Originally Posted by zut
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.

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.

mrrealtime 03-04-2006 08:39 AM

Quote:

Originally Posted by jere7my
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.

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

enipla 03-04-2006 10:30 AM

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

Quote:

Originally Posted by Wikipedia
By contrast, a closed (but not isolated) system can exchange heat and work, but not matter, with its surroundings.

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.

Quote:

Originally Posted by Cecil
Here's the original question: "A plane is standing on a runway that can move (some sort of band conveyer).

A runway that can move.

Hmmmm….. A runway that can move.

Quote:

Originally Posted by original question
The plane moves in one direction, while the conveyer moves in the opposite direction.

“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.

zut 03-04-2006 10:48 AM

Quote:

Originally Posted by enipla
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.

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.

Quote:

Originally Posted by jere7my
Agreed, with a quibble...

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.

VicSarjoo 03-04-2006 01:18 PM

Simple & Elegant Explanation - Plane Takes Off
 
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.

timmerov2 03-04-2006 03:28 PM

Quote:

Originally Posted by VicSarjoo
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.

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/...1&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. ;->

enipla 03-04-2006 06:31 PM

Quote:

Originally Posted by zut
Bad analogy, because the treadmill in this case isn't reacting to the plane's motion. That changes the problem considerably.

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.

VicSarjoo 03-04-2006 07:31 PM

I leave geeking to party now.
 
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

Paradoxic 03-04-2006 08:21 PM

ground speed? DOH!
 
Quote:

Originally Posted by enipla
Let’s say the airplane needs 100mph ground speed to take off...

This is what happens when you consider this as a thought experiment and not a physics equation. What if I've got a 100mph tail wind?

No airplane requires ground speed to take off. It requires airflow across it's wings of sufficient velocity to take off - or, to put it another way, it requires air speed, not ground speed.

The answer is not intuitive, and this fails as a thought problem unless you actually do the math.
  • Force generated by the engines directed in one direction.
  • Force generated by the treadmill in the opposite direction (how about a treadmill driven by the same engines as the plane uses?)
  • The forces meet and cancel out, at the wheel assembly (w/hot tires) and ground friction.

zut 03-04-2006 10:01 PM

Quote:

Originally Posted by enipla
No. It’s nearly a perfect analogy.

No it's not, really. Changing the treadmill from one that reacts to the plane wheel speed to a simple constant-speed treadmill is changing the meaning of the question.
Quote:

Originally Posted by enipla
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.

I already explained in my previous post about the two different interpretations of the question. As did flight. As did timmerov2.

As far as I can tell, there isn't anyone in this thread who disputes the fact that, if you interpret the question as meaning that the plane fuselage moves forward and the wheels spin at double speed, the plane will take off. There's nothing wrong with that interpretation, and nothing wrong with that answer.

However, much of the rest of the discussion in this thread centers around the alternate interpretation. There's nothing wrong with this alternate interpretation, either, but it does lead to different conclusions.

Quote:

Originally Posted by Paradoxic
The answer is not intuitive, and this fails as a thought problem unless you actually do the math.
  • Force generated by the engines directed in one direction.
  • Force generated by the treadmill in the opposite direction (how about a treadmill driven by the same engines as the plane uses?)
  • The forces meet and cancel out, at the wheel assembly (w/hot tires) and ground friction.

I'm still not sure if I agree with you, or you agree with me. Agreed that the plane requires airspeed, and the answer is not intuitive. However, you keep mentioning a treadmill "driven by the same engines as the plane uses," which makes no sense to me. It makes no difference what the power source driving the treadmill is, as long as it's large enough to provide the power necessary for accelerating the belt.

If you assume a treadmill capable of large velocities and accelerations, and assume that it tracks the wheel speed of the plane, the belt will supply a force to the plane to counteract the engine thrust. However, this requires some coupling through the wheels--the wheel inertia, for example. The scenario you're giving is a bit oversimplified because it skips the step describing the coupling mechanism.

Who_me? 03-05-2006 04:56 AM

Jeeze, not again...

The first problem is, as I see it, that the original problem set up impossible circumstances. Setting up a problem with impossibilities is kind of like dividing by zero... it sort of invalidates the whole thing.

There is no way for a conveyor belt to match exactly the speed of the rotation of the wheels, plus even if there was, that would set up a feedback loop with both the wheels and the belt going infinite miles per hour. Also, no matter what anyone thinks, the friction from the wheels turning would still be much less than the thrust of the engine... unless they were spinning at an infinite speed, I guess.

I just though of something... what if you locked the wheels altogether and poured water on the treadmill... then froze it. Hey! Then the plane could take off!

enipla 03-05-2006 09:52 AM

Quote:

Originally Posted by enipla
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.

Quote:

Originally Posted by paridoxic
ground speed? DOH!
This is what happens when you consider this as a thought experiment and not a physics equation. What if I've got a 100mph tail wind?


As you can see, I’ve already taken that into account. I’m looking at this as a physics question. The people that don’t believe the plane will take off (BF#2) see this as a thought experiment.

Quote:

Originally Posted by paridoxic
Force generated by the treadmill in the opposite direction (how about a treadmill driven by the same engines as the plane uses?)

Perfect. See below.
Quote:

Originally Posted by enipla
Originally Posted by enipla
No. It’s nearly a perfect analogy.

Quote:

Originally Posted by zut
No it's not, really. Changing the treadmill from one that reacts to the plane wheel speed to a simple constant-speed treadmill is changing the meaning of the question.

Where does the question state ‘Wheel Speed’?
Quote:

Originally Posted by ”The Question”
This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same.

bolding mine
Sorry, but you’re the one changing the question.

And in my analogy, I assumed that it was understood that as the toy car speed up, so would the treadmill. Still works.

To expand on my example of a toy car pulled down the runway……

Lets say the treadmill has rollers 6” in diameter. Attach a string from the toy car to a pully the exact same diameter, that is fastened to the ground. When we turn the pully, the toy car moves forward.

Now. Connect that pully to the 6” roller of the tread mill. But cross the belt that connects them. As the pully move clockwise, the roller moves counter clockwise.

Now, crank on the pully. It pulls the car forward. It moves the treadmill backward at the same speed the car is moving forward.

I don’t like this analogy as much as the person pulling the string, since this is a closed system. And the real world stuff would be two closed systems.

enipla 03-05-2006 10:43 AM

[quote= “The Question”)This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same.[/quote]

Actually, my example above with the pulley that pulls the toy car, and at the same time drives the conveyor backwards answers the question. And satisfies the constraints of the question as well.

The plane moves. If the plane can move, it can take off.

Hell, tie a cable on the back of the plane to the rear roller of the conveyor (you’ll have to wrap it underneath. See it?) As the plane moves forward, it pulls the conveyor backwards under its wheels. Here is another system that satisfies the constraints of the question and the plane takes off.

zut 03-05-2006 11:28 AM

Quote:

Originally Posted by enipla
Where does the question state ‘Wheel Speed’?
bolding mine
Sorry, but you’re the one changing the question.

Already covered:
Quote:

Originally Posted by zut
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.

From the first column:
Quote:

Originally Posted by Cecil
As you point out, one problem here is the wording of the question. Your version straightforwardly states that the conveyor moves backward at the same rate that the plane moves forward. If the plane's forward speed is 100 miles per hour, the conveyor rolls 100 MPH backward, and the wheels rotate at 200 MPH. Assuming you've got Indy-car-quality tires and wheel bearings, no problem. However, some versions put matters this way: "The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation."

You're assuming one version of the question. Nothing wrong with that. Paradoxic is assuming another. Nothing wrong with that either.

I suppose you could argue that your interpretation is more semantically correct for the given problem statement. However, note that in response to the same problem statement, Cecil himself explicitly states that "if the treadmill continues to accelerate" then "the plane stands still and doesn't take off." Admittedly a confusing flip-flop on his part, but it does illustrate the different interpretations of the question.

enipla 03-05-2006 12:05 PM

Quote:

Originally Posted by God help me I went back to the original thread
Imagine a plane is sitting on a massive conveyor belt, as wide and as long as a runway, and intends to take off. The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation.

I went way way back to the original question. Before Cecil addressed it.

I was wrong about the ‘plane speed/wheel speed’ thing. The original question states wheel speed.

I think someone in that thread showed that it is impossible for the treadmill to match the wheel speed unless everything was at a full stop. Meaning not movin/spinning at all. Unless, the conveyor could continue to accelerate and keep the plane from moving. To look at the question that way, the question answers itself. The only way for the planes wheel speed can match the conveyor speed is if the plane is not moving relative to the ground. So obviously, it would not take off.

Not sure what the question was really asking. If it concerned the ability of a plane to take off while not moving but with the engines at full power, the answer is so obvious that I can’t imagine anyone seriously asking it (as it turns out some folks didn’t get that either :stunned: ). Why even include the conveyor part at all?

Anyway, I thought that my example of the plane actually hooked to the conveyor pulling it underneath itself as it went forward at least showed a system where the conveyor matched the speed of the plane. Does this also mean the speed of the wheels?

:shrug:.

Depends. :D

Have a nice Sunday.

David Simmons 03-05-2006 05:09 PM

Quote:

Originally Posted by enipla
I went way way back to the original question. Before Cecil addressed it.

I was wrong about the ‘plane speed/wheel speed’ thing. The original question states wheel speed.

I think someone in that thread showed that it is impossible for the treadmill to match the wheel speed unless everything was at a full stop.

I'm quite sure the last statement is true. The wheel speed is always the aircraft speed plus the conveyer speed so unless the plane's speed is zero the conveyor and wheel can't have the same speed. it appears to be an unstable system. In order for the conveyer belt to move, the plane has to move. Once the plane moves in the slightest then the conveyer rapidly accelerates to its limiting speed, or to destruction whichever occurs first.

zut 03-05-2006 08:17 PM

Quote:

Originally Posted by enipla
Not sure what the question was really asking. If it concerned the ability of a plane to take off while not moving but with the engines at full power, the answer is so obvious that I can’t imagine anyone seriously asking it (as it turns out some folks didn’t get that either :stunned: ). Why even include the conveyor part at all?

For me, personally, I thought the answer to the question where the plane is allowed to move (with wheels spinning) was so obvious that the question wasn't worth asking--the answer is clearly what you stated, and how can anyone argue with that? In any case, interpretation of the question is key.

Quote:

Originally Posted by David Simmons
it appears to be an unstable system. In order for the conveyer belt to move, the plane has to move. Once the plane moves in the slightest then the conveyer rapidly accelerates to its limiting speed, or to destruction whichever occurs first.

Unstable in the sense that the system quickly bumps up against some real physical limitations. Stable, though, in the sense that the conveyor could theoretically hold the plane in position to within an arbitrarily small distance, and small perturbations in thrust only require small changes in belt acceleration to hold it in place. Until the system bumps into the real physical limitations, of course.

David Simmons 03-05-2006 08:46 PM

Quote:

Originally Posted by zut
Stable, though, in the sense that the conveyor could theoretically hold the plane in position to within an arbitrarily small distance, and small perturbations in thrust only require small changes in belt acceleration to hold it in place. Until the system bumps into the real physical limitations, of course.

You'll have to go through this process for me step by step. If the wheel speed (in the conveyer matches wheel speed case) is the plane speed plus the conveyor speed, how can the conveyor ever be going fast enough? It seems to me that if the plane moves at all, the converyer will just go to its maximum speed and stay there.

Plane and conveyor speed are relative to the ground. Wheel speed is the peripheral speed of the wheel.

wheel speed = plane speed + conveyor speed

conveyor speed = wheel speed - plane speed

Until the conveyor is going fast enough for the force required to accelerate the wheel equals the engine thrust the plane speed won't be zero. The conveyor speed then can't match the wheel speed because it is slower by the plane speed. Based on computations in this post I don't see how the required wheel acceleration force can be achieved before any reasonable, or even unreasonable, system limits out.

There is the problem of bearing friction. Such friction is usually taken to be a constant times the velocity. This will make the force opposing engine thrust grow faster than just the force required to accelerate the wheel alone. That reduces the required wheel speed for balanced forces. However the forces tending to throws the tire off the rim and the tire speed limit will probably be reached before force balance is reached even with friction helping the retarding force.

zut 03-05-2006 10:06 PM

Quote:

Originally Posted by David Simmons
You'll have to go through this process for me step by step. If the wheel speed (in the conveyer matches wheel speed case) is the plane speed plus the conveyor speed, how can the conveyor ever be going fast enough? It seems to me that if the plane moves at all, the converyer will just go to its maximum speed and stay there.

But remember that the conveyor responds to engine thrust with a change in acceleration, not just a change in velocity. For example, you calculated a nominal required acceleration in your linked post, based on some mass and thrust assumptions.

If we ignore physical limitations for a second, then what would happen in a perfect system would be that the plane's engines would light off, the treadmill would respond with the constant acceleration that you calculated, all the forces would balance, and the plane wouldn't move. Right?

In a less-than-perfect control system, there would be some lag in the response. So the plane's engines would light off and the plane would surge forward slightly before the treadmill had a chance to respond. The control system would register the increased wheel speed, and respond with a larger treadmill acceleration than the steady-state acceleration you calculated. The increased acceleration would transmit more force to the plane, slowing it down, until plane speed = 0. Then wheel speed = conveyor speed (even though both are still accelerating!) and everyone's happy.
Quote:

Originally Posted by David Simmons
There is the problem of bearing friction. Such friction is usually taken to be a constant times the velocity. This will make the force opposing engine thrust grow faster than just the force required to accelerate the wheel alone. That reduces the required wheel speed for balanced forces. However the forces tending to throws the tire off the rim and the tire speed limit will probably be reached before force balance is reached even with friction helping the retarding force.

Can't argue there. Interpreting the problem this way (with an ever-accelerating belt keeping the plane stationary) is pretty clearly a thought experiment, and not practical at all. Unless...I recall someone in the previous thread suggesting an RC model outfitted with very heavy wheels. It's possible something like that would be test-able.

David Simmons 03-05-2006 10:43 PM

Quote:

Originally Posted by zut
But remember that the conveyor responds to engine thrust with a change in acceleration, not just a change in velocity. For example, you calculated a nominal required acceleration in your linked post, based on some mass and thrust assumptions.

If we ignore physical limitations for a second, then what would happen in a perfect system would be that the plane's engines would light off, the treadmill would respond with the constant acceleration that you calculated, all the forces would balance, and the plane wouldn't move. Right?

In a less-than-perfect control system, there would be some lag in the response. So the plane's engines would light off and the plane would surge forward slightly before the treadmill had a chance to respond. The control system would register the increased wheel speed, and respond with a larger treadmill acceleration than the steady-state acceleration you calculated. The increased acceleration would transmit more force to the plane, slowing it down, until plane speed = 0. Then wheel speed = conveyor speed (even though both are still accelerating!) and everyone's happy.

I would like to see a computation. I think you are describing a Type 1 velocity system that will track a constant velocity with zero error and I'm not sure that this setup fits that description.

zut 03-06-2006 07:49 AM

I'm assuming a system where the force from the engine thrust is balanced by the force transmitted through the wheels due to the acceleration of the treadmill (and the inertia of the wheels).

I'm not sure what calculation to do here, because the most applicable calculation is the one that you did yourself in the other current thread. Just take a look at that calculation and ask yourself what would happen if the treadmill accelerated more than the acceleration required to keep the plane stationary.

David Simmons 03-06-2006 08:20 AM

Quote:

Originally Posted by zut
I'm assuming a system where the force from the engine thrust is balanced by the force transmitted through the wheels due to the acceleration of the treadmill (and the inertia of the wheels).

I'm not sure what calculation to do here, because the most applicable calculation is the one that you did yourself in the other current thread. Just take a look at that calculation and ask yourself what would happen if the treadmill accelerated more than the acceleration required to keep the plane stationary.

I think you're right. No matter the details of the mechanization, an acceleration of 1245 rad/sec2 is needed to balance a thrust of 500 lb given the assumed wheel inertia. That means that the wheel and the conveyor will almost instantly, say within 1.5 sec., reach entirely unrealistic speeds. As far as I'm concerned that makes BR#2 a fantasy world scenario.

zut 03-06-2006 10:03 AM

Well, I prefer "the realm of a thought experiment" to "a fantasy world scenario," but, yeah, we're talking about a situation that isn't really testable on a real aircraft given real-world limitations.

Ethilrist 03-06-2006 11:45 AM

This never ceases to amaze me. I think, in the first article he wrote, Cecil pointed out that this was intended as a thought experiment, not to find out whether or not the plane would take off, but to find out how people look at puzzles and try to solve them.

Everybody's right.
Everybody's wrong.

Interpetation #1: Assuming real-world materials and a vaguely-stated supposition that the runway moves backwards as fast as the plane moves forwards, the plane takes off, because making the wheels spin twice as fast doesn't exert enough force to stop the plane. BR #1.

Interpretation #2: Assuming non-real-world materials and the iron-clad fact that somehow the treadmill exerts enough force to keep the plane from moving, the plane does not take off, because non-moving planes don't take off without a lot of wind, and that's not in the puzzle. BR #2.

So, for all of you arguing that real-world materials behave a particular way and even if you melted the tires off the plane, it would still screech down the treadmill and take off, congratulations. You're right.

And for all of you arguing that the question states that the treadmill keeps the plane from moving, congratulations. You're right, too.

And for all of you arguing that the treadmill, moving fast enough to satisfy group #2, generates enough wind due to the laws of contagion such that the wind passing over the wings causes the plane to lift off, you're right, too, but you're cheating. :)

bup 03-06-2006 12:08 PM

Quote:

Originally Posted by jere7my
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.

I see at as asking a question which contains a paradox, like Cecil's first column says.

It's like the question, "A car travels 50 miles of a hundred mile trip at 30 mph. How fast does it need to go for the last fifty miles to average 60 mph?"

Implicit in the question is an impossibility.

I guess a more accurate answer would be 'mu,' but I'm more inclined to say you can't make a treadmill that stops a plane's forward movement, therefore the plane takes off.

David Simmons 03-06-2006 11:26 PM

Quote:

Originally Posted by zut
Well, I prefer "the realm of a thought experiment" to "a fantasy world scenario," but, yeah, we're talking about a situation that isn't really testable on a real aircraft given real-world limitations.

I've been thinking over this thought experiment in my spare time. Since I am retired I have a lot of spare time.

I think it's a lousy thought experiment. Such experiments are those that you can do theoretically but not practically. I don't think you can even do this one theoretically.

Take the case where the only force opposing the engine thrust is the acceleration of the wheels. In order to stop the plane indefinitely you must constantly accelerate the wheels and that is not possible.

In the case where viscous friction is also included the scenario would be to run the wheel speed up until wheel acceleration force plus the viscous friction force balances the thrust. If the acceleration is then stopped the force immediately becomes merely the friction force and the plane moves forward. If the acceleration is not stopped the plane moves backward but here again you can't maintain constant acceleration indefinitely.

I think BR#2 is a loser.

zut 03-07-2006 07:42 AM

Quote:

Originally Posted by David Simmons
I think it's a lousy thought experiment. Such experiments are those that you can do theoretically but not practically. I don't think you can even do this one theoretically.

Take the case where the only force opposing the engine thrust is the acceleration of the wheels. In order to stop the plane indefinitely you must constantly accelerate the wheels and that is not possible.

Why not? The realistic limiting factors that I can think of are: a) material limitations of the tire and bearings and belt and so forth, b) velocity limit of the motor or whatever power source moves the treadmill, c) compressibility effects (sonic shock waves) when the conveyor exceeds the speed of sound, and d) speed of light.

To turn this into a thought experiment, I'd be comfortable ignoring all of these (and Chronos pointed out that the last one--the speed of light--isn't a problem).

In any case, a thought experiment is useful in exploring the relative magnitudes and effects of various causes. To my mind, this airplane question is similar to "what happens when you approach the speed of light?" When you ask that question, you don't get hung up on how long it takes, or what kind of vehicle you travel in, or where the fuel comes from, or any of those kind of questions.

wildwriter 03-07-2006 07:57 AM

[QUOTE=VicSarjoo]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.

No, this is NOT AT ALL like this sort of situation. To create the situation of your "set of steel chains" the plan has to move through the air, to create the lift. Until the plane reaches a speed relative to the air mass around it (or, equivalently, to a stationary observer also not being moved by the treadmill) the wheels sit on the conveyor belt and tend to get moved backward alolng with the surface of the conveyor belt.

This is not a force-vector calculation (except ina VERY abstract way) and it is not a closed-vs-open environment. It is a semantic/logic exercise and subject to the same problems of any other logic problem. The key difficulty is sometimes called "assumptions contrary to fact" wherein we have put forth assumptions (that it is possible to build a conveyor belt that can move at a speed opposite to (and synchronize with) a forward motion of a plane) that are contrary to fact.

wildwriter 03-07-2006 08:03 AM

OBTW, when the pilot stands on his brakes and revs the engines, I don't THINK he's generating a force equal to the total thrust of the engines, just enough force to stop the wheels from turning. Maybe it has something to do with inertia, et al, but it is beyond my aging Physics education.

flight 03-07-2006 11:31 AM

Quote:

Originally Posted by wildwriter
OBTW, when the pilot stands on his brakes and revs the engines, I don't THINK he's generating a force equal to the total thrust of the engines, just enough force to stop the wheels from turning. Maybe it has something to do with inertia, et al, but it is beyond my aging Physics education.

Enough force to stop the wheels from turning is exactly the same thing as the total thrust of the engine (assuming the engine is horizontal). Most planes can produce more thrust when at higher speeds though.

Chronos 03-07-2006 01:47 PM

Quote:

Fact 4: The conveyer, as it moves faster, exerts more and more backwards force upon the wheels, and this force is not inconsequential.
Could you please give us the equation for frictional force as a function of relative speed? Because all of the models for dry friction with which I'm familiar have the force being approximately constant with speed. In other words, it's possible to construct a system such that the frictional force would be enough to prevent the plane from taking off, but such an airplane wouldn't be able to take off from a normal runway, either.

jere7my 03-07-2006 02:50 PM

Quote:

Originally Posted by zut
To turn this into a thought experiment, I'd be comfortable ignoring all of these (and Chronos pointed out that the last one--the speed of light--isn't a problem).

I'm not actually convinced of that, since the wheels are rotating, and thus a non-inertial frame of reference. General relativity would come into play at some point. For that matter, proper acceleration is only applicable for someone attached to the conveyor belt; passengers in the plane would see Lorentz effects as the belt accelerated relative to their frame of reference. The belt would be able to accelerate as long as it wanted without bumping into c, but the motionless plane would see the speed of the belt top out at c, long before which it would contract until it was too small for the plane to stand on. :P (I don't want to think about what would happen to the wheels, rotating up to c.)

Paradoxic 03-07-2006 09:31 PM

The Plane Cannot Take Off
 
Quote:

Originally Posted by Chronos
Could you please give us the equation for frictional force as a function of relative speed? Because all of the models for dry friction with which I'm familiar have the force being approximately constant with speed. In other words, it's possible to construct a system such that the frictional force would be enough to prevent the plane from taking off, but such an airplane wouldn't be able to take off from a normal runway, either.

Well, I'm trying not to get too much into this without taking away from my present work (let's hear it for PowerPoint!). But O.K., without having to drag out my first year Engineering Statics and Dynamics text book, here's a little something to chew on, using a 747. Data is taken from the Boeing website, as well as Goodyear Aviation.

Using one common 747 configuration, the aircraft at takeoff is 130 tons. ~80% of that weight is distributed on the aircraft's 16 main gear tires, which are sizable, 49" tall and 19" wide - the designation is H49x19.0-22, for aircraft tires the second number is the nominal width, not aspect ratio as in passenger cars. (oh, each tire costs $3,600).

Length of the tire's ground footprint under load is 20% of it's circumference.
49" x PI x 20% = 31"
footprint is 31" x 19.0" = 589in˛


With 65 tons of downward force on each of the main tires, there is

65/589 = 220lbs/in˛

on the contact surfaces. Now, at this point I haven't gotten the time to look up the coefficient of friction (µ) between the tires and the tarmac, but it's pretty damn high... µ is the tangential force required to create slippage, divided by 220lbs/in˛. It's gonna be pretty damned high, high enough that there will be no slipping of the tires on the tarmac. So we know that the tires and the tarmac will roll together, with a resultant significant backwards force on the tires.. This means that a sufficiently powerful treadmill can provide enough backwards force to keep the plane on the ground, because the rolling resistance plus to friction between the tarmac and the treadmill is too significant to overcome.

Now, as far as the landing gear assembly is concerned, guess what? The frictional resistance of the tire assemblies (ball bearings, etc) is pretty considerable until the plane starts going down the runaway and some lift is provided by the wings. As the plane's wind speed increases, the plane's relative weight on the wheel assemblies decreases, and frictional resistance decreases, etc.

What does this all mean?

If the plane, from the very get-go, does not generate any air speed, the frictional resistance of the landing gear, added to the backwards force generated by the treadmill - it's possible that, on a treadmill, with the plane at full thrust, the plane will not only be stationary, but the wheels may be spinning at significantly less than they would at takeoff speed - maybe even a speed relative to the treadmill of less than 50mph, far less than the average takeoff speed of 155mph.

According to Goodyear aviation: "Heavy loads and high speeds cause the heat generation in tires to exceed that of all other tires and can have a very detrimental effect...rubber dissipates heat slowly...for this reason, aircraft tires and only be used intermittently.". So, rotate the tires too fast and they burn up. Think about that.

So, a 747 cannot take off, nor can any other commercial ariliner. But guess what? Start plugging in these figures for a light and powerful plane, and the problem appears differently.... a commerical airliner can't take off, but maybe a jet fighter can?

The moral of this story is that people that like non-engineering thought experiments shouldn't build airplanes.

David Simmons 03-07-2006 10:31 PM

Quote:

Originally Posted by zut
Why not? The realistic limiting factors that I can think of are: a) material limitations of the tire and bearings and belt and so forth, b) velocity limit of the motor or whatever power source moves the treadmill, c) compressibility effects (sonic shock waves) when the conveyor exceeds the speed of sound, and d) speed of light.

To turn this into a thought experiment, I'd be comfortable ignoring all of these (and Chronos pointed out that the last one--the speed of light--isn't a problem).

In any case, a thought experiment is useful in exploring the relative magnitudes and effects of various causes. To my mind, this airplane question is similar to "what happens when you approach the speed of light?" When you ask that question, you don't get hung up on how long it takes, or what kind of vehicle you travel in, or where the fuel comes from, or any of those kind of questions.

I think it's finally getting through to me that you think BR#2 is a neat problem.

zut 03-08-2006 12:15 PM

Yeah, pretty much. Exactly what happens isn't intuitively obvious, I don't think. There's also a lot of effects in play, so you can get into a whole string of, "what happens?" "Well, this does." "But ignoring that, what happens?" "Well, this does." "But ignoring that, what happens?"

I think it's sort of an interesting exploration of forces and motions, examining them by pushing the thought experiment to its boundaries. I imagine most other people think of it as a tiresome exercise in semantics, but my enthusiasm is undampened. :)

Chronos 03-08-2006 03:46 PM

Very impressive, Paradoxic. But I didn't see any speed dependance anywhere in there. We are agreed, of course, that there is some amount of friction in the landing gears. I think we're also agreed that a plane on a normal runway can overcome that friction, start moving, accelerate forward, and eventually take off. I assert that the friction in the treadmill case will be no greater than the friction in the runway case, and that the plane will therefore still accelerate forward and eventually take off. You assert, however, that the friction will increase with increasing relative speed. You still haven't given any support to that assertation, however.

NoelG 03-12-2006 02:11 AM

First post (Yay, me.)

K, on this topic: so many wasted words for such a simple question. When conducting mind experiments, one doesn't worry the mechanics. Thus, Einstein could hop on a train that could approach the speed of light, etc.

The simple and obvious answer to the OP is: Given that a treadmill can be constructed to keep the plane stationary, the plane will not take off. No lift is being generated, lift coming from airflow over the wings. QED.

flight 03-13-2006 01:22 PM

Quote:

Originally Posted by NoelG
First post (Yay, me.)

K, on this topic: so many wasted words for such a simple question. When conducting mind experiments, one doesn't worry the mechanics. Thus, Einstein could hop on a train that could approach the speed of light, etc.

The simple and obvious answer to the OP is: Given that a treadmill can be constructed to keep the plane stationary, the plane will not take off. No lift is being generated, lift coming from airflow over the wings. QED.

First, welcome to the boards. Second, reading the whole thread would help to avoid problems like this. As has been stated by multiple level headed people you are answering a different question than the people who think the plane will take off. You are defining the speed of the plane relative to the moving surface of the treadmill. The others are defining it relative to an immobile object. Think on this young one, and return with wisdom.


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