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david.atwell

The Airplane-Treadmill Conundrum

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Here's a wild new internet phenomenon- this is the problem, as originally posited:

An airplane is placed upon a very large treadmill with a very powerful motor. The treadmill is built to match the speed of the airplane exactly in the opposite direction. The airplane is then brought to full power by a qualified pilot.

Will the airplane take off?

There is a correct answer to this question, as well as a practical answer to this question. I'll tell you what it is later, but first, what do you think?

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:?

I am going to use very basic knowledge here without prior knowing of anything about this...

First of all, a windmill is used to gather energy from the wind using the turbines. The engine in the turbines may or may not allow for the windmill to go as fast as the airplane, but for the benefit of this question it can. Thus yes, the plane can take off unless there are insane of amounts of wind.

Now if it was a very large fan it would be different :)

I iz dumbs.

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That's a treadmill, not a windmill. A treadmill is something like this:

fitnex-t30-treadmill.jpg

If Mythbusters proved it, I guess it's true. But from my primitive-to-none-physics/aerodynamics-educated mind, I can't seem to find how. Isn't the airplane's velocity that determines whether it will fly or not? And if it's not moving, shouldn't it have a velocity of 0 (v of plane - v of treadmill = 0?) Just a shot in the dark. :?

Edit:

NASA[/url]":1m45oqg0]

NO MOTION, NO LIFT

Lift is generated by the difference in velocity between the solid object and the fluid. There must be motion between the object and the fluid: no motion, no lift. It makes no difference whether the object moves through a static fluid, or the fluid moves past a static solid object. Lift acts perpendicular to the motion. Drag acts in the direction opposed to the motion.

So, apparently, it has to be moving in some direction.... :|

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Whoops! My bad.

Anyways, the plane in that case is moving.Two forces are acting upon each other, two opposing forces that is. However, the plane is still moving as fast as it was as though it were on the ground.

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For the purposes of this discussion, we will assume the wheels are frictionless.

Airplane wheels are not powered. A plane's motion, including taxiing, is generated by it's jet/propeller/turbine/warp nacell/whatever. The velocity of the wheels is nearly irrelevant to the motion of the plane, other than a small amount of friction. The force from the treadmill will act in opposition to the wheels, yes, but that just means the wheels will be spinning twice as fast as they normally would. Since the wheels are the only point of contact with the treadmill, and they are frictionless, the net force on the wheels does not transfer to the plane itself; it just makes them spin faster.The plane's acceleration will be the same as it would have been without the treadmill. The plane takes off as normal. Whoever built the treadmill is very angry that they wasted all that money on a half-baked diabolical scheme.

In actuality, there will be friction between the wheels and their axles (and/or between the axles and the plane, as the case may be) as well as friction between the tire and the treadmill (otherwise they wouldn't spin), but that's just peanuts to the forces produced by the plane's method of propulsion. The jet/whatever is orders of magnitude more powerful than that friction. Unless the plane is rubber-band powered.

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Or we could think about this a different way, as both are still. v of plane - v of treadmill would be correct (I think), however, what if both are already nough? The, there would be nothing to hinder the plane, allowing it to gather velocity and take off.

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There's a bit of ambiguity in the problem. It says that the treadmill matches exactly the speed of the airplane in the opposite direction. Now, does it mean the speed of the airplane's fuselage, or the radial speed of the airplane's friction-laden wheels?

pyrochild's got the practical answer, based upon the assumption in the original problem: that "speed of the airplane" means "speed of the airplane's fuselage." Yes, the plane will take off; airplanes are not given thrust through their wheels, but through the jet engines or propellers attached to the fuselage, which are designed to give the plane airspeed, not ground speed. So no matter what goes on underneath the plane, the engines are independent; they will accelerate the plane to takeoff speed easily, just as they ordinarily would, with the wheels spinning wildly at twice the speed they would be going on a flat, unmoving tarmac.

But what is the other answer, based upon the other reading of the problem - that "speed of the airplane" means the same thing that your car's speedometer measures - the radial speed of the wheels?

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An airplane on a treadmill!? :shock:

You'd need a big enough treadmill to even fit an airplane on one :P

That aside, if

that "speed of the airplane" means the same thing that your car's speedometer measures - the radial speed of the wheels?

then the speedometer would still show the correct speed, because just because the treadmill is going in the exact opposite direction at the same speed, it doesn't mean that the wheels are rotating slowly, but infact at the speed at which the turbine is making the wheels spin. I think that makes sense. :?

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I don't get it.

From what I remember about how things fly, they depend on the lift from their wings in order to rise into the air. The lift comes from the air that is flowing over the aerodynamically built wings. Now from what I understand the plane is going one way and the treadmill is going the other way at the same speed, so therefore the fuselage would be standing still. Where would the air movement come from to give the air-o-plane the lift that it needs.

The only thing that I can think of is that maybe the jet, prop, or that other thing that pyrochild mention along with the action of the treadmill dragging some air along with it, would give it enough wind to get some lift.

I have to wounder, who thinks of these things?? :?

(Who paid for the treadmill?) :shock:

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Other solutions:

~I'm sick of these motherhugging airplanes on these motherhugging treadmills.

(How odd. Firefox marks the second as misspelled, but not the first.)

~The question is irrelevant. Everything is better on treadmills. Case closed.

~What about a helicopter on a turntable?!

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In your alternate scenario interpretation, would the wheels' and treadmill's velocity not attempt to approach infinity?

Ding ding ding! :-) In the alternate scenario, the treadmill would be wired to attempt matching the plane's ground speed - thus, as the airplane accelerated, the treadmill would accelerate twice as much (one measure to account for the additional speed of the plane, the other to account for the additional speed the treadmill is imparting to it). The friction of the wheels would hold the airplane more or less stationary as the treadmill accelerated to infinity, opened a hole in the space-time continuum, and vanished into hyperspace.

Or, more realistically, as the treadmill accelerated to the breaking point and shattered itself. :-) (Yes, very aware of the irony inherent in discussing the "realism" of a runway-sized treadmill. Very aware. :-))

EDIT: What about the helicopter on the treadmill? ;-)

EDIT2: Right, turntable. That makes more sense. :-) Helicopter on a treadmill would get destroyed at one end, most likely...

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I find it singularly strange that there are STILL debates raging about whether the airplane would take off. You'd think that all the people who came up with the right answer (i.e. pyrochild) would have gotten through to the rest after all this time...

Helicopter on a turntable, now... that is interesting. I assume you mean that the turntable is turning in the opposite direction of the helicopter's rotors at the same RPM? So, if the rotors are going counterclockwise at 4000 RPM, the turntable is spinning clockwise at 4000 RPM? In that scenario, I don't think the helicopter would take off. A helicopter's lift comes from its rotor pushing the air downwards, causing the rotors, and hence the helicopter, to move upwards. However, if the turntable is spinning opposite the rotors, the rotors' angular velocity with respect to the ground (and the air) would be null. Hence, since the rotors are not moving in the air, the air doesn't move, so the helicopter doesn't go up, and all you end up with is one dizzy helicopter pilot. This, of course, ignores the effects of the rotation on the helicopter (i.e., the thing should fly apart if it's spinning that fast), and the effect of the helicopter's rotation on the surrounding air (i.e., it would create a lot of wind in other directions). The effect would be the same if a propeller-driven aircraft were to be rotated around its longitudinal axis at the same RPM the propeller was rotating, and in the opposite direction.

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But what if the rotors were locked into stationary position relative to the copter, and the turntable spun in the direction the rotors should be going at a sufficient speed? Obviously, there'd need to be some sort of clamping device in place to allow it to continue gaining momentum throughout. But still...

Oh, and pyrotechniques - I got the correct answer, too. :-)

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The airplane teadnill thing wouldn't work, there is no air moving past the wings, therefore no lift. Simple physics! :D

Well, you're right and you're wrong. Have you even read the rest of this thread?

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Actually, it would work, because there is air moving past the wings. If, that is, you're using a propeller-driven plane. The propeller creates a wind, running air over the wings. Oh, and it moves the plane forward, too, but we won't go into that right now.

It works with a jet too. The thing is, a jet's engine is pushing against the air, not the ground, and since the air isn't moving, the plane moves forward. In order to stop the plane from moving forward, you would need to blow a 150-or-so mile per hour wind at the plane. If you did that, it would just rise straight up, because of the lift created by the wings.

@david.atwell: Well yeah, it would work if the rotors were locked, although if the copter was clamped to the turntable, it would have to pull the turntable up with it... and, as soon as it stopped rotating fast enough, the whole setup would fall down again. Yeesh. Just tie my brain in knots, why doncha? :D

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Argh, all these airplane/treadmill threads are so incredibly frustrating. The plane WILL accellerate as normal, (perhaps ever-so-slightly slower due to the increased friction from the treadmill) and take off. The wheels will be spinning twice as fast as normal, but this is irrelevent as the engine is delivering power to a propeller, which is dragging its way through stationary air.

As for a helicopter on a turntable, if you ignore the effects of the tail rotor trying to keep the heli from madly spinning, and also ignore the fact that spinning the entire helicopter at high speeds will probably throw the whole thing apart, no, it won't take off. (if the turntable is going the opposite direction to the rotor at the same RPM as the rotor) The reason being that the engine is trying to rotate the rotor at X rpm, and if you spin the heli in the opposite direction at the same speed, you end up with a stationary rotor blade and a very very rapidly spinning helicopter.

To make things even more confusing, if you put wheels on the helicopter in such a way as it could turn or pivot around it's axis, and factor in the tail rotor again, the turntable will spin but the heli will stay facing the same way (the tail rotor will counteract both the force of the main rotor trying to turn the heli, and the added friction of the wheels rolling on the turntable) Therefore, you end up with a rapidly spinning turntable, a stationary heli that is in contact with the table with some sort of circular array of wheels, and the main rotor will rotate as normal, allowing it to take off.

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Okay, the ultimate: Treadmill. Plane on top. Turntable on that. Helicopter on that. Which blows up first?

(Answer: my brain)

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