747 on a treadmill conundrum..

[lifesagift]

All that matters in this conundrum is the speed of the plane's wing relative to the air.

That's it.

As long as the speed of the air over the wing is below the level to achieve the necessary lift, plane won't become airborne, no matter how fast those wheels are a-spinnin'.

This is kinda basic.

The conundrum knows that air needs to pass over the wings in order to create lift.. we all know that. So the question is how is the air (pushed by the engines) related to the treadmill (which is governed by the wheel speed)?

[JesusHChrist]

The conundrum knows that air needs to pass over the wings in order to create lift.. we all know that. So the question is how is the air (pushed by the engines) related to the treadmill (which is governed by the wheel speed)?

No, that is not the question. From the OP:

"Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?"

If wheel and treadmill speed match, velocity of the wing through the air = 0. How could it be otherwise?

Sounds like you're getting things mixed up.

Again, basic.

[Diablo]

From the OP:
"The conveyor belt is designed to exactly match the speed of the wheels"

Case 1: the reference for the speed of the wheel is the tire contact patch. The treadmill top surface is stationary with respect to the airport lounge. I'll call the stationary treadmill "the runway." This is a normal takeoff.
The contact patch of the tire's speed matches the (stationary) treadmill as the plane accelerates away due to thrust from its propeller, fan or reaction motor (jet or rocket) and lifts off normally. The wheel rotates at airframe speed/wheel circumference (in radians/2pi sec).

Case 2: the reference for the speed of the wheel is the axle of the wheel which translates at the same speed as the airframe as the plane accelerates away and lifts off normally. The top of the treadmill also translates at airframe speed. The wheel does not rotate.

Case 3: the reference frame for the speed of the wheel & treadmill is the topmost point on the wheel circumference. For this point to be stationary with respect to the treadmill contact patch, the wheel cannot rotate. This case must be identical to case 2 where the treadmill runs at airframe speed. The aircraft accelerates away and lifts off normally.

Other cases involving rotated reference frames are too silly and will not be considered. I'm also ignoring the difference between velocity and speed for case 3.

In each case the treadmill has to be long enough for the plane to reach liftoff speed, v2min or greater.

I just take case 1, and I think that this is the instantaneous speed which is zero, but you would have to consider the speed over time, so this is not valid.

[lifesagift]

The fact that people with great scientific ability are disagreeing suggests that we should maybe look at the question rather than the answer?!

[Diablo]

I suppose there's another point. The OP doesn't mention the issue of the wind speed. If a gale was blowing hard enough then the plane would get enough lift to take off and could turn around and fly off.

---

There's another way to look at this. Maybe the OP means that the wheels on the conveyor move at the same speed - angular velocity - as the plane's wheels. But one or the other may be bigger so would move further. If the plane's wheels were a bigger diameter then for each revolution they would move further and the plane would accelerate and take off. If it was the other way around the plane would go backwards and crash into the control tower.

[pocaracas]

The way I remember hearing about this "problem", a few years ago, it was with a treadmill that moves backwards so as to counteract the forward motion of the plane. It doesn't matter if it rolls as fast as the wheels, or faster than the wheels, it just moves at the speed of the plane itself, in an attempt to push it backwards, thus keeping the air around the wings always the same and, thus, the plane wouldn't take off.

However, if we assume that the plane is already moving, there is already some air going through the wings. The plane is moving due to its own engine, so, no matter how fast the treadmill pushes back, considering the wheels are free to roll as fast as they may (no drag in the axles), the plane will keep moving through the air and the treadmill's influence will be negligible.

Even if the plane starts from a resting state, the treadmill should not influence the motion of the plane through the air, because it will just cause the wheels to spin twice as fast. Yes there will be a grater drag force, keeping the plane attached to the treadmill, but remember, the treadmill will only move as fast as the plane, never faster. The plane will never go backwards, due to the treadmill. So, sooner or later, the engine's power overcomes the ground drag force and the plane moves forward through the air.

[Jenny A]

All that matters in this conundrum is the speed of the plane's wing relative to the air.

That's it.

As long as the speed of the air over the wing is below the level to achieve the necessary lift, plane won't become airborne, no matter how fast those wheels are a-spinnin'.

This is kinda basic.

Yep. That's why I like my kite analogy. If there's wind, the kite flies. If there isn't you can get it in the air by running across the ground because that moves the kite relative to the air. But, if there's no wind and you run on a tread mill the kite will still just lay there. Set up a big enough fan and you can lift either the plane or the kite.

[lifesagift]

The conundrum knows that air needs to pass over the wings in order to create lift.. we all know that. So the question is how is the air (pushed by the engines) related to the treadmill (which is governed by the wheel speed)?

No, that is not the question. From the OP:

"Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?"

If wheel and treadmill speed match, velocity of the wing through the air = 0. How could it be otherwise?

Sounds like you're getting things mixed up.

Again, basic.

Lol I am the OP and the questioner

[FreeTony]

Even if the plane starts from a resting state, the treadmill should not influence the motion of the plane through the air, because it will just cause the wheels to spin twice as fast. Yes there will be a grater drag force, keeping the plane attached to the treadmill, but remember, the treadmill will only move as fast as the plane, never faster. The plane will never go backwards, due to the treadmill. So, sooner or later, the engine's power overcomes the ground drag force and the plane moves forward through the air.

The treadmill will influence the drag on the plane. It will blow the tyres up if they spin too fast.

As I said in a previous post, if the treadmill has to try and match the speed of the tyres, it will rapidly tend to an infinite speed, not just double. As per previous post:

1. Plane starts to roll down the treadmill. Wheel speed is say 1mph, treadmill is zero.
2. Treamill reacts by rolling 1mph backwards.
3. Since they are in contact, this must increase the wheel speed.
4. Which in turn means the treadmill must again try and match the speed.
5. This loop rapidly reaches a speed that will blow the tyres up. Even if they don't blow up, the friction increase will be enough to sttop the aircraft.


If however the treadmill starts at a backwards speed of Vrotate (the speed the a/c lifts off) and maintains this, the aircraft will probably take off fine as long as the tyres can handle it. In this case the maximum tyre speed is 2*Vrotate.

[lifesagift]

I think as someone said.. treadmill = tide = groundspeed, therefore the treadmill/wheel thing is designed to rule out the contact with the floor..