Aircraft and conveyor belt (again)

[Observer]

I was very interested in the recent (now locked) thread on the 777 incident that re-opened the debate on the jet plane on conveyor belt problem. Congratulations to a number of posters who gave very clear explanations and illustrations of the correct answer - being, of course, that the aircraft will take off. However, I think there is an inherent paradox in the problem (touched on by JohnD) and also believe the assertion that aircrraft wheel rotational speed will be 2 x aircraft forward speed is wrong. I acknowledge the very helpful analogy, given by BAS, of the super-engineered toy car on a supermarket checkout conveyor belt, which (i found) makes it much easier to visualise what's happening.

So - imagine the toy car with frictionless wheel bearings on the belt. I am standing alongside the belt and supporting the rear of the car so it cannot move backwards. I start to walk forwards at 2 mph. Necessarily the car, supported by the 'thrust' of my hand moves along the belt at the same speed, so its wheels have a rotational speed of 2 mph. In the conditions given by the OP, the belt must match the wheel speed so the belt starts to move at 2 mph in the opposite direction. However, the thrust I impart by supporting the car while walking forward is still maintained so the wheel rotational speed must increase to belt speed + 2 mph and so the belt speed must also increase and so on. It seems that if I maintain a forward thrust on the car, the belt speed and the wheel rotational speed will continue increasing towards infinity. That's the paradox.

But, if the belt is continuously accelerating towards infinite speed, the rotational speed of the wheels at any moment must be equal to belt speed PLUS the forward velocity imparted by the 'thrust' of my forward movement (i.e not 2 x the forward velocity).

Not absolutely sure I've got this right but still finding this interesting.

 

[Observer]

This has degenerated and is now closed

Correction. That thread had degenerated and was closed. This is a new one. I'm not arguing against the consensus (if it was a consensus) of the original thread but looking to lengthen the debate on the detail of what's happening to the wheel speed. A related but not identical issue. Do you have moderator powers or are you acting as self-appointed censor?

 

[imamartian]

Thinking about this from a different perspective, surely the jet on the conveyor acts exactly as a hovercraft (i.e. the wheels and conveyor have a combined zero friction), and would therefore move forward and take off.

However, if the air around and behind the jet was in some way attached to the conveyor and moving back with it (which is where some peoples' reasoning gets stuck), then the thrust from the jet would be negated and the jet would not move.

 

[Trazor]

You have to remember, that in order to overcome the friction in the wheels, and hold station, the jets need only apply a very small amount of power.

Once the jet engines are opened up to anywhere near full power , off goes the jet.

 

[imamartian]

also, if the wheels on the plane were drive wheels, then the forward motion would be negated by the conveyor.

another analogy...you are the plane and you are running on a treadmill. As fast as you run the treadmill matches your speed, so you stay in the same spot. Someone comes up behind you and pushes you in the back with a broom handle, you would be pushed forwards and fall off the front of the treadmill, however fast your legs ran.

 

[Jesus H. Christ]

I get the distinct feeling that someone will be summoning me soon.

 

[Observer]

Hovercraft on conveyor belts and runners on treadmills are distinctly inapt analogies for the original problem, which in any event was not the point of this thread. Stick with the toy car on the supermarket conveyor belt. It makes it really easy to work out what's happening.

 

[Softus]

It seems that if I maintain a forward thrust on the car, the belt speed and the wheel rotational speed will continue increasing towards infinity. That's the paradox.

The car is moving at a fixed speed (relative to, say, the end of the belt).

The wheels are going at whatever speed they are going.

The belt is going at whatever speed the wheels are going.

How, exactly, is that a paradox?

JHC

 

[Jesus H. Christ]

Aha! I just knew it.

Truly, I am Spartacus.

And now I must go. Spice racks don't make themselves.

 

[imamartian]

With respect, i did not say a hovercraft on a conveyor belt.

And secondly telling me my analogy is inapt without an explanation why; and also telling me to stick to another analogy because i'll understand it better i find quite patronising.

And then you tell me i missed the point of the thread when it's entitled "Aircraft and conveyor belt (again)"

I am as entitled to post to any thread as you are. The mods will decide if i have done so incorrectly. So please don't tell me what and where to post !

 

[Observer]

It seems that if I maintain a forward thrust on the car, the belt speed and the wheel rotational speed will continue increasing towards infinity. That's the paradox.

The car is moving at a fixed speed (relative to, say, the end of the belt).

The wheels are going at whatever speed they are going.

The belt is going at whatever speed the wheels are going.

How, exactly, is that a paradox?

JHC

Because, with sufficient thrust to keep the car/aircraft moving fowards relative to the belt, the rotational speed of the wheels will necessarily be faster than the speed of the belt and the belt will keep trying to catch up and so the wheels will continue accelerating. And so on in a closed loop. Or have I missed something?

 

[Observer]

With respect, i did not say a hovercraft on a conveyor belt.

And secondly telling me my analogy is inapt without an explanation why; and also telling me to stick to another analogy because i'll understand it better i find quite patronising.

And then you tell me i missed the point of the thread when it's entitled "Aircraft and conveyor belt (again)"

I am as entitled to post to any thread as you are. The mods will decide if i have done so incorrectly. So please don't tell me what and where to post !

You didn't - so I withdraw that part of my response. The hovercraft analogy is (imo) inapt because a hovercraft is never in contact with the ground. The runner on a treadmill analogy is (imo) inapt because a runner is using an entirely different traction force to achieve motion. I apologise if I appeared patronising.

 

[Softus]

Or have I missed something?

Well, I see your point, but I don't see any merit in it.

If we take it that you're right, i.e. that the speed of the wheels and belt will approach infinite, then the landing gear will clearly overheat/collapse/explode, in which case the plane won't take off.

Perhaps you think this subtlety was intended as the real challenge, and that whole debate about engine thrust versus frictional losses was provoked merely as a smoke screen for the mechanically naive.

Maybe it was, but do you really care? Is is really that interesting?

 

[tim west]

If the wheel bearings really had no friction whatsoever then the axle would not be supported by them, you need friction to hold something up, any substance used within a bearing housing whether compressed gas or ice or other would have some friction, same as the tyres themselves there has to be resistance otherwise they'd fall through the ground, though on a molecular level whether it could be called repelling rather than resistance is debatable.

 

[Trazor]

Because, with sufficient thrust to keep the car/aircraft moving fowards relative to the belt, the rotational speed of the wheels will necessarily be faster than the speed of the belt and the belt will keep trying to catch up and so the wheels will continue accelerating. And so on in a closed loop. Or have I missed something?

What you have missed is the fact that the wheels are freewheeling, and the force applied to the plane to drag it backwards is very small.

Imagine that the wheel bearings have no friction, then the plane would hold station without any thrust from the jets. Now if you apply any thrust via the jets, this must move the plane forwards.

The amount of friction in the wheels is so small in reality, that the jet engines easily overcome this.

Going back to your toy car on its conveyor, with your hand stopping it moving backwards. What ever speed the conveyor now does, you will still only need the same amount of force to hold the car still, any extra force you apply must have a reaction ie. forwards motion.