B Can a 747 Take Off on a Conveyor Belt?

[sophiecentaur]

a constraint that vplane=0vplane=0v_{plane} = 0

How can it be suggested that plane can equal 0 if the wheels are free to rotate? This is getting out of hand.

disagrees about what point to make about it

What point is there to make except that, unless the wheels break up, the plane can take of. If the wheels are ideal then there is no difference between the stationary runway and a moving conveyor, (forward or reverse).
We seem to be indulging one or more contributors who seem to think that the wheels have a fundamental contribution to the situation. How many more times must it be pointed out that they don't?

 

[jbriggs444]

How can it be suggested that plane can equal 0 if the wheels are free to rotate?

Because that is the implication of the problem statement -- if read in a particular (and fairly natural) way. You take a seemingly reasonable premise, follow it to an absurd conclusion. That's the way "reductio ad absurdum" works.

What point is there to make except that

92 points, apparently.

 

[Clausen]

No..

Thank you! You have restored my faith in humanity. I was beginning to think that there isn’t anyone here who would give a straight answer.

Of course the answer to this question has no bearing whatsoever on the question of whether the aircraft will in take off.

Taken by itself, this question alone does not say much about the ability of the plane to take off, but now that we have at least one small area of agreement, maybe we can expand that area a bit?

Can you please answer my follow up question in the same straight forward manner?

Remember, all of this follows from the first question, which I will repeat here for convenient reference:

If during a given time interval, a wheel with a circumference of 1 meter rotates through one revolution in trying to roll to the left, on a conveyor belt that is moving to the right a distance of one meter, all with respect to a fixed point on the ground, does the center hub of that wheel move with respect to that fixed point?

You answered "No", and I totally agree.

Now, the follow up:

If I now push on the hub of that wheel so that the hub Does move to the left (with respect to the same reference) and at the same time the rotation of the wheel and the speed of the belt is (somehow) maintained as in question 1, would the wheel need to be sliding on the belt as it moves?

I think it would have to be. What do you think?

This may not be clear how it applies to the 747 on the conveyor belt but once I have your answer to this I will explain how it all fits together. Please have a bit of patience.

 

[Clausen]

Your honor, may I cross-examine the witness?

If one attempted to restrain the motion of the hubs on set of 747 wheels by means of a rearward acceleration of the conveyor belt upon which they ride, what acceleration would be required to match the static thrust of the craft's engines operating at takeoff power?

For how long could said acceleration be maintained without the wheels disintegrating?

Would you say that a 747 can operate its engines at such a power setting for longer than that?If it please the court, I now draw your attention to the question of statutory construction...

https://www.law.cornell.edu/wex/statutory_construction

"Courts generally steer clear of any interpretation that would create an absurd result which the Legislature did not intend."

I notice you didn't answer the question.

I will answer yours but can you please wait until I have Nugatory's answer to my follow up?

 

[jbriggs444]

I notice you didn't answer the question.

Because the answer is not controversial. That is not our point of disagreement.

 

[sophiecentaur]

It struck me that the misconception here is a bit like asking how long it would take to stop a bicycle by back-pedalling with an ideal freewheel.

 

[Nidum]

If you have two different physical systems then one system one can only affect the other if there is a coupling mechanism between the two . The degree to which one system affects the other system depends on the characteristics and effectiveness of any such coupling mechanism .

In our present problem we have one physical system which is the aeroplane and another physical system which is the conveyor belt .

The only coupling mechanism between these two systems is the feeble one which comes from small interaction forces due to friction¹ .

The two systems thus act and behave almost independently . No action of the conveyor belt can have any significant effect on the forces acting on or the motion of the aeroplane during the take off run .

The aeroplane takes off normally and the conveyor belt does whatever it likes .

Note 1 : The fact that the wheels may turn faster than normal has no significant effect on the aeroplane motion or the forces acting on the aeroplane .

 

[Nidum]

This does not affect the actual problem answer but interesting to note that the conveyor cannot be running at constant speed . It must start at zero speed and accelerate at the same rate as the aeroplane . Not driven by the plane though - there would have to be an independent motor/engine , sensors to detect aeroplane speed and a control system .

Completely bonkers .

 

[Janus]

Bandersnatch has already touched on the key issue of this problem. If you assume ideal conditions, the instant the jet tries to move forward, the conveyor belt speed and rpm of the wheels would become infinite.
It suffers from the problem many scenarios do when you try to apply ideal conditions to them.
If you short an ideal battery with a perfect conductor, you get a similar problem. you should have zero voltage across a perfect short, but an ideal battery will always maintain a set voltage. So do you have zero voltage across the conductor and no current, or battery voltage across it and infinite current (Actually, using ohms law, I= V/R, which in this case is a division by 0, and is undefined. )
In the real world, the conveyor belt would not have an infinite power supply available to drive it, so it's top speed would be limited by that. The axle friction of the airplane's wheels would not be zero, so there would be a maximum speed at which they could spin before over heating and seizing up or failing for some other reason. In addition, the conveyor belt would not be able to instantly adjust to the tire rotation speed and would always lag behind by some amount. And even with a sufficient power supply, the conveyor belt would have limits on the strains it could endure.

It is only by knowing the real limitations of all the systems involved that you would be able to come up with an answer to this question under those conditions.

 

[TurtleMeister]

There was another thread on this topic that I posted in a few years ago but I can't seem to find it. It surprises me that so many people get this wrong, even the pilot in the Myth Busters video. I think Nidum did a good job of describing the situation in his/her previous two posts. Barring obvious physical constraints, such as tire maximum ratings, the conveyor direction and speed has no bearing on whether the plane can take off or not.

 

[willem2]

Actually, If you accelerate the conveyor fast enough, it can stop the plane!
A force is needed to make the wheels rotate, The wheels can only start to rotate by a force of the conveyor on the wheels.
We have \alpha = \frac {T} {I} where T is the torque from the conveyor on the wheels, I the moment of inertia of the wheels.
Substituting I = \frac {1}{2} m r^2 for the moment of inertia of a cylinder, where m is the mass of the wheels.
and T = F r where F is the thrust of the engines that the conveyor must balance.
and a = \alpha r where a is the linear acceleration, we get: <br /> a = \frac {2 F} {m}. m ~ 1500 kg. (8 tyres of 184 kg) F = 250 kN . so<br /> a ~ 333 m/s^2. So if we only keep on accelerating the conveyor at this rate, the plane can't take off.<br /> <br /> A problem might be that the conveyor might cause so much wind, that the plane can take off at 0 ground speed anyway.

 

[sophiecentaur]

Barring obvious physical constraints,

I am fully with you there. This is what I can't understand. The constraints are really obvious and seem to be blinding people to the basic Physics of the situation. We had a similar problem on the 'filling buckets' problem, where people also couldn't let themselves just deal with the basic logical Physi,cs. These problems can always be analysed and analysed to death and there's no fun or (more to the point) there is no ANSWER.
How could we ever teach Science to kids if we littered the curriculum with practical details from the start about Wire Resistance, Non-uniform Gravity on Earth, Friction, Heat loss / gain? The poor devils would never feel able to predict anything because some smart alec would introduce a footling reason why they will be wrong. We are Physicists (aren't we?) and - just like Engineers, we start on a problem in the simplest possible way.

 

[Janus]

I am fully with you there. This is what I can't understand. The constraints are really obvious and seem to be blinding people to the basic Physics of the situation. We had a similar problem on the 'filling buckets' problem, where people also couldn't let themselves just deal with the basic logical Physi,cs. These problems can always be analysed and analysed to death and there's no fun or (more to the point) there is no ANSWER.
How could we ever teach Science to kids if we littered the curriculum with practical details from the start about Wire Resistance, Non-uniform Gravity on Earth, Friction, Heat loss / gain? The poor devils would never feel able to predict anything because some smart alec would introduce a footling reason why they will be wrong. We are Physicists (aren't we?) and - just like Engineers, we start on a problem in the simplest possible way.

The problem is that there are scenarios that if they are reduced too far, lead to nonsense results.

 

[jbriggs444]

a = \frac {2 F} {m}. m ~ 1500 kg. (8 tyres of 184 kg) F = 250 kN . so
a ~ 333 m/s^2. So if we only keep on accelerating the conveyor at this rate, the plane can't take off.
.

A 747 has 16 tires (4 each on 4 pylons) and each of 4 engines has around 250 kN thrust. However, I agree that the result is in the right ballpark and results in the wheels reaching their rated max speed in a fraction of a second.

Edit: I like your figure for tire+wheel mass. It's better than the one I used up-thread.

 

[sophiecentaur]

From OP:

The conveyor best is designed to exactly match the speed of the wheels, but run in the opposite direction.

results in the wheels reaching their rated max speed in a fraction of a second.

How does that follow? The wording in the OP is a bit vague but I read it as meaning that "match" means the conveyor surface always goes backwards at the same speed as the wheels are going forward (i.e. the bearings and the rest of the plane). The acceleration is just twice that of the plane over the ground. Where does the "fraction of a second" come from? The plane takes many seconds to accelerate to takeoff speed.

 

[A.T.]

... but I read it as ...

The whole thread in five words.

 

[sophiecentaur]

But I, personally, have been declaring how I read it. I think many of the loopy contributions have not ever stated their full case. 'It stands to reason' is never a full case.
I think we should demand that answers should include how the OP has been actually understood by the contributor and, where appropriate, some reference of book work. Some of the assumptions in this thread have been so 'intuitive'.

 

[jbriggs444]

I think many of the loopy contributions have not ever stated their full case. 'It stands to reason' is never a full case.

If you chase back the sub-thread to which you responded, you will find.

Actually, If you accelerate the conveyor fast enough, it can stop the plane!

@willem2 proceeded to perform a computation containing actual physics and arrived at a figure for how fast the conveyor would need to accelerate rearward so that the friction required to spin up the wheels to match the conveyor speed would match the forward thrust of the aircraft engines.
I then responded with agreement that..

result is in the right ballpark and results in the wheels reaching their rated max speed in a fraction of a second.

To which you responded that you did not see how it followed.

Looks to me like it follows.Now @willem2 did not quote the post to which he was responding, but his post immediately followed one by @TurtleMeister:

Barring obvious physical constraints, such as tire maximum ratings, the conveyor direction and speed has no bearing on whether the plane can take off or not.

@willem2's post seems to be on-point in a rebuttal of this claim. [Arguably a rebuttal which was already accounted for under the "obvious physical constraints" exception].

Edit to add a final clarification:

None of this should be read as disagreement with your (@sophiecentaur) sound advice to state assumptions first and calculations after.

 

[sophiecentaur]

Looks to me like it follows.

We are clearly at cross purposes here. I realize you are not a total loony (just enough to want to contribute as regularly as I do on PF!) by the sentence:

The conveyor best is designed to exactly match the speed of the wheels, but run in the opposite direction

If you are suggesting that means to drag the wheels back so fast that they lock up and drag the plane backwards then what you are claiming will follow.
But I do not understand what 'matching' means, in your interpretation.
The way the original question uses 'matching', surely implies that the linear speed of the conveyor would be equal and opposite to what would be the tangential speed of the wheel if it were on the ground. That is actually equal and opposite to the instantaneous speed of the plane over the ground. I have made that clear more than once. Are you saying that is the wrong interpretation? The wheels are free to rotate - just as if it were on ice, in an ideal case. No force (lateral) is exerted on the plane, even if you move the ice backwards.

 

[A.T.]

But I do not understand what 'matching' means, in your interpretation.

I think that sub-thread you reply to is not about guessing what "matching" means anymore, but simply asking what would the belt have to do, in order to stop the plane based on rotational inertia of the wheels.

 

[sophiecentaur]

None of this should be read as disagreement with your (@sophiecentaur) sound advice to state assumptions first and calculations after.

Sorry, I missed this last line of the post.
But I still take issue with the requirement to accelerate things in a fraction of a second. The air speed of the plane is the variable that determines what the conveyor needs to be doing.

 

[sophiecentaur]

I think that sub-thread you reply to is not about guessing what "matching" means anymore, but simply asking what would the belt have to do to stop the plane, based on rotational inertia of the wheels.

That could be true but what a fruitless conversation, in the light of almost absolute ignorance of the values of all the variables involved. I would totally agree that the experiment is a nonsense but it is scaled down very easily with a low speed prop aircraft and the Physics point is proved.
Like I said earlier, how easy is it to stop a bicycle by back-pedalling with a freewheel hub?

 

[TurtleMeister]

I think we should demand that answers should include how the OP has been actually understood by the contributor

Here's the op:

Imagine a 747 sitting on a large conveyor belt, as long and as wide as the runway.
The conveyor best is designed to exactly match the speed of the wheels, but run in the opposite direction.

CAN THE PLANE TAKE OFF?

The op is not clear, and in my opinion, is the cause of the confusion. So here is my interpretation, the way I think it should be stated:

Imagine a 747 sitting on a large conveyor belt, as long and as wide as the runway. The conveyor belt is designed to match the normal takeoff speed of the aircraft but in the opposite direction. Can the plane take off?

I think this is the interpretation used in the Myth Busters video.

 

[Jake Miguel]

The tire wheel speed is not really related to what the plane needs to fly; over 100mph of wind to make the lift. The plane will have no care for tire wheel speed. If there was a gale of 160mph, then it would be possible for the 747 to hover over a fixed position with respect to the ground. Planes don't have care for ground speed, but rather air speed is what they crave.

 

[jbriggs444]

Sorry, I missed this last line of the post.
But I still take issue with the requirement to accelerate things in a fraction of a second. The air speed of the plane is the variable that determines what the conveyor needs to be doing.

Reductio Ad Absurdum.

You persist in arguing against the perceived conclusion (the absurd requirement) instead of the real conclusion (that the interpretation that led to that requirement was unintended or that the problem is badly posed).

 

[Janus]

From OP:
How does that follow? The wording in the OP is a bit vague but I read it as meaning that "match" means the conveyor surface always goes backwards at the same speed as the wheels are going forward (i.e. the bearings and the rest of the plane).

The problem I have with that interpretation of the intent of the question is the equivalent of asking that if you put a car on a treadmill that travels backwards at the same speed as the car is moving forward relative to the ground, can the car move forward? It is a bit nonsensical, because the car moves forward as one of the initial postulates of the problem. It just seems more reasonable that the problem is assuming that the treadmill moves backwards at the same speed as the tread of the wheels move relative to the car or plane. Now you can reasonably ask whether or not the car or plane can move forward.

 

[sophiecentaur]

The problem I have with that interpretation of the intent of the question is the equivalent of asking that if you put a car on a treadmill that travels backwards at the same speed as the car is moving forward relative to the ground, can the car move forward?

But it isn't th3e same thing at all. In the car scenario, the motion of the car is determined entirely by the power delivered via the wheels. Forces on the wheels are an essential part of the propulsion. If the 'road' moves backwards at the tangential speed of the wheels, the car goes nowhere. That is just not the case with an aeroplane.

the same speed as the tread of the wheels move relative to the car or plane.

You can't include them in the same model. What drives the car forward? What drives the plane forward?
I suggested earlier that people in doubt should draw a free body diagram. Is the force on the plane in any way dependent on the wheels (in the absence of friction losses)?
I was all ready to drop this and I read your post. It's just not right.

 

[Janus]

But it isn't th3e same thing at all. In the car scenario, the motion of the car is determined entirely by the power delivered via the wheels. Forces on the wheels are an essential part of the propulsion. If the 'road' moves backwards at the tangential speed of the wheels, the car goes nowhere. That is just not the case with an aeroplane.

You can't include them in the same model. What drives the car forward? What drives the plane forward?
I suggested earlier that people in doubt should draw a free body diagram. Is the force on the plane in any way dependent on the wheels (in the absence of friction losses)?
I was all ready to drop this and I read your post. It's just not right.

But isn't that point? In the case where the treadmill moves at the same speed as ground speed, the results are the same for car and plane. When it's the same speed as tangential speed of the wheel, then obviously the car doesn't move relative to the ground as it is propelled by the wheels. The plane is not propelled by its wheels and the problem becomes more reliant on the constraints applied. Since the intent of the question is to present a perceived conundrum. It does not seem likely that the scenario with the trivial answer is what was meant

 

[sophiecentaur]

the problem becomes more reliant on the constraints applied.

This is just more over analysis. You can keep changing the goalposts for another 119 posts but who will it help? The original question was obviously about the simplest case. Why not deal with just that?

 

[DaveC426913]

I can't believe this is still under discussion.

It doesn't matter what the wheels are doing - it's not a car.
A plane gets its forward motion via the air, using props or jets.
Once the thrust of the engines rises, the plane will accelerate with respect to the air - and eventually gain lift via air, as usual - no matter what the wheels are doing.

You can do whatever you want with the conveyor - move it as fast as you want - till the wheels blow, and the landing gear is worn down to nubs - the plane is still going to move forward, given sufficient thrust.

Full stop.