Can a 747 Take Off on a Conveyor Belt?

[A.T.]

I believe Tito did mention white wheels, though... How did you attribute that quote to me, anyway ? .

Sorry, corrected now.

 

[Clausen]

The problem statement? Anyway, if it isn't about real life, the answer could be literally anything, including "yes it can take off if aliens teleport it into space."*.

Let's not get silly here. Lots of problems are posted as brain teasers that have little or nothing to do with real life. All I am trying to do is work within the constraints posed by the problem, regardless of how realistic those constraints are.

In either case, just for the record, regardless of the problem as stated, do you agree that a *real* airplane on a *real* conveyor can take off?

Yes, I have already stated that in my first post in this thread and I pointed out that in a real life situation there is no relation between the speed the wheels are turning and the ability of the plane to take off. Then, I followed that up by saying if we are to strictly be constrained by the condition that the belt moves backwards as fast as the wheels turn forwards, the only way the plane can advance is by sliding while the wheels are spinning. You still have not shown that to be wrong.

Just because nobody's bothered to try it doesn't mean they couldn't. Mythbusters used a Cessna because that is easier/cheaper. Also, the scenario can be made functionally equivalent using wind: The plane is sitting stationary on the ground (speed of ground = speed of wheels = 0), with a 160kt headwind lifting it off the ground.

You don't like the problem posed in the OP, so you want to change it into something you like. The OP specified a 747 on a conveyor belt. Nothing else.

The problem, as specified, is mathematically/grammatically flawed/incomplete: 1=2 unless 0=0, in which case the problem is pointless (a plane sitting on the ground in no wind, with its engine off doesn't take off -- so what?)..

OK. Then your answer is the question is flawed and unanswerable as stated. I am fine with that. But if you say, as others have, that the plane will definitely take off, under the constraints given, you need to explain how it does that. I say it can only do that if the wheels slide while spinning at the specified rate as dictated by the belt. That solution is in accord with all of the constraints of the problem.

Right. So with the engines at full throttle, what is keeping the plane stationary?.

I never said it will remain stationary. I said it will move forward with the wheels spinning and sliding and I gave some numbers but I do not know what the coefficient of resistance will be for a spinning and sliding wheel. Do you? I think it would be interesting to know that.

Or by breaking the problem because the problem is flawed. That's how it works in real life.

My life is not dictated to by your life. Instead of breaking the problem and solving something else, I try to come up with a solution to the problem as it is stated.

Because the problem is ill-posed, "correct" is debatable in terms of how the problem statement works. But in terms of how real life works, the issue is not debatable: a plane on a conveyor can take off. So any answer that interprets the problem statement to yield an answer of no requires making up other assumptions and adding them to the problem*.

So does any answer that yields a YES. You are all making up your own assumptions.

 

[A.T.]

Then, I followed that up by saying if we are to strictly be constrained by the condition that the belt moves backwards as fast as the wheels turn forwards,

Which makes no sense, because you are equating a linear velocity (belt moves) to an angular velocity (wheels turn). Also, specifying linear velocities requires a reference frame.

You are all making up your own assumptions.

But your assumptions are just as unclear as the original problem statement.

 

[sophiecentaur]

Which makes no sense, because you are equating a linear velocity (belt moves) to an angular velocity (wheels turn). Also, specifying linear velocities requires a reference frame.But your assumptions are just as unclear as the original problem statement.

Was the linear vs angular thing of any real importance? I think we all managed to cope with that.
Actually, the Original Problem was perfectly well specified. The only 'assumption' was that the conveyor belt is at least as long as the runway and that it's surface surface would be the same and would actually support the plane. What went wrong was that, later, people inserted some of their own constraints and also a few misconceptions about what factors were in fact of any importance.

 

[rootone]

I think the thing that has been learned here has nothing to do with planes.
It is however a good demonstration that if a question is imprecise then a precise answer cannot be expected.
In fact people may tend to make their own assumptions concerning imprecision in the question, and answer accordingly.
In my case had pictured the plane being somehow fixed in place with engine(s) off,
so the conveyor would have no effect other than turning the wheels.

 

[sophiecentaur]

imprecision in the question,

No imprecision in the question; you can assume the required takeoff airspeed of a specified aircraft. The only thing that was missing was the common sense of people who ignored the fact that is by far and away the major factor. The fact that there are 70+ posts goes to show how imprecisely many people tend to think about straightforward physical situations and how easily they can be misdirected. (That's how magicians earn their money.
If in doubt, draw a free body diagram and the answer will leap out of the page at you.

 

[jbriggs444]

Was the linear vs angular thing of any real importance? I think we all managed to cope with that.
Actually, the Original Problem was perfectly well specified. The only 'assumption' was that the conveyor belt is at least as long as the runway and that it's surface surface would be the same and would actually support the plane.

There is another assumption buried in the problem statement. That statement can be read in two parts.
1. There is a constraint that $v_{conveyor} = -v_{wheels}$
2. That constraint can be realized by appropriate engineering of a hypothetical conveyor.
For reference:

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

No such conveyor design is feasible when applied to a 747 on a takeoff run with the brakes off.

[Barring the unreasonable case where the wheels are made spin so rapidly that they disintegrate. That's another assumption buried in the problem statement: That a 747 on a takeoff run has wheels]

 

[sophiecentaur]

No such conveyor design is feasible when applied to a 747 on a takeoff run with the brakes off.

When you think of the serious consideration that the Space Enthusiasts give to some of their proposed schemes, I don't reckon it would be that unthinkable. But, if the OP had thought again about the question, he/she could have suggested a Learjet or Hawk and all the same theory would have applied. And all the same misconceptions would have been laid to rest.

[Barring the unreasonable case where the wheels are made spin so rapidly that they disintegrate.

I wouldn't imagine that any aircraft or car wheel assembly couldn't cope with twice normal operating speed.

 

[jbriggs444]

When you think of the serious consideration that the Space Enthusiasts give to some of their proposed schemes, I don't reckon it would be that unthinkable. But, if the OP had thought again about the question, he/she could have suggested a Learjet or Hawk and all the same theory would have applied. And all the same misconceptions would have been laid to rest.

I wouldn't imagine that any aircraft or car wheel assembly couldn't cope with twice normal operating speed.

There's another assumption you're bringing in -- that it's not wheel rotation speed that matters, but aircraft ground speed instead.

 

[sophiecentaur]

There's another assumption you're bringing in -- that it's not wheel rotation speed that matters, but aircraft ground speed instead.

That isn't an "assumption"; it's a consequence of the wording of the OP.
The wheel over conveyor speed will only be twice the wheel over ground speed (= takeoff airspeed). How could you suggest that it wouldn't be twice if the conveyor has equal and opposite velocity?

 

[jbriggs444]

That isn't an "assumption"; it's a consequence of the wording of the OP.
The wheel over conveyor speed will only be twice the wheel over ground speed (= takeoff airspeed). How could you suggest that it wouldn't be twice if the conveyor has equal and opposite velocity?

We are in violent agreement. The reason that it's a consequence of the wording of the OP is because of the absurdity of the alternative.

 

[sophiecentaur]

I am relieved about that. I mis-read your double-negative type structure.

 

[sophiecentaur]

83 now.

 

[Clausen]

83 now.

I don't like odd numbers.I would just like to have some of you people on a witness stand, under oath, and pose this question to you:

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?

A simple Yes or No will do please, no hemming and hawing.

 

[jbriggs444]

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?

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."

 

[sophiecentaur]

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

To restrain the hubs, you would have to be moving the conveyor Forward and not Backwards. There would be no 'restraint' involved. the plane would just be moving forward, through the air and the conveyor would be keeping up with it. The wheels would have no tangential force on them.

 

[Nugatory]

does the center hub of that wheel move with respect to that fixed point?

A simple Yes or No will do please, no hemming and hawing.

No.

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

 

[jbriggs444]

To restrain the hubs, you would have to be moving the conveyor Forward and not Backwards. There would be no 'restraint' involved. the plane would just be moving forward, through the air and the conveyor would be keeping up with it. The wheels would have no tangential force on them.

If one were to attempt to restrain the hubs from moving forward, one would need the conveyor to be accelerating rearward.

This would be quite difficult, given the tendency of the wheels to freely roll with a forward angular acceleration, however a significant rearward force could be achievable with a sufficiently huge acceleration.

Accelerating the conveyor forward would instead result in a forward force on the contact patch at the bottom of the tires and an associated forward force on the hubs. It would not impede the forward motion of the craft.

Note that I am including the moment of inertia of the wheels in the model. The required tangential force on each wheel is equal to $\frac{static\_thrust}{number\_of\_wheels}$ and the resulting angular acceleration $\alpha = \frac{r_{wheel}\ static\_thrust}{number\_of\_wheels\ I_{wheel}}$ of each wheel would be rather large.

Edit: Back of the envelope...

747 static thrust is around 1,000,000 N, tire radius about 0.6 m, tire mass somewhat over 100 kg.
Estimated moment of inertia per wheel = $100\ 0.6^2$ ~= 36 kg m^2. 16 wheels on the main landing gear plus whatever is under the nose. Call it 1000 rad/s^2, give or take a factor of 2.

At their max rated speed (235 mph or around 100 m/s), a 0.6m radius tire will be rotating at about 30 rad/s. That means about 30 milliseconds into the takeoff attempt, the tires will be at their max rated speed -- well before the engines have even finished spooling up.

[Which calculation puts a similar upper bound on the duration of that chirp they dub into movies when the wheels touch down during landing of a jet aircraft -- a nice sanity check]

 

[sophiecentaur]

If one were to attempt to restrain the hubs from moving forward, one would need the conveyor to be accelerating rearward.

This needs clearing up. The plane is going forward so, to stop the wheels from rotating, the runway would need to go forward. That's true but not relevant to the OP which stipulates that the conveyor is going backwards at an equal speed to the forward speed of the wheels. At least, that's how I understood the OP. There is no wonder that we are shouting at each other if we interpreted the OP in directly opposite ways.
BUT, because the craft has wheels, which are very low drag (by design), it makes no difference one way or the other to the ability to take off. A forward moving conveyor would make things slightly easier but that's obvious (isn't it?).
Haha. 88 posts and we're only just getting down to what the OP meant. Good ole PF.

 

[jbriggs444]

This needs clearing up. The plane is going forward so, to stop the wheels from rotating,

Note the wording of the claim: "attempt to keep the hubs from moving forward". I am not speaking here of the rotational velocity, $\omega$ of the hubs but of their ground-relative linear velocity, $v$

It is, of course, the ground-relative linear velocity (and, hence, the air-relative linear velocity) that is of relevance to take-off. However, the constraint mentioned in the problem statement is more direct than that.

The constraint in the problem (if read as @Clausen would have it) is that $$v_{conveyor} = - \omega_{wheel} r_{wheel}$$.
The free-wheeling constraint I believe you adhere to (and with which I agree) is that $$v_{plane} = \omega_{wheel} r_{wheel} + v_{conveyor}$$
Putting those together, we have that $$v_{plane} = 0$$.
As I read the problem, a constraint that $v_{plane} = 0$ must be achieved entirely by design of the conveyor. This means that the plane must be restrained from forward motion by the rearward acceleration of the conveyor alone.

I am trying to point out that this chain of reasoning leads to an absurdity. So the starting interpretation that $v_{conveyor} = -\omega_{wheel} r_{wheel}$ is thrown into doubt.

[My suspicion is that pretty much everyone here agrees about the physics but disagrees about what point to make about it].

 

[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 [itex] a = \alpha r [itex] where a is the linear acceleration, we get:
$a = \frac {2 F} {m}$. 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 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.