A jet plane on a large treadmill

[mcski]

Plane goes back in time.

To 1984 and meets KITT...On a truck

[jono]

False premise. The system isn't designed to control airspeed. Wheelspeed = -treadmill speed. That's the equation. Nothing else has been defined. Thrust doesn't control the force of the treadmill!

If wheelspeed cancels out treadmill speed, then the problem becomes only whether the engines can overcome the friction between the wheels and the treadmill.

Edit to attempt to be concise: that wasn't a premise at all, I arrived at that conclusion by applying Newtonian mechanics to the plane and wheels given the stated characteristic of the treadmill.

Long version: You are still neglecting the inertia of the wheels. That doesn't work when the treadmill potentially has to accelerate Fast to keep up with the slightest rotation of the wheels and then doubles their speed and thus it's own ad infinitum. Fortunately calculus offers a better way to understand that scenario. (And fortunately is probably an understatement; where would the world be without the ability to understand impossibly powerful treadmills?!?) As such, the limit of the plane's forward velocity (airspeed, sans wind) is zero even as thrust is applied because even rocking forward results in a huge force being applied by the treadmill. But if the treadmill applies more force than the thrust the plane moves backwards, reversing the trend of the wheels and cutting back on the treadmill speed. The definition requires that the treadmill move at whatever speed is needed to match the wheels and (because the plane can't move without requiring infinite acceleration of the wheels and a correspondingly infinite force to do that) also whatever force is needed to match the thrust. Therefore, those two variables are "controlled" by the engines and the wheels. Via this magical control mechanism with instantaneous response.

Once you account for the inertia of the wheels instead of assuming that the thrust only has to overcome friction you notice that you can practically ignore friction and arrive at the conclusion that the thrust can still be matched by the treadmill and therefore, according to the given fact that it matches wheel speed, the thrust will be matched as well. The energy thus used will go a little bit to friction and mostly to spinning up the wheels. Until they explode or seize and stop spinning, at which point the treadmill does likewise and the plane slides away toward liftoff.

[M111S]

More to the point, when it lands on the treadmill does it stop immediately ?

[philippeR]

Come on ! This thread needs to be in the hall of fame.

[Mustonen]

Edit to attempt to be concise: that wasn't a premise at all, I arrived at that conclusion by applying Newtonian mechanics to the plane and wheels given the stated characteristic of the treadmill.

Long version: You are still neglecting the inertia of the wheels. That doesn't work when the treadmill potentially has to accelerate Fast to keep up with the slightest rotation of the wheels and then doubles their speed and thus it's own ad infinitum. Fortunately calculus offers a better way to understand that scenario. (And fortunately is probably an understatement; where would the world be without the ability to understand impossibly powerful treadmills?!?) As such, the limit of the plane's forward velocity (airspeed, sans wind) is zero even as thrust is applied because even rocking forward results in a huge force being applied by the treadmill. But if the treadmill applies more force than the thrust the plane moves backwards, reversing the trend of the wheels and cutting back on the treadmill speed. The definition requires that the treadmill move at whatever speed is needed to match the wheels and (because the plane can't move without requiring infinite acceleration of the wheels and a correspondingly infinite force to do that) also whatever force is needed to match the thrust. Therefore, those two variables are "controlled" by the engines and the wheels. Via this magical control mechanism with instantaneous response.

Once you account for the inertia of the wheels instead of assuming that the thrust only has to overcome friction you notice that you can practically ignore friction and arrive at the conclusion that the thrust can still be matched by the treadmill and therefore, according to the given fact that it matches wheel speed, the thrust will be matched as well. The energy thus used will go a little bit to friction and mostly to spinning up the wheels. Until they explode or seize and stop spinning, at which point the treadmill does likewise and the plane slides away toward liftoff.

No. The inertia of the wheels is a factor in re: the treadmill speed matching mechanism (makes instant infinity impossible without infinite force) but has nothing to do with the jet engine acting on the airplane outside of the mechanism itself effectively locking the wheels.

Imagine pushing on a cart that has a giant lead wheel mounted to it so that it could spin freely, and that every time you moved the cart, the wheel would spin instantly in the opposite direction at a high speed (lacks context, which is clue #1 you're on the wrong track). What effect do you think the wheel's spinning inertia will have on your ability to push the cart?

Again, you're adding stuff to the problem that doesn't logically make any sense. Wheelspeed = -treadmill speed. That literally means the wheels can only slide forward, not roll. They'll slide at the same rate, no matter how fast they're spinning. I contend that this impossible machine will result in a locked wheel, but it doesn't actually matter; if you're cool with infinite force and infinite velocities, whatever, have fun.

[jono]

Infinite forces/speeds are specifically not involved, because the wheels and tires have non-zero mass.

We do not disagree on the major points, though. Eventually the thing slides after the wheels blow up, but it sounds like you're saying the conditions of the problem require the brakes to be applied. That's not true. There is no mention of locking the wheels and if they spin on real bearings or frictionless ones the mass of the wheels holds the plane still either way as the wheels accelerate to failure. After which, sliding.

The giant lead wheels would take a lot of force to move, right? That's just a proportional change. These wheels take a lot of force to accelerate to failure--all the engines' thrust, as it happens. Because in order to achieve the condition that treadmill speed = -wheel speed the force applied to the wheels will be equal to the thrust; if it's less the plane goes forward and it would never be more because exceeding thrust would mean going faster than the wheels, which it never does.

[Mustonen]

No, I don't care about brakes. Do you?

My point with the wheel is that the rotation of the wheel has nothing to do with your ability to move it. You aren't connected to the wheel. The rotational inertia doesn't affect anything. Neither is the jet engine connected to the airplane's wheels, at least in terms of rotational inertia. The mass of the wheels is still a factor, but probably mostly irrelevant in the context of everything else going on.

"Accelerate to failure" is irrelevant. The failure of the wheel/treadmill mechanism isn't particularly important outside of the difference it makes to the amount of friction the jet engines have to overcome.

You're still over complicating it. The treadmill is impossible, but that impossibility doesn't logically extend to the application of power by the jet engine. The treadmill isn't powered by the jet engine, even indirectly. Some mysterious force works to match speed of treadmill and wheels; whether that force holds everything static or if infinite things happen (which isn't any more logical, mind you that infinity isn't actually a number, like "10" or "1 million") the end result is the same.

Again, the treadmill is Not designed to keep the plane stationary, it's only designed to cancel out the ability of the wheels to provide a frictionless interface.

[Telenater]

Proof that the U.S. school systems are failing the country.

[jono]

The engines and the wheels are connected to the plane, the wheels through bearings that might as well be frictionless. That indicates that the sum of the forces between the wheels and strut/plane in the horizontal (x) is zero. We also agree that the plane does not roll forward, and being connected by static friction to the treadmill, the tires do not slide unless their friction coefficient falls below 0.835 (see DJSapp's calcs).

Actually writing the equations we have

F (x, treadmill) - F (x, bearing) = 0

Or simply,
F (x, treadmill) = F (x, bearing)

Similarly, since the plane does not accelerate relative to the wheels, the sum of the forces on the plane (ex.wheels) are known to be:

F (x, bearing) - F (Thrust) = 0

Or simply,
F (x, bearing) = F (Thrust)

Therefore F (x, treadmill) = F (Thrust)

So where does all that force go? Into accelerating the wheels according to T = I * alpha where T = F (Thrust) x tire radius, or

alpha = F (Thrust)*(r)/(Izz, wheels)

Until the tires fail.

It's not that there's a connection specifically, it's that the feedback mechanism, whatever it is, can't do its job without conforming to the above equations. In order to keep the speeds matched the force between the tires and the treadmill will be equal to the thrust as long as the plane is stationary. The radial force in the wheel bearings exactly counters the thrust, which is what keeps the plane stationary in the force realm, while in the velocity realm the same is achieved (and that's where the feedback connection lies, of course) by matching speeds. So in a way I'm stating this backwards, but the result is the same, as it must always be.

[mcski]

Other than being a complete asshat, what is it you are trying to prove?

[jono]

No, I don't care about brakes. Do you?

Obviously I'm assuming brakes are off (although using a little brake and/or less than full throttle would slow wheel acceleration and increase the time to failure). I was just responding to your assertion that the impossible machine would result in locked wheels. Hopefully you can see how the only way that would be true is if the brakes were locked; if the wheels are free to spin they will spin...fast.

[jono]

Other than being a complete asshat, what is it you are trying to prove?

What is the thread about? I'm saying that with this slight variation on the original (the treadmill matches wheel speed in the opposite direction, rather than plane speed) if the only impossible machine in the system is the treadmill then starting the plane from rest and applying thrust the plane will remain stationary while the wheels and treadmill accelerate quite rapidly until the tires and maybe the wheels explode. Then the plane slides on down the treadmill, picking up speed on its stubs and throwing off sparks until it has enough speed to lift off. The math behind that might require a degree or a course in dynamics or at least insane dedication to the study of mixing possible and impossible machines, and maybe that's not your cup of tea. But if you could watch it happen you'd see a plane start its engines and sit there for a brief moment before the tires flew apart as the treadmill sped up really dramatically and then stopped when whatever's left of the axles froze solidly to the struts. Then the plane would take off, a little slowly and with the aforementioned sparks. You'd be wearing safety glasses, of course, so you wouldn't shoot your eye out. The plane probably lands next to KITT on the trailer cruising along somewhere on I-5, but that's just wild speculation. The rest is absolute fact, as is obvious to the most casual observer.

[Svengali]

Proof that the U.S. school systems are failing the country.

Well at least they can feel better about their speelling


and that they got telenater to post!

[iceman]

It's essentially a frictionless surface, the plane takes off, yeesh.

[GeezerSteve]

Yeah, of course it takes off.

The wheel bearings spinning faster and faster and failing theory ignores OP's original assumption:

This conveyor has a control system that tracks the plane's speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction).

"Plane's speed" = air speed. Applying this assumption, if takeoff air speed is 185mph, the wheels would spin per ground/treadmill approach speed of 370mph. No biggie. Certainly nowhere near the speed necessary to burn up the bearings.

[jono]

It's essentially a frictionless surface, the plane takes off, yeesh.

Originally, yes. I could be wrong, but I didn't think anyone was still hung up on the original. It's when the treadmill offers to match the speed of the wheels that the frictionless assumption fails.

[GeezerSteve]

I didn't think anyone was still hung up on the original.

Hung up? WTF? This thread is about a thought experiment based on the assumptions in the OP. Duh. Anyone can win an argument if he can change the assumptions.

[iceman]

Yeah I wasn't aware the question had changed but I did skip about 35 pages of the thread so hey.

[jono]

This is going around the book face now Attachment 190120

This post from grskier changed the question and with it the answer...sort of.

[MagnificentUnicorn]

I have a new perspective on this; if the treadmill could operate at infinite speeds with infinite acceleration, but the plane was limited by normal material / bearing capabilities the wheel bearings or wheels would fail, exploding or melting before the plane ever moved forward an inch. In a perfect treadmill system the wheels would almost instantaneously accelerate to failure speeds, rendering it .

If the wheels materials were made perfect and able to accelerate to infinite speeds it would still sit in place if friction still existed.

Remove friction and the plane takes off.

Im drunk / buzzing on a cigar.

This was my answer years ago and many pages back

In order for the plane to take off it must be able to achieve forward motion. If the treadmill perfectly opposed the thrust of the plane the wheel bearings melt and the plane does a nose dive

The engines and the wheels are connected to the plane, the wheels through bearings that might as well be frictionless. That indicates that the sum of the forces between the wheels and strut/plane in the horizontal (x) is zero. We also agree that the plane does not roll forward, and being connected by static friction to the treadmill, the tires do not slide unless their friction coefficient falls below 0.835 (see DJSapp's calcs).

Actually writing the equations we have

F (x, treadmill) - F (x, bearing) = 0

Or simply,
F (x, treadmill) = F (x, bearing)

Similarly, since the plane does not accelerate relative to the wheels, the sum of the forces on the plane (ex.wheels) are known to be:

F (x, bearing) - F (Thrust) = 0

Or simply,
F (x, bearing) = F (Thrust)

Therefore F (x, treadmill) = F (Thrust)

So where does all that force go? Into accelerating the wheels according to T = I * alpha where T = F (Thrust) x tire radius, or

alpha = F (Thrust)*(r)/(Izz, wheels)

Until the tires fail.

It's not that there's a connection specifically, it's that the feedback mechanism, whatever it is, can't do its job without conforming to the above equations. In order to keep the speeds matched the force between the tires and the treadmill will be equal to the thrust as long as the plane is stationary. The radial force in the wheel bearings exactly counters the thrust, which is what keeps the plane stationary in the force realm, while in the velocity realm the same is achieved (and that's where the feedback connection lies, of course) by matching speeds. So in a way I'm stating this backwards, but the result is the same, as it must always be.

Dude you might be a rocket scientist or something but the only way the wheels on an airplane roll are by forward or backward motion of the plane. In this scenario of a treadmill that can instantly match wheel speed(impossible) the only way the treadmill can match wheel speed is if the plane is moving forward down the treadmill for take off. For a 747 with a takeoff speed of 180 mph the treadmill would only have to go that fast. The plane would still be moving forward because it moves by thrust acting on the atmosphere not powered wheels. It will take off. Even in the theoretically impossible scenario of a treadmill with infinite speed it wouldn't matter because a 747 takes off around 180 mph.

[2stix]

Simple answer. Lock the brakes, wheels don't turn, treadmill moves in direction of plane and plane takes off. 0 wheel speed. Profit?

[Flyoverland Captive]

^^^ Winning!

[jono]

Simple answer. Lock the brakes, wheels don't turn, treadmill moves in direction of plane and plane takes off. 0 wheel speed. Profit?

I was thinking that, too, but unfortunately if the treadmill matches the wheel speed it will be locked when you lock the brakes, not freewheeling.

Aaronwright, I was not referring to the OP scenario but to the one that grskier quoted where the treadmill matches the wheel speed. In that scenario wheel speed = treadmill speed so the plane never moves, it doesn't matter how you power it (that issue was resolved on page 2 or something). You can prove the scenario I outlined a little less rigorously/more intuitively if you consider the treadmill's feedback system to be slightly imperfect so that it lets the plane rock forward before responding. In that case the response would be a massive force which would push the plane back (because it would be more than the thrust). Then it would have to correct again, and basically the plane would move fore and aft really slowly as the wheels went faster and faster. Assuming the control is perfect you'd never see the plane move.

Bottom line, if the treadmill is going to keep pace with the wheels it has to keep the plane from rolling forward (because rolling forward means the wheels are going faster than the treadmill). That requires a massive force (equal to the thrust since the only forces acting on the plane are thrust and the wheel bearings) which spins the wheels really fast. If the wheels could handle it they could be spun up to ludicrous speed, but if we assume they're real they just explode.

You can ignore the math if that's not something you're familiar with, I put it there because Mustonen seemed like he'd get it that way better.

I think the more common hangup for this example is that while people are familiar with F = m*a, the torsional/rotational equivalent of T = I*alpha doesn't see much use so most people aren't ready to do anything but neglect the mass of the wheels. That just happens to fail when a very large torque is applied to the wheels.

[mcski]

Will the plane take off if David Hasselhoff sticks a finger in your ass?

[jono]

I couldn't begin to guess; how'd that work for you?