Plane on a Treadmill, redux

Cecil's Columns/Staff Reports
21

Unsurprisingly, I agree with Cecil here, but let’s look for a moment at a real-world picture of BR#2, the situation in which the treadmill accelerates quickly enough to cancel out the thrust from the engines.

The engines of the plane are at full throttle, straining with thousands of pounds of thrust against the incredible forces being exerted tangentially on its wheels—it would be a bit like stopping a condor from flying away by rubbing the tip of one of its claws with an emory board, only moreso. Any tiny irregularity in the surface of the treadmill would be magnified by the incredible speeds it achieved; if the airplane’s tires hit the slightest bump they would be jounced upward, the plane would lose contact with the treadmill, and it would move forward. If the wheels were not made of some kind of unobtanium, they would melt before your eyes, and a shower of sparks would erupt from the landing struts as they hit the treadmill—and, soon enough, the entire plane would be ground away to scrap. Finally, since all the backwards forces are being applied in shear to the very bottom of the system, the airplane would be rotationally unstable: there would be a net rotational force that would flip the airplane onto its nose, at which point the engines would shove the airplane down into the treadmill, vastly increasing the treadmill’s force, and the airplane would be flung by its nose off the back of the treadmill to crash.

Most of these are practical concerns, which Cecil rightly dismisses, though I think they contribute helpfully to a picture of the situation. But the last is intrinsic to the setup—if the treadmill applies force at the bottom, and the engines apply force in the middle, there will be a massive net rotational force. (You can try this at home, with a full glass of milk—push at the bottom with one finger, and in the middle from the opposite side with another finger, and let us know what happens.) This is exacerbated by the fact that the wheels convey all the force to the airplane via their axles—i.e., rotationally—and if you counteract that by making them frictionless then you decouple them from the airplane entirely and the treadmill has no effect. However you frame it, unless your airplane is a very unusual design, you wind up with the airplane being flipped nose-down around its front axle. However stable the three-point resting state may be, if the forces are sufficient to counteract the engines at full throttle, they’ll be sufficient to flip the plane.

22

Hey look at that, I made it into Cecil’s column!

23

I thought this whole thing was just a semantic puzzle like teachers give students to get them to comprehend what they read. If the puzzle says the plane is moving forward then it is clearly not standing still. When it is standing still the treadmill is not turned on. If the plane is going 50 mph forward then the treadmill may be going 50 mph in other direction
but the plane is moving inn relation to an onlooker. If a car is moving forward while the treadmill beneath it is going in the opp direction it just means the car is using more power
to go forward then normal but it is still moving forward. The puzzle did not say treadmill
exerting enough force on wheels or whatever to cancel whatever power vehicle using,just
going backwards at same speed vehicle is going forwards. But then maybe I need a
teacher to help me compehend what I read.

24

Alternately, you could reframe the question like so: “Some guy is standing next to a car with a belt sander, holding it against the driver-side door. The sander is programmed to exactly match the speed of the car, but in the opposite direction. Can the belt sander stop the car from driving off?”

This is a more or less equivalent situation, since both the car and the plane have access to their motive force. Yes, it would technically be possible for the belt sander to run so incredibly fast that the car would be prevented from moving, but 1) it would have to be a magic belt sander to do it, 2) it would have to be a magic (but not frictionless) car door to withstand it, and 3) the car would have a net rotational force, and would fishtail around the point of contact with the sander.

25

(Sorry to reply to myself here, but…)

  1. The belt sander would have to move much much faster than the top speed of the car—orders of magntitude faster than 100 mph or whatever. It is this, I think, that makes the secondary interpretation of the question moot—it is hard to see how the speed of the sander/treadmill could be called “exactly the same” as the speed of the vehicle when it’s moving so much faster than the vehicle ever could.
26

God help me for IANA geek, but DanBlather said:

“They do, it’s called a catapult. But they don’t make it strong enough to launch the plane by itself, they just make it strong enough to work in conjunction with the engines so that the plane will take off in a reasonably short distance.”

This does not help me. I’ve seen those catapults in person and believe me that bird is moving with alacrity in a New York minute. It seems to me that if the Navy could do it with a conveyor belt, the carriers would be designed radically differently and for the better.

This does not appear to me to be a valid metaphor.

27

If we assume windspeeds (relative to the ground) in excess of hurricane speeds, then yes, the (completely stationary) plane topples over. Stop the treadmill and call your insurance agent.

But seriously…

Point #1: A plane’s wings must travel THROUGH the air to generate lift.

Agreed? Good…

Point #2: If the plane is held to zero mph (relative to the ground) than the wings are not moving forward THROUGH the air.

Agreed? Good…

Add Point #1 and Point #2 and you get zero lift.

Note: Yes, I understand that ‘a car propels itself with tires and a plane with it’s engines’ - but the plane’s wings still need to go forward through the air to generate lift. If the treadmill keeps the plane from moving forward (relative to the ground and relative to the air) then there is no lift.

We are, of course, assuming that the plane’s engines aren’t somehow moving all of the air AROUND the plane, right?

Assuming zero headwind (air not moving relative to ground) and plane not moving (relative to ground) - and you get plane not moving relative to air. No lift.

28

Two points:

  1. The Navy would still need a long-ass treadmill, because the plane will end up moving forward (relative to the ship) even though the treadmill’s going (excluding BR#2, which would require tires that absorb tons of friction but don’t burst or anything).

  2. The Navy doesn’t do it because it’s a hypothetical Freakin’ treadmill! Imagine the cost to even attempt to construct a treadmill that could hold an airplane, let alone magically move as fast as it perceived the plane to be moving (and in the real world would fail anyway).

For those still struggling with BR#1, imagine a small conveyor belt humming along, and you hold a hot wheels car on it. You hold it still. The conveyor belt speeds up. What happens? The car stays still, the wheels just spin faster. You can even move the car in the opposite direction of the direction of the treadmill, and if the conveyor belt speeds up, you’ll still be moving the car at the same speed - those tires will just spin faster.

The only sort-of problem is if the conveyor belt went so damned fast that the friction on the tires created acceleration opposite to yours, but in the real world the tires could never withstand it.

29

Correct. But that’s not what happens. The plane still moves forward.

Never happen. The treadmill can’t stop the plane from moving forward - the tires are just free-wheeling faster than they normally would.

30

[QUOTE=bup]

The Navy doesn’t do it because it’s a hypothetical Freakin’ treadmill! Imagine the cost to even attempt to construct a treadmill that could hold an airplane, let alone magically move as fast as it perceived the plane to be moving (and in the real world would fail anyway).
QUOTE]

I understand the hypothetical nature of the question. Putting a man on the moon at one time fell into the same category.

My point is, if a plane could theoretically become airborne on some sort of mechanism that obviated a long runway (which is, I believe the underlying assumption here), someone would have attempted it in practice (which I must assume from this protracted discussion has not occurred.)

31

Unfortunately, the contrast between the car and the plane isn’t as great as it seems to be at first blush. Normally, at a certain accelerator pedal position, the car’s engine delivers a certain amount of power, which is usually used to overcome friction, tire hysteresis, aerodynamic drag, and so forth. The resulting car velocity is the velocity at which all the forces balance.

If the car is on a treadmill, and doesn’t accelerate, the same thing has to happen–the forces need to balance. However, in this case, the most important component of power loss–aerodynamic drag–isn’t present. So one of two things will happen. Either the driver has to back off on pedal position (thereby artificially reducing the engine power output) or the engine will keep speeding up, likely to some place well past the redline speed. If the engine doesn’t blow, the actual power output will drop significantly, until it matches the friction and hysteresis losses.

Either way, you’re reducing the automobile’s power output, something you’re not doing with the plane. This is still, as you say, a fundamental difference between the two modes of propulsion, but it’s not as obvious as it first appears.

I don’t think I disagree with you in meaning, other than to say that you don’t need to “infinitely” increase anything. Substantially, yes; infinitely, no. That distinction was one of Cecil’s errors in the original column.

I disagree. My sense is that the horizontal distance between the center of mass and the front landing gear is much more than the vertical distance. That means it should take a force higher than the weight to send the plane ass-over-teakettle. Most airplanes have a thrust-to-weight ratio less than one, so it shouldn’t be close.

Additionally, frictionless wheels will not decouple the plane from the treadmill, because of the inertial forces. I go into more detail here.

I suspect that means you posted way too many times in that thread. In fact, I just checked, and only one idiot posted more times in that thread than you did…

32

:stuck_out_tongue:

I will not get involved in this one though, or at least I keep telling myself that.

33

Nope. The plane moves forward just as fast as ever, and needs as much real estate to get airborne.

34

Both true, but the question is not whether an airplane can fly with no forward motion, but whether a treadmill in the described situation can create the condition for your Point #2.

35

Ho. ly. cow.

The light just came on. I get it.

Thank you for suffering a fool so gracefully.

36

Bup, can we agree that there is SOME friction (from tires, bearings, whatever) caused by the rotating wheels? And this will put SOME restraint on the forward motion of the plane?

And can we agree that the faster the treadmill moves, the more friction is generated and the greater the restraint is?

Then we only have to calculate HOW MUCH friction and HOW FAST the treadmill needs to move to generate the amount of friction necessary to counteract the attempted forward motion of the plane.

Practical? Hardly. The treadmill would have to be going near the speed of light, methinks; the tires would melt, the bearings fuse, and the whole landing gear would go up in a puff of smoke and flame. But there is a THEORETICAL number that we could calculate.

37

There’s still a mistake in Cecil’s column. Near the bottom, he says

The speed of light does not impose a limitation. The “constant acceleration” the belt needs is not the familiar non-relativistic acceleration (which would eventually run into relativistic limitations), but a quantity called proper acceleration. At low speeds, the two are equivalent, but there’s no limit on how high a proper speed can get (proper speed is speed times the relativistic factor gamma). So a belt with sufficiently high proper acceleration would asymptotically approach the speed of light, but never exceed it, and could keep the plane in place.

There are, however, other limitations which do arise, and which would arise long before you got to relativistic speeds. For starters, the plane would eventually take off from moving air entrained to the treadmill, flowing over the wings. I think that an exponentially-accelerating treadmill might (I say “might”, but I have no intention to calculate it) be able to overcome this, too, but a merely constant acceleration won’t.

38

Just saw this after posting:

No, we can’t agree on that. Friction in the real world is, of course, very complicated, and in real situations, it can be dependant on speed. But it does not necessarily increase with increasing relative speed, and in the simplest models of friction, it actually stays constant for any nonzero speed (for zero relative speed, friction can actually be greater than for any case with relative motion).

39

BR#2 makes it clear that if the treadmill exerts as much backwards force as the engines exert forward force, the plane doesn’t move, and won’t take off. The relevant question is whether a treadmill could do this.

We could call BR#3 that there is a limit to the force that can possibly be exerted by the treadmill on the plane. The treadmill exerts force on the plane by applying
friction to the tires, so the maximum backwards force it can exert will be the weight of the plane, times the coefficient of friction for tires on treadmill (probably around 0.9). Essentially, this means that any plane whose engines can exert enough force to lift 90% of its weight straight up would be able to take off on even the most aggressively accelerating treadmill.

The really weird part is that you can make it a bit easier for the plane to take off by stopping the wheels suddenly. They will start to skid, so the friction will depend on the coefficient of kinetic friction, which is generally lower than the coefficient of static friction. Counterintuitive (hit the brakes to go faster?), but it
should work.

This whole problem is actually equivalent to asking “can a plane take off if its wheels are jammed and can’t rotate”, and that the answer depends on the plane’s thrust-to-weight ratio.

40

In that case, all bets are off. I certainly don’t expect a linear relationship between increasing speed and increasing friction, but if a positive correlation can become a negative one at some uncertain value, how can this question be answered at all?

In other words, if, as the speed increases, the amount of friction that can be applied to the wheels by the treadmill is insufficient to counterbalance the resulting forward motion, then the wheels cease to be a significant factor, the air speed takes over and the plane takes off.