Er, what? The conveyor does not move, yet it does? You’re not saying that the entire conveyor is uprooted and dragged down the runway, stuck to the plane, are you? That’s an interesting take on the problem. Posit a stake connecting the treadmill frame (not belt) to the Earth.
If you assume massless wheels and limited friction. Or limitations on the treadmill. Or that “matching wheel speed” means “matching hub speed” as opposed to “matching tire circumference speed.” It’s all about the assumptions.
The plane experiences forward thrust. Not the same as moving. And posit some sort of sensor/feedback system that reacts when the plane moves a small distance epsilon, for whatever small distance you like, down to infinitesimally small.
The wheels would have to be spinning fast enough that just the friction between the wheels and the axle would be enough to counter the jet engines. I suspect it would be much greater than 400mph.
Again, I feel that this answer goes more against the OP than mine does.
No, that the belt will rotate forward with the plane. The wheels will not rotate and the conveyor will not be rotating compared to the wheels.
The sensor/feedback system is only possible if you have friction at the axle. Minus that, which I take to be the implication, and you are in a catch-22 situation. The plane can’t move without the wheels moving, but there is no friction to apply anywhere to hold the plane in place on the conveyor.
If the plane moves forward then the treadmill has stopped doing its job. The only way for the plane to move forward is if the wheels are turning faster than the treadmill is. Therefore, one of either two things happens. The plane generates a heap of thrust but doesn’t move forwards because the treadmill and wheels are spinning so incredibly fast that they actually create enough rearward force via friction to counterbalance the thrust. Or, we say that there is no friction in the system in which case the treadmill and wheels end up spinning infinitely fast, the plane moves forwards anyway, and the universe implodes destroying the plane, the treadmill, and the wheels.
Nothing in the OP says that the conveyor belt has to keep the plane in place. It just says that the belt moves at a retrograde speed equal to the forward motion of the wheel. The wheels are totally passive and unpowered, therefore their behavior is dictated by traction. You do realize that aircraft wheels aren’t powered, right?
You have just become person number #101 laboring under the misconception that planes work like cars. Aircraft wheels are not powered. Airplanes don’t have transmissions. The wheels turn passively, let’s all chant it together…
Ah. You’re interpeting “conveyor speed” as “speed with respect to the wheels.” I don’t think anyone else is doing that; they’re using “speed with respect to the ground.” Under your interpretation, the conveyor seeks to hold the wheel rotation to zero, yes? Yet another interpretation.
Not exactly true. The “answer” depends on what assumptions you make. I should point out that there’s nothing inherently wrong with deep and profound ignorance; that’s what the whole educational system is here for. Personally, though, I find aggressive ignorance to be counterproductive. You obviously have a different perception.
Only in a sense. Rather I am viewing the conveyor belt as not being a mechanized belt that moves itself to stay in synch, but rather it is like one of the jogging in place ones where it is your inertia forcing it to move backwards.
For real life ones, we need to slant the belts upwards in order for this to work as the belt has intertia and friction about its tumblers. But if you removed that inertia and friction, you could have it perfectly horizontal and if you ran on it, you would stay in place: as the OP requested. But when you are running on it, you are trying to force yourself forward by pushing on the belt. When you put a plane on it, you are pushing against the air. Since the plane starts at rest, its wheels are not initially moving. When you push it forward, the belt has no inertia nor friction to fight itself being drawn forward with the plane.
The wheel moves to the left at 1 mph and the top of the conveyor belt moves to the right at 1 mph. Sum total, 0, the wheel did not move. If the wheel did not move, how could traction start it moving?
So the conveyor belt unilaterally moved itself backwards, turning the wheel. Fine, but no matter how fast you turn the wheel, if the conveyor is going to move backwords at the exact speed the plane would have moved forward then the plane cannot move. Given friction between the wheel and the plane, this is a possible scenario, but requires that friction. If you remove it, there is no speed at which the conveyor could go to equal the forward movement of the plane. If the wheel does physically move to the left it is moving faster to the left than the conveyor belt is moving to the right.
If the wheel is free to rotate, and traction exists, then the wheel will turn against the treadmill. The force of the treadmill is translated to the rotational energy of the wheel, instead of moving the plane backward.
No, it isn’t. The wheel and the belt would diverge from a fixed point at the same speed. The wheel spins twice as fast as it would if the belt weren’t stationary. I’m still not sure you get that the wheel is a passive agent in this case.
Or maybe you’re using the terms “moving” and “spinning” interchangeably, in which case it’s impossible to understand what you mean.
While you’re chanting, let’s strap you to some roller skates and put you on a treadmill. You are offered a helmet, but you refuse it since the wheels turn passively. We’re gonna turn this treadmill on… aren’t you worried about friction? Your response: “Wheels turn passively! Wheels turn passively! Let’s all chant it together! Wheels turn passively! Hey, why am I moving?”
Errrr… you forgot about the thrust from the jet engine strapped to my butt. Wheels still turn passively, a lot faster, and I don’t move forward.
However, I do accept the helmet, because if I’m not coordinated enough to keep my butt pointed precisely against the direction of the treadmill then it’s all going to end in tears.
Yes I know how they work, I fly them for a living.
If you read my post more thoroughly you’ll see that I allow for two possibilities.
We allow for friction in which case, the treadmill ends up going fast enough that the friction is enough to counter the several thousand pounds of thrust coming from the engine.
We say that the system has no friction in which case the plane accelerates down the treadmill and takes off.
The second option is the one that I think of as the “obviously correct” answer in that it is probably the one the poser of the question is looking for. However, the “obviously correct” answer requires that one of the conditions of the experiment not be met, i.e., the moment the plane moves forwards the treadmill is no longer matching the speed of the wheels. This means that the thought experiment, as posed, is seriously flawed and not really worth thinking too hard about.
There is another possibility. IF you could get a treadmill to accelerate fast enough, then once the wheels were turning at a couple of hundred knots, the tires would fail, and the plane would collapse onto its undercarriage struts. Now that it is no longer on wheels it will lose control and turn into a mangled wreck.
Picture a wheel freely rolling down the road at (hub velocity) 4mph - I dunno, maybe it fell off a truck or something. It approaches the back end of the treadmill which is running 2mph (surface velocity) in the opposite direction. The wheel runs up onto the treadmill. Now the wheel is running at 2mph (hub velocity) in its original direction, and the treadmill surface is moving at 2mph in the opposite direction. But the wheel still moves forward, relative to both the ground and the treadmill.
Or are you arguing that the wheel is then moving at 4mph (hub speed) * relative to the surface of the conveyor belt* so the belt should speed up (assuming a powered belt) to 4mph; that would hold the wheel still with respect to the ground, provided that there were no other forces on the wheel. But the engine applies a force to the wheel that will accelerate the wheel hub forward barring an equal opposing force.
