I’m starting to think that I screwed the pooch on this problem from the beginning.
I was making the assumption that the treadmill could not exert a force on the plane, (assuming frictionless wheel bearings), and that the whole situation was equivalent to the plane taking off from a frictionless surface. After reading treis posts, however, I’m pretty sure my original assumption was incorrect. I want to think about it some more, but I’m going to bed now.
The situation described in the OP is literally impossible. Not just physically impossible–logically impossible. The treadmill can’t match the speed of the airplane’s wheels. Say the plane’s engine starts, applying a force which would cause the wheels to turn under normal circumstances. After a short interval, the wheel should be going at some speed, let’s call it “1.” The treadmill, however, will have accelerated to catch up, and will be going at speed “1” in the opposite direction. But this very process of acceleration will have caused the wheel to speed up past “1” and onto “2.” So the treadmill keeps accelerating through to “2.” But, of course, again this means the wheel will be accelerated past “2” to “4.”
And so on ad infinitum.
It’s not that the wheel would be spinning infinitely fast or anything like that. It’s just that it turns out the treadmill can never catch up to the wheel. Since the OP scenario says teh treadmill matches the wheel’s speed, it turns out the scenario is logically impossible. It’s self-contradictory.
Everything follows from a contradiction. So everyone is right.
-FrL-
Except wait. I’m confused because I know a car can be on a treadmill which automatically matches the car’s speed. I don’t know what the difference is between the car situation and the airplane situation which should be relevant to this question. I don’t see how the source of the applied force would make a difference as to whether the treadmill could keep up with the wheels, logically speaking.
I don’t understand how the treadmill could be exerting force on anything other than the wheels, “assuming frictionless wheel bearings.” If the wheel bearings are frictionless, then presumably they’re not transferring any force from the wheel to the plane, are they?
What is Treis’ insight which I am not understanding?
It looks like you are saying that the “speed of the wheels” means the speed with which the whole plane is going forward. But if that’s what it means then there really isn’t even a problem to think about. I think everyone would readily agree that if the plane can move forward then it can take off.
The bolded portion of the OP applies a kinematic constraint to the system, regardless of friction in the bearings, how much thrust the engines can produce, or even if the plane is replaced with a wheel-driven vehicle. It’s given the tangential velocity of the wheel at the point of contact is equal and opposite to the linear motion of the conveyor, relative to some outside reference frame (v[sub]t[/sub] = -v[sub]conv[/sub]). If the wheel is to maintain rolling contact with the conveyor, the hub of the wheel must remain fixed in space, and by definition, the plane can not move.
Unless the wheels are skidding – which is technically possible, even if highly unlikely under this relationship. The simplest case would be locking the brakes. In this case, v[sub]t[/sub] = -v[sub]conv[/sub] = 0, as the wheels are not turning, and the conveyor is not moving. If the engines produce enough thrust to overcome static friction, the aircraft will accelerate.
Xema and others, this problem is subtly different from that described in the AVweb column posted by ElvisL1ves. In that scenario, the conveyor matches the velocity of the aircraft over the ground, rather than the tangential velocity of the wheel. In that case, the aircraft is now defined as moving, and the wheels will indeed spin with a tangential velocity twice that of the aircraft and conveyor’s linear velocity.
Something like this could actually be designed. You would need some kind of feedback from the treadmill to the plane’s engine so that the treadmill could limit the plane’s speed to one it could match. The simplest way would be to just disable the engine and match speeds at zero.
Or make the surface of the treadmill out of some super-sticky glue so that the wheels can’t roll no matter how much force is applied. Then you can match speeds at zero again
I don’t really see how the OP’s scenario would apply to “real” aircraft, either. You could drive a car on a treadmill because the wheels are driven by the engine – they rotate because the engine applies a torque to the driveshaft. Rolling contact between the wheels and the treadmill will drive the treadmill at the same velocity as the contact surface of the tires – the wheel’s tangential velocity.
In contrast, aircraft engines (jets and props) push huge amounts of air backwards, with the resulting reaction pushing the plane forward. As there is no connection between the engine and wheels (they free-wheel), any rotation of the wheels has to be the result of either the aircraft moving over the ground, or the ground moving under the aircraft. Since the defining relation in this problem does not allow the aircraft to move, this means that the treadmill has to initiate motion – it moves backwards, and the engines slowly ramp up to keep the airplane from moving backwards.
I think the way something like this would be designed, the treadmill wouldn’t ask the wheel “how fast are you rolling?” and the match that speed. What it would do instead is ask “where are you?” and if it’s in not where it should be then the treadmill would change speeds to move it back to where it was. The treadmill would have to be capable of greater acceleration that the wheel is in order to be sure that it could always do that. Sample several times per second and you should be fine.
This isn’t perfect but it illustrates the basic effect.
For those of you without a fast connection, guy has an electric fan mounted on a skateboard. Fan blowing air back. So that’s the airplane.
Skateboard is sitting on a long sheet of paper. That’s the conveyor belt.
Fan is turned on and skateboard released. Guy begins pulling the paper out from under the skateboard wheels. Despite this the skateboard moves forward relative to the fixed floor continuously.
Also note that when the skateboard reaches the end of the paper it doesn’t speed up or slow down relative to the floor.
The speed of the airplane relative to the fixed ground is pretty much determined by the engine. That’s it. The conveyor belt makes no real difference.
As I was writing my first post in this thread I wrote that there was two possible ways to interpet the problem. This was the other way and from the time of writing out the first scenario I had forgotten what I thought the second one was. If this is the case then you have the same situation as the train example. It is possible to do this for a while and the plane will not take off.
Actually they happen at the same instant. As you note the airplane is not allowed to move and that includes backwards. The acceleration of the treadmill equals the force of the engines*the radius of the wheels squared/Iwheels.
Right the treadmill will be constantly increasing its acceleration. What physically is happening is that all of energy from the jets engines are being transferred into the wheels. They spin faster and faster until at some point they hit a physical limit and the whole thing goes kablooie. The only way to put energy in the wheels is to apply a torque and in this case that is friction.
The wheel is attached by a strut to the airplane and exerts a force that way. If you look at either of the pdf files I drew you would see that in order for the wheel to not accelerate backwards there must be a force opposing the frictional force. This comes from through the strut from the engines.
The only way for that to happen is for the wheels to slip and I don’t see any indication of that.
I think that there’s still a certain amount of confusion about the velocities involved; when I posed this problem to my g/f she made some of the same assumptions at first.
v1 is the velocity of the plane itself, that is the velocity of its center of mass wrt the fixed earth.
v2 is the tangential velocity of the point on the wheel that is touching the conveyor belt, also wrt the fixed earth. Barring slipping, v2 must have the same absolute value (but different direction) as the speed that the conveyor belt is running.
v1 and v2 are not linked - read my thought experiment of the rollerblader on a treadmill to see what I mean. v2 can be 1,000mph while v1 is 1mph.
The center of mass of the wheel is moving at v1 (since it’s fastened to the plane) but that has nothing to do with v2.
v1 is the only one that matters when seeing if the plane can take off.
For what it’s worth, as I was reading robby’s initial attempt to describe the system, I wondered if it would be possible to modify the model for easier cogitation while still capturing the same physics. However, as should soon be obvious, I know next to nothing about physics. So, I’d like to know whether this makes any sense. Start with the airplane sitting on the treadmill. Treadmill starts, carrying the plane tail-first with it; accelerates smoothly to the takeoff speed of the airplane (we, uh, assume an ideal aircraft shape that doesn’t get ripped to pieces traveling backward through the air like this ;)); and holds at that speed. Now the airplane applies thrust (prop, jet, whatever, plus assumptions as before about the airframe not being torn apart). Relative to the treadmill, the airplane moves with increasing speed up the belt, but relative to a stationary object, the airplane gradually decelerates, stops moving backward, and then accelerates forward. And, eventually, takes off.
Right?
It is, quite possibly, the perfect Straight Dope thread.
If there is no slipping, and v2 is the tangential speed of the wheels, and the speed of the belt is -v2 then v1 has to be v2-v2 = 0. If v1 is not zero, then the speed of the belt can’t be -v2, or the wheel has to be skidding. There isn’t any other way.
This is wrong you cannot drive a car on a treadmill with a constant speed while applying a torque to the wheels. See this Dyno Machine for an example. You can see the front wheel is fixed otherwise the motorcycle would accelerate forward.
