shiggins, I just realized one of the major things that is stopping you from understanding this model in terms of physics and the forces acting on the bodies. You say the wheels are freerolling, ie. not driven. You are only partially right in that. The wheels are free rolling as long as the plane is sitting there. As soon as the engines are powered up and produce thrust, they are no longer free rolling. They act like driven wheels from then on, as long as they are in contact with a surface and experiencing friction.
NOOOOO for the love of god NO! That is not what is stopping me, it's what is stopping you. They are not driven, ever. They are freerolling wheels, if they were driven, it would go against the very definition of freerolling. They react to the motion of the system, they do not cause it. They're always freerolling.
You are correct in that they don't ever have any drive mechanism attahced to them. But they do have thrust forces acting on them. That's the point I was trying to make. Free rolling was not the best word. If you think in terms of engines attached to them, then they are always free rolling. However, you are assuming that because they don't have engines attached to them, they will never have forces acting on them. This is simply not the case. With me so far?
That's not what I am assuming. I know there are forces acting on them. However, I know that being freerolling, the forces acting on the aircraft, through the wheels, caused by the conveyer moving in the opposite direction is not enough to overcome those forces acting in the direction of motion.
The question said the plane was initially at rest. The wheels are therefore not spining. The treadmill is not spinning. The engines fire up. The plane, due to the thurst forces acting on it, tries to move forward. The tires try to rotate. As they do, the treadmill surface moves backward, inch for inch, exactly the amount the tires rotate. So the tires have no translational motion. Zero. And it stays zero because as they speed up, so does the treadmill going in the opposite direction. If they are spinning faster, then so is the treadmill. Therefore, their speeds still match. If their speeds match, the plane has no translational motion. It starts at zero and stays at zero translational speed. No change in speed means no translational acceleration. Zero initial translational speed plus zero translational acceleration equals zero final translational speed. The rotational velocity keeps increasing. That's what I said to begin with. Where are you getting your translational acceleration from?
An unopposed force acting on a mass results in acceleration, right? Since you think the plane will accelerate translationally, and keep its wheels with it, wouldn't the wheels need to move up the treadmill? If they do, then their speed is not equal and opposite to the treadmill, is it? The question says that the wheel and treadmill speeds stay equal and opposite at all times. Since this is a given condition, how can the wheels move up the treadmill? Their rotational speed is always matched by the treadmill. Always. If this is the case, and the question says it is, there can never be any translational speed. Ever. So, does the plane leave its wheels behind? If the wheels don't move translationally, how can the plane?
So, I'm driving my car, the wheel has a translational velocity of 5mph, it also has a rotational velocity corresponding to that 5mph. You're saying that the rotational velocity can be nothing other than that value to produce a 5mph translational velocity? What happens if I get a little trigger happy, and start doing a burnout. I'm moving forward at 5mph, but my wheels are spinning a lot faster than they were when I was simply driving at 5mph? So, the rotational and translational motion of a wheel must be related, but not always directly proportional. It depends on the contact surface of the tire, is there slippage as in the burnout, is there a conveyer belt as in our question.
You're not paying attention. The original question is absolute garbage. It's nonsense. I gave the correct wording for the question, you read it, I know you did. You have clearly chosen to ignore it. Just so you don't ask, the reason the original question is garbage is in your response. Out of the two velocities you could have chosen to have the treadmill match, you chose rotational. If the treadmill matches the rotational velocity of the wheel, the whole system goes to hell. You have to choose the translational velocity, or the velocity of the aircraft. Which is what the question originally asked, before it was posted on this forum in its butcherd form.
I have always worked with my understanding of the question. Can you post the actual wording of the original question, and your wording, here? Now? Thanks.
Original Question http://ferrarichat.com/forum/showpost.php?p=135395153&postcount=1 Correct Wording. http://ferrarichat.com/forum/showpost.php?p=135415605&postcount=555
Original question: Imagine a plane is sat on the beginning of a massive conveyor belt/travelator type arrangement, as wide and as long as a runway, and intends to take off. The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation. There is no wind. Can the plane take off? Your rewording: "On a day with absolutely calm wind, a plane is standing on a runway that can move (some sort of band conveyor). The plane moves in one direction, while the conveyor moves in the opposite direction. The conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction). Can the airplane ever take off?"
What is the difference in the wording? What is the original question missing? The only thing I can see is that the original question says it exactly matches the speed of the wheels in the opposite direction, and you have said it matches the speed without specifying what item's speed you are referring to. It seems to me the original question is worded better in that it is more specific? What am I missing?
The speed of the treadmill is dependent on the speed of the airplane, not the speed of the wheel. Or rather, the translational speed of the wheel, not the rotational speed. You can interpret the original question in that way, but many, including yourself, did not.
But you didn't say it matches the speed of the airplane in your rewording. You just said it matches the speed. Speed of what? Passersby? Ship on the horizon? Airplane (which is what you meant)? How is it a better/more exact wording? And why is the original question garbage? Because it does not say whether it is matching the wheel's rotational or translational speed? Honestly tyring to understand why you called it garbage.
I didn't reword the question. It's the wording, as far as I can tell, of the question from the first time it was presented. Sure, you can come up with any number of reference frames with the new wording too. Speed with respect to the ground? or realtive to a cow orbiting the earth on the back of Jesus' dog. Realistically, one would pick either groundspeed, or airspeed. Both yield the same result. In the original wording of this thread, you can chose translational speed, which is possible, and has the plane flying, or rotational speed, which is not possible and leads to pages and pages of discussion.
OK, I think we are getting somewhere. Why can the conveyor belt not match the rotational speed of the wheel in the opposite direction?
Simple physics, the forces need to sum in order for the plane not to move. The thrust force is much greater than the friction force. So as the conveyor attempts to match the wheel speed then the wheel speed would continually "out accelerate" the conveyor belt until a catastropic failure in the wheel assembly - which prevents the plane from taking off.
Except as the tires speed up, so does the treadmill in the opposite direction. You now have a new system with the tires and treadmill going at a faster speed, but still no forward motion because the treadmill cheated and sped up. It always cheats and keeps up. Damn treadmill! How is the plane supposed to move forward? It doesn't. For the plane to move forward, the tires have to speed up to cover more ground than the amount of treadmill they are covering at that moment. Right? If you run to stand still on a treadmill, and then you want to move up the treadmill, you have to run faster, right? What if the treadmill sped up at the same time you did? You run faster but you still don't move up, right? Do you agree that if the tires and treadmill speed up equally in opposite directions, the tire's position on the treadmill will not change? If so, can you explain how the plane moves forward if the wheels don't? Does it lean forward? Does it separate from the wheels? Does it stretch?
The question states that there is no delay. The speeds match at all times. Imagine it's a very long treadmill of superstrong, super light carbon fiber, sitting on frictionless bearings and powered by a jet engines firing simultaneously with the plane's jet. So there is no out acceleration. If tehre was, you would be right. But there isn't. You can't change the situation as laid out in the question.
Let's look at the very instant that the wheel has it's first rotational velocity, the conveyer will match that. But, the conveyer is adding rotational velocity to the wheel, so the wheel has a new rotational velocity which is the result of the forward motion of the plane, and the effect of the conveyer, the conveyer has to match this. Again, increasing the the wheels rotational velocity, and on it goes, a vicious cycle which heads towards infinity. It's an infinite loop because the reaction of the conveyer is driven by the reaction of the wheel, which is inturn, driven by the reaction of the conveyer.
The reaction is instantaneous. If there was the slightest delay, then you would be right. But the question's wording implies that it is always matched, not always "caught up to". There is no out acceleration. Now how does that affect your prediction of what happens to the plane. Does it take off or not? If you say it does, then you are saying it moves forward. In order to move forward, the wheel speed has to be greater than the treadmill speed. The question states this cannot be. So your answer cannot be right. My answer has no conflicts. The wheel speed matches the treadmill speed in the opposite direction. At all times. This results in the wheels spinning faster adn faster as the engine provides more thrust. But there is no translational speed. The plane does not move. No movement, no airflow, no lift, no takeoff. Everything is consistent in my answer. I can back it up with scalars, vectors, component forces, resultant forces, frames of reference, Newton's laws, everything. Yet you claim I must be wrong and you must be right. Explain how the wheels move forward translationally on a surface that is always matching their speed? You can't. Because it can't happen. All the wheels do is spin in place. That means the plane does not move. Repeat after me: No movement, no airflow, no lift, no takeoff.
Yes, but then the situation is impossible. I mentioned this on the first page of the first iteration of the thread. The problem statement gives a condition which is not physically possible.
OK, I couldn't make it through the whole thread it was like punching myself in the face over and over - but, perhaps the equasion originally posted can be simplified - what it seems like Ash, Payne, myself, et al are assuming is the the conveyor is acting to prevent the plane moving forward - that is it's principal duty, which was, I assumed, why it was introduced into the equasion. Original poster, please jump in and correct me if I'm wrong... If I'm correct, let's make this a little less dynamic in terms of everyone's assumptions. Instead of a conveyor belt, assume the plane is sitting on frictionless roller drums and is effectively prevented from moving forward at all - but allowed to rise if enough lift is generated. I say this because it sounds like what the original poster intended as the situation - a plane with no movement trying to lift off only by generating thrust. That being the case, it seems pretty obvious that the plane does not lift off (at least most planes will not generate enough thrust alone to lift them - they require the air moving over the wings to generate lift - it's not a rocket). I think what most people are getting caught up in are the variables associated with the conveyor, but I read the initial post to mean that the plane had a groundspeed of 0 and was using the conveyor as a way to achieve this...
Of course it is. Ever seen two gears spinning against each other? Go stand on a treadmill with rollerblades. It's possible, isn't it?
