<p>Just think of the hula hoop thing, I’m sure you have all done it.</p>
<p>Hold a hula hoop in front of you perpandicular to your shoulders. toss it away from you, but spin it towards you on the top. It goes forward, hits the ground, and continues to go forward until it eventually halts its forward momentum from the ground friction and rolls towards you. The hoop goes in both directions on the ground while spinning towards you the entire time.</p>
<p>Now imagine that you could apply the throwing force, or let’s say the airplane’s engines against the air, constantly. It would spin towards you, but move away from you. Now imagine you could apply the throwing force increasingly. The hoop would still spin towards you, but accellerate away from you. Even with the moving treadmill, an airplane should be able to apply enough thrust against the air to keep it moving forwards while its wheels spin backwards.</p>
<p>Intuition is misleading. I wrote out the equations. The plane can’t move if the wheels don’t slip. If you think that’s not so, you should write equations too.</p>
<p>It’s simpler than that. Ben, at what point was it decided that the wheels aren’t allowed to rotate? I must have missed that post. If the wheels were rigidly attached to the airplane, then yes, your last post would be correct. In fact, airplanes often “run up” their engines at very high power levels and are able to hold the aircraft still with just the brakes. I am not arguing the holding power of static friction of the tires against the runway. The tires will be able to hold back a force of the static coefficient times the normal force at the contact point. That could be a huge number. </p>
<p>However, you need to remember that the wheels can spin freely. In this case, the tires will be spinning twice as fast as they normally would (since the ground is moving at the same speed backwards that the airplane would normally move forward). The force the moving runway exerts on the tires is not opposite of the direction the wheels would normally spin. They are in the same direction. The wheels would normally spin forward, but the ground spins the wheel in the same direction (its just moving at twice the speed as compared to a normal runway).</p>
<p>Yes the hula hoop slips on the floor, but in this problem, the treadmill in a sense, slips on the wheels</p>
<p>The point is, something can move forward while spinning backwards. And since the airplane is relatively independent of the wheels, who cares if the wheels don’t slip. With the hulahoop, the hoop is the entire structure. Not so of the airplane.</p>
<p>I see what you’re saying, sky, but you could explain one thing? How is the following possible: </p>
<p>(1) the wheels are attached to the airplane; (i.e. the center of mass of the airplane is a constant distance away from the center of mass of each wheel. </p>
<p>(2) each tire is rolling without slipping against the treadmill (i.e. the velocity of the tire matches the velocity of the treadmill at their point of contact)</p>
<p>(3) the airplane moves with respect to a tree.</p>
<p>It is just a simple mathematical fact that these three things cannot happen at once. There is no mechanics involved whatsoever, just geometric facts about relative positions of objects. Do you claim that the above three statements can all be true at once?</p>
<p>What does the distance between the aircraft c.g. and the wheels have to do with anything? This is a one dimensional problem, and any moments about the aircraft c.g. are not illustrative to any aspect of this problem. </p>
<p>I do not understand why all of you are over complicating the problem so much. One other poster is trying to solve this using energy methods (conservation of energy), and now you are trying to introduce moments (I’m assuming that since you are talking about forces acting at a distance from the c.g.). </p>
<p>Simply put, treat this as a simple point mass problem. There is absolutely no reason to treat it in any other manner. You have a point mass, with a thrust applied forward, and any opposing forces due to the moving runway acting opposite of that thrust. </p>
<p>Yes, since the tires are free to spin (assuming a reasonable bearing friction), then the tires can spin at any speed necessary to match the relative speed between the runway and the aircraft (which will simply be twice as fast as a normal takeoff, ie. you have the airplane moving 100 mph forward, and the runway moving 100mph backward, leaving the wheels to spin at 200mph (both actions are causing the wheels to spin “forward” relative to the pilot). </p>
<p>I do not see why you keep questioning the topic of slipping wheels. Ofcourse they are not slipping, they are free to spin. Wheels will only slip once the static friction has been overcome. As long as the wheels remain spinning, the static friction will be constant. (If for instance, you lock the wheels and overcome the static friction, the dynamic or sliding friction will become the new friction between the wheels and the runway (which is less than the static friction). </p>
<p>The airplane will move relative to a fixed observer at the side of the runway/treadmill simply because of Newtonian physics. You have a net forward force, and therefore the plane will continue to accelerate.</p>
<p>The force the runway exerts on the wheels, and subsequently (but not equal to) the force the spinning wheels exert on the plane are MUCH much less than the force of the propulsion system of the plane moving the airplane forward.</p>
<p>You apparently don’t want to engage with simple geometry. The situation you describe is simply not possible – I’ve written out equations of motion for wheels, treadmill, and airplane, and there is no way for the thing you claim to be true. No matter how many paragraphs are written.</p>
<p>(One way to make this easy to think about is replace the treadmill by a wheel the same size as the airplane wheel. Unless there is slippage, the airplane cannot move.)</p>
<p>What geometry is there? This is a one dimensional problem. </p>
<p>There is only ONE equation to solve here, and that is F=ma for the c.g. (assumed as a point mass) of the airplane. If the plane c.g. has a net force (which it does, obviously), then it accelerates. If it accelerates, it will get to an airspeed where the lift is sufficient to take off. It’s as simple as that. </p>
<p>I have done my best to explain it to you. You have done nothing to contribute to this thread except to claim that it is not possible with no proof. What forces are you using? </p>
<p>My feeling is, that you are using the frictional force of the runways against the tire as the opposing force at the center of gravity of the plane. This is not the case… Since you are not appreciating anything I am saying… I am not going to explain this any further to you… but I think you will realize what you are doing wrong…</p>
<p>Okay, a clarification: if we take away the assumption that the treadmill moves backwards at the same rate as the wheels spin, the airplane can take off. But that assumption was clearly stated in the first post, and so that would be a different problem.</p>
<p>And what force does this moving runway (as exactly stated in the problem) exert on the airplane? Does it oppose the force of the engine? (Equal magnitude, opposing the line of action of the thrust…)</p>
<p>Yes, if the conditions of the problem were to hold, this runway would have to exert a backward force sufficient to counterbalance the jet engine thrust. (Which is possible. In certain conditions, the frictional force of the road on the wheel is backwards.)</p>
<p>Remember, we are arguing over the problem as stated in the first post, which says that the runway moves in the opposite direction of the wheel motion, at the same velocity.</p>
<p>But that’s the point… the runway DOESN’T have to counterbalance the thrust of the jet (or any kind) of engine! The runway can only exert a (relatively) constant thrust due to wheel friction (rolling, bearing, etc, etc, etc). It doesn’t matter if the runway/treadmill is moving backwards twice as fast as the airplane would otherwise move forward… the force due to the runway on the airplane isn’t all that different. </p>
<p>When was it said that the treadmill had to hold the airplane back? It didn’t… and that is exactly the point! </p>
<p>Yes… the frictional force of the runway to the tires does oppose the direction of motion… but they are still much lower than the force due to the thrust of the engine. </p>
<p>The conditions of the problem only state that the runway moves backward at the same speed that the airplane is moving forward. If the airplane (relative to a fixed observer … say … in the control tower) is moving 20 mph forward, the treadmill would be moving 20 mph (relative to the fixed observer) backward… the airplane would still be moving forward, but the wheels would be spinning twice as fast.</p>
<p>No No No… no one ever said anything about relative to wheel motion…
I will cut and paste the problem statement once again:</p>
<p>An airplane is sitting on an enormous treadmill. As the plane starts its engines, the treadmill runs in the opposite direction at the same speed the plane is moving. Can the plane take off?</p>
<p>We are talking about the treadmill moving backward at the same speed the airplane CG is moving forward…</p>
<p>I will repeat what your professors have inevitably told you many times… “READ THE PROBLEM CAREFULLY!”</p>
<p>God, some of you guys are idiots. Ben Golub, I used to respect you. I don’t anymore (not just because of this).</p>
<p>I have one question for you: if the wheel was 100% efficient (no bearing friction), would the treadmill’s speed even affect the plane’s velocity?</p>
<p>If the friction between the plane and the plane’s wheel is negligible compared to the friction between the treadmill and the plane’s wheel, then the speed of the treadmill is irrelevant. If that were the case, a treadmill that started moving under a stationary plane would not make the plane move. The wheels, however, would move.</p>
<p>Note: if you meant that the plane’s wheel has friction with neither the plane nor the treadmill, then absolutely nothing would happen.</p>
<p>“God, some of you guys are idiots. Ben Golub, I used to respect you. I don’t anymore (not just because of this).”</p>
<p>lmao… would you like a hug?</p>
<p>Anyway, the plane doesn’t take off, because it is stationary relative to the ground and to the air, despite moving relative to the treads. That is the point of the treadmill, not for it to move at an arbitrary constant speed (which of course would allow the plane to move forward and take off, by reason of the same frictions and forces that let it take off if it is on a stationary treadmill - aka a runway).</p>
<p>River Phoenix: just because you’re stationary relative to the ground doesn’t mean you’re stationary relative to the air. Think: fans and vacuum cleaners.</p>
<p>It’s already been established that a propellor could not come close to providing sufficient lift over the wings. I mean, we could assume that it does, but we could also assume that the answer is “yes” if we wanted. If you read the thread, you’ll note that I’ve already suggested the possibility that the treadmill is inside a wind tunnel.
A “harmless” riddle indeed - the only reason it inspires so much response is that it is so trivial and yet so vaguely worded.</p>
<p>No, the riddle is not as trivial as you claim it is. You simply do not understand the mechanics of the riddle. Although they are simple, you are not grasping the situation at hand. The situation is NOTHING like what you describe. The riddle is not vague, and it is not meant to be any kind of trick question. And it does not require any strange assumption like 100mph headwind to be valid. </p>
<p>The only reason it inspires so much confusion and response is because most people are using their experience in automobiles to answer the question. In an automobile, the wheels must spin in the direction of travel and at the same speed as the car is traveling. The force for the acceleration of a car comes from the wheels, and thus it must be so. </p>
<p>In an airplane, the wheels can spin freely, and are therefore independent of the motion of the airplane. Since the force to accelerate an airplane doesn’t come from the tires, and the forces on the airplane due to the landing gear are small, the airplane is free to accelerate and is not held back by the treadmill/moving runway.</p>