The problem we are all looking at is the paradox of the treadmill speed matching the wheel speed. This mathematical definition seems to prevent the forward motion of the plane as the forward motion of the plane equals the forward motion of the wheel which by definition is equal to the backwards motion of the treadmill, which counteract the forward motion of the plane and the wheel, so nothing can move. But thrust from the engine requires that something must move with significant energy…?
I see three possible resolutions to this paradox…
-
The paradox is the result of mis-interpreting the problem: The treadmill matches the the speed of the forward motion of the wheel (the center of the wheel in a forward motion vs the rotational velocity of the outside of the wheel - Translational Velocity vs Surface Rotational Velocity). Interpreted this way, the treadmill does not stop the plane from moving forward, just doubles the rotational speed of the wheel. Now wheel rotation speed is equal to plane forward speed + treadmill backward speed = twice the forward speed of the center of the wheel.
-
As discussed in a earlier post, the energy of the thrust is converted completely into angular momentum in the tires, given the amount of energy in the thrust, the wheels would very quickly disintegrate.
-
Something in the problem statement must be invalidated by the paradox. We could say the definition of the problem states that the plane must never move to meet all the conditions. This means that the thrust from the engine is countered by a force applied through the wheels or that the engine applies no thrust. The problem gives us no reason to assume that the plane is malfuntioning. The assumtion in the problem is that the engine thrust will cause the plane to start to move and that the treadmill will react, if the plane does not move, the treadmill will never move - and if the treadmill never moves, then what holds the plane in place? Therefore the plane must move. This paradox necessarily breaks the rule that the treadmill speed matches the wheel speed. This rule is broken so we must now read it as "the treadmill tries it best to keep the plane is place and to match the wheel speed. It no longer perfectly matches speeds and the plane will take off. The thrust energy can not be negated and must go somewhere.
Most thought problems like this are intended to be simple and elegant rather than getting down to dirty details like tire failure rates, the appreciable effects of negligible friction at infinite speeds. Look for the simple problem intended.
Most people in this thread take the stance that friction and angular momentum of the tires should be taken as negligible. I agree.
I also think we need to ignore local malformation of the treadmill and tires. Assume the tires are perfectly round, the bearings perfect, the treadmill as flat as a tarmac.
Also assume that the tires will survive and exaggerated rotational velocity due to the treadmill (to practical a consideration for a physics thought problem).
Given the above assumptions of a “perfect universe” the treadmill will have no effect on the movement of the plane no matter how fast it is moving. No friction, no angular momentum - it is as if the plane we sitting on a perfectly frictionless surface. The only reason we have to think that the plane will not move is that the OP seems to say that it will through the paradox. A paradox is a paradox because it can’t exist in real life (or in physics thought problems). The resolution to the paradox is to ignore it because the treadmill has no effect on the plane’s motion so we don’t care that the paradox says the plane can’t move. The only way the plane would be held still is that the treadmill is effecting it. We just stated that the treadmill can’t effect it. Therefore, the paradox in the problem is not that the treadmill with match the speed of the wheels, but rather the paradox is that the treadmill will effect the plane at all. It can’t, ignore the treadmill and all its rules, drop it and all consideration of it from the problem and we see that the plane takes off normally.
Analogy: A car on a perfectly frictionless level surface can not move. A plane can take off quite well in those circumstances (seaplane, iceplane) Friction with the ground works against the plane where is it needed for the car to move.
A car on a treadmill in the same problem will go nowhere as it can push only against the treadmill. This would bring the treadmill back into play but not the paradox as the engine thrust in a car is not pushing the car forward, it is applying rotation energy to the tires. The car does not need to move forward for the treadmill to start moving (whereas with the plane, the plane must move first). The treadmill matches the movement of the tires and thus the friction required for the car to move does not exist.
I like the rollerblades on the running treadmill analogy as well - very poetic.
I personally think #1 most likely. I have also seen this same problem debated on other sites and the OP is that the treadmill matches the forward speed of the plane (rather than the wheel, from which the confusion lies - Translational or Rotational). Also, the “speed of the tires” would be normally measured as Translational in dist/time or Ratational in rotations/time. The speed of the treadmill would be measured in dist/time. The units match makes me think that the statement speed of wheels matched the speed of th treadmill is referring to the Translation speed of the wheel.