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#1
03-03-2006, 03:26 PM
 Paradoxic Guest Join Date: Mar 2006 Posts: 7
Plane on a Treadmill - SOLUTION

MODERATOR EDIT: Please note that this thread was started in 2006, has been revived several times since. Please be careful about what you're responding to, since it could be way (way!) old. -- CKDH

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Cecil is wrong. The plane does not take off. This is why:

Original Statement:
"A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?"

Keep in mind that no system is frictionless. We are assuming real systems, and the conveyer as described is well within present capabilities. It can be built.

Fact 1: The force generated by an engine is limited.
Fact 2: Friction is a force
Fact 3: The mass of the airplane is an "inertial force"
Fact 3: The frictional force of the wheel assemblies against the ground, whenever referred to below, contains the friction of the tires against the ground, as well as the wheel assembly (bearings, axle, etc). This also includes the downward force of the mass of the airplane upon the entire assembly. This entire frictional force is not inconsequential.

Fact 4: The conveyer, as it moves faster, exerts more and more backwards force upon the wheels, and this force is not inconsequential. Cecil dismissed this, but this is considerable and is a significant part of the frictional force of the wheel assemblies. Unfortunately, this is Cecil's big physics flaw. The backwards force of the conveyer is equal to the force generated by the conveyor system. THIS is what everyone has ignored, and this data is required in order for the system to be closed. You cannot ignore the force generated by the conveyor system itself. It is equal to the backwards frictional force against the wheels. You cannot ignore the force of the conveyor system itself. The conveyor is moving, a force is being generated to create that motion, and that force, the force to move the conveyor upon which a big hulking mass of airplane is sitting, is large and considerable.

As the below shows, it is the force generated by the conveyor system that prevents the plane from achieving lift.

Fact: The plane has no horizontal motion.
Item: For the plane to have horizontal motion, the force of the engines must be greater than:

[The big hulking inertial mass of the plane] + [frictional force of the wheel assemblies with the ground (including conveyor system)]

If the control system tracks the rotation of the plane's wheels exactly, then it follows that the full, force of the engines at that moment have force exactly equal to the frictional force of the wheel assemblies against the ground (bearings, ground frictional force, etc). Therefore, as long as the treadmill tracks as in the original statement, the full force of the engines are exactly equal to the frictional force of the wheel assemblies against the ground. (it may take some time to realize this). As the constant force of the engines are equal to the frictional force of the wheel assembly and the ground friction, no force is available to counteract the inertial force of the mass of the airplane.

Fact: In the original statement, the velocity of the plane relative to the conveyer, is not the same as the velocity of the plane relative to the air.
Item: No matter how fast the conveyer moves, with or without anything on it, it won't make the air move any faster six feet above it. Therefore, velocity of the air over the wings is independent of the conveyor speed.

Fact: A plane takes off not by it's speed relative to the ground, but by the velocity of the air relative to it's wings.
Item: We can agree that, in the original statement, a plane without wings will not take off. See the next Fact for a further explanation.

FACT: THE AIR FLOW OF THE ENGINES ALONE ARE NOT SUFFICIENT TO PROVIDE LIFT!
Proof: An airplane at the end of a runway as it's about to take off, regardless of whether the airplane is propeller driven, or has engines mounted on it's wings, is at full throttle/full thrust. At the beginning of the runway, this full throttle/thrust is not sufficient to pull enough air over the wings for the plane to take off - if it did, planes would take off immediately and would not require a runway.

This exercise can be compared to an airplane sitting at the beginning of a runway with it's engines at full throttle.

A plane's engines provide forward motion to increase the flow of air relative to the wings. The air flow provided by the engines alone are not sufficient to provide lift. Forward motion is required. If the force of the engines is equal to the frictional force of the wheel assemblies and the conveyer assemblies, then there is no forward motion. Without forward motion, the airflow generated by the engines alone are insufficient to create enough airflow across the wings to provide lift.

(If we can talk about frictionless surfaces, infinite thrust engines, etc. we change the experiment.)

Last edited by C K Dexter Haven; 01-03-2014 at 09:48 AM.