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I've posed the question in this particular way to avoid the ambiguity usually found in the posing of the "airplane on a treadmill" puzzle, e.g.

I'm not specifying how the treadmill is controlled but asking if it can be controlled in such a way that the thrust of the plane's engine is countered with an equal and opposite force. Assume the wheel bearings are frictionless and the wheels rotate freely. Please justify your answer.

[EDIT] Idealize the problem such that we can ignore rolling resistance.

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2 Answers

The answer is not really, not in any practical way. The force on the airplane from the propellor is not balanced by anything from the wheels when you exclude friction. The wheels just slide on the treadmill. If you have wheels with contact friction that have a huge moment of inertia, like enormous flywheels, and you have contact friction but no rolling friction in the axle, then it is technically possible to accelerate the treadmill an enormous amount to give a force at the contact point of the wheels to the ground which is equal to the force on the airplane from the propellor. This will keep the airplane stationary, since the net force on the airplane will be zero.

This force produces a torque

$$ F R$$

were R is the radius of the wheel, and F is the force from the propellor (the two have to balance to keep the airplane from moving forward), and this leads the wheels to accelerate with an angular acceleration of

$$ \dot{\omega} = {FR\over I} $$

Which for normal wheels will be something of order $F\over MR$, i.e. all the force from the propellor is going to spin the wheels. This ridiculous angular acceleration is unrealizable for real airplanes, considering the small fraction of airplane mass in the wheel and the small radius of the wheel.

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The $\alpha$ is ridiculous! But there's something else I thought about too that is separate from those particular physical considerations. The plane's engine power is constant for constant thrust but, whatever is driving the tread must produce linearly increasing with time power to keep the plane from moving. Sooner or later, any physical system driving the tread hits it's finite power limit and then the angular acceleration can no longer counteract the thrust. The treadmill can only delay the plane's motion. –  Alfred Centauri Jul 19 '12 at 3:12
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yes it is possible. You have to account for the Tire Friction and Rolling resistance. It is a bit complicated math but you can resort to experiment. Attach a spring balance to the nose and measure the resistance offered by the tires while the treadmill is spinning at desired speed. Now make sure that engine produces just enough thrust to cancel the resistance and there it is, your model airplane is stationary on a running treadmill. In reality, when the engine starts, it produces surplus thrust even while it is idling that it is almost impossible to hold it down without brakes. So a full scale aircraft with engines turned on, without brakes, will find its way out of the treadmill.

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I'll edit my question based on your answer. –  Alfred Centauri Jul 18 '12 at 22:41
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