Originally posted by: randay
Originally posted by: jagec
At that point you CAN actually match the so-called "wheel speed" of the plane with the treadmill.
You can never match treadmill speed with wheel speed. The wheel will always be moving faster then the treadmill. x=y=x+y
Disclaimer: Stopping an airplane from taking off by using a treadmill involves some rather unrealistic assumptions about the power of the treadmill and the robustness of the wheels.
BUT assuming a treadmill of arbitrarily high power, and indestructible wheels of nonzero mass (I'll throw frictionless bearings in there to keep it simple), the treadmill can slow down the plane by speeding up the wheels.
So,
we put a "speed sensor" with a wireless transmitter into the hub of one wheel. The treadmill is designed to match this measured speed in reverse.
Now the plane pulses its engines for a brief second, which causes it to begin moving forward with a small, constant speed. The treadmill accelerates to try and match this speed. The extra force is exerted on the wheels, which have a certain angular momentum. As long as the treadmill continues to accelerate and provide that force, it couples with an equal force at the hub to cause a torque and induce angular acceleration in the wheels. Well, that force at the hub is borrowed from the plane's momentum, since the engines are off. Eventually all of the momentum term is eaten up, the plane comes to a "stop", and the treadmill is now matching the wheel speed. Now that acceleration has stopped, the force on the wheels (and therefore angular acceleration) is back to nil, so the plane will sit in place for forever, barring frictional losses or further engine or treadmill inputs.
Another example: Get a Matchbox car, and one of those flywheel toy cars. Tie a string to the front of the Matchbox car, and measure the force required to keep it stationary on a treadmill. This will be a very small initial force as the wheels accelerate to the treadmill speed, and then an even smaller constant force to compensate for frictional losses. Repeat the experiment with the flywheel car. There will be a fairly substantial initial force as both the wheels and the flywheel accelerate to the treadmill speed, and then a much smaller constant force to compensate for frictional losses (slightly greater than the matchbox car, but a similar order of magnitude). Now accelerate the treadmill on both cars. The matchbox car's force will increase slightly, due to a tiny amount of angular momentum and slowly increasing frictional losses. The flywheel car's force will increase much more, and will remain at a constant high point (slowly increasing due to friction) as the treadmill maintains acceleration.
The way that the treadmill can overcome the force of the engines is through constant acceleration. I believe that once the plane runs out of fuel, the amount of energy tied up in angular momentum in its wheels will be roughly double the amount of energy that was extracted from the fuel (the treadmill contributes an equal amount of force in the opposite direction).
Originally posted by: smack Down
Physics and reality really have nothing to do with the question. It is all in how you define the control system on the treadmill. Make it match the ground speed the plane, or car takes off. Make it match the wheel speed plane or car stays put. Any idiot should be able to understand that.
Physics and reality have everything to do with control systems. The only reason that it is
possible to design a control system which can keep the plane on the ground is because of a bizarre second-order force coupling (ignoring friction). If the plane was on skis, we assumed a frictionless environment, and the control system was ordered to "keep the plane stationary", it could
not. You can't
define the control system as working in way that defies physics, and call it a physics problem.
Originally posted by: Cerpin Taxt
I don't disagree, but I guess my point is that there must be an interval between the point where the disparity begins between the speeds of the wheels and treadmill, and the point where those second-order effects begin to affect the speed of the plane as you described. Thus, during the interval, the pre-conditions of the original problem are violated, making those preconditions impossible in operational terms. In other words, there isn't a real-world scenario where the preconditions would hold true at all times. Absent some qualifier in the original problem that the preconditions need not hold true during that interval, the preconditions are basically impossible.
Yes, there is an interval, but that just means that your error signal changed...which is how control systems work. Temperature controllers don't keep the room at EXACTLY 70 degrees, they fluctuate between slightly below and slightly above that, with the amount of fluctuation depending on the room, your heating apparatus, and the quality, type, and calibration of the controller.
Similarly, if the treadmill is ordered to "match the wheel speed", it will ramp up the acceleration of the treadmill (slowing down the plane via angular momentum) until it overshoots the mark and the plane actually starts moving backwards (while the treadmill continues to accelerate). Then it will decrease the acceleration until the plane catches up and starts moving forward again. And so on. To human eyes it may remain motionless, but that's just the hallmark of a good control system.