That damn airplane-on-a-treadmill problem has come up again, thanks to the New York Times, aided and abetted by Boing Boing. For some reason, this problem inevitably produces very heated responses, such as this one. It doesn't help that the problem is frequently mis-stated to explicitly have the airplane stading still relative to the ground.
The key question here is whether the plane moves relative to the ground (not the treadmill) or not-- if it does, then the plane will eventually take off, and if it doesn't, it won't. The main source of confusion then seems to be what, exactly, the wheels are doing in this scenario.
I tend to think that Rob Knop has it right, particularly when he says that it's a stupid and badly-formed hypothetical. I really hate these sorts of problems-- if you're going to try to come up with physcis-based brain teasers, keep it simple and elegant.
And if you're going to do silly things with treadmills, set it to music.
Once you nail down the assumptions, it's easy to work out. Trouble is, nobody can ever seem to agree on which set of assumptions is 'obviously' the one intended by the wording of the problem.
What if we put a million monkeys typing random postings onto on a blog while on a treadmill powered by their keystrokes. Will that blog ever get off the ground?
Jonathan: depends how many times it's submitted to Dig....
:p
OK, does the jello have bananas or oranges in it?
I tend to think that Rob Knop has it right, particularly when he says that it's a stupid and badly-formed hypothetical.
I don't agree. Part of the "fun" (some people enjoy them, anyway) of such a problem is to clarify exactly what are the hidden assumptions, what are the ambiguities, etc. Once the problem has been restated in a perfectly clear, unambiguous, mathematically precise way, the answer is often obvious. But the trick is how to restate it.
I just find all the "magic treadmill" and "magic wheel bearing" discussion obnoxious. It's just a variant on Zeno's Paradox, the way I see it, and just like Zeno it fails in the real world.
In defense of my heated response, I read Knop's explanation and now I see where the problem is.
Mark Fraunfelder and I actually conversed a good bit over his BoingBoing post and kept on talking past eachother, but I think we finally reached a treadmill detentes. You're right, it's about the assumptions, I operated on the assumption that you just used the example of a treadmill to keep the plane at a velocity of zero relative to the ground's reference frame to emphasize that it's lift, not thrust, that allows most airplanes to fly (I assumed we weren't talking about military jets as well). Mark (in a typical engineer fashion) was arguing over the poor choice of using a treadmill for this task and was rightly pointing out that unless you greatly increased the rolling friction of the wheels, the treadmill simply won't slow down the plane.
So Knop's post was very helpful in just clearly stating the false assumptions being made by both sides (I happily acknowledge my sloppy acceptance of a treadmill), and allowing us to figure out where we had diverged.
I think it is safe to say that if you had an infinite number of monkeys banging on an infinite number of typewriters, you would very shortly have an infinite number of jammed typewriters wallowing in an infinite pile of monkey crap.
And while wallowing does describe a lot of Hamlet, I'm not sure I see the connection.
Re: the treadmill thing, though, it is kinda fun to think about what the treadmill itself does. However, any assumption about a massless wheel or a zero-load treadmill is going to break down immediately. To do a reasonable analysis, you must assign some sort of inertial to the wheel and/or the treadmill. Which perhaps destroys the fun, because it's nice to talk about infinite accelerations and molten lava and so forth, but it's no longer physically reasonable.
Plus, there's the relativistic correction....
For fixing a plane in place, chain it down.... Then, no it doesn't take off. Even if you chain to a wall behind it, and have frictionless pulleys that allow the chain to move freely up and down, just not forward, the plane won't take off. No question there. The whole treadmill thing is just confusing.
As Dave Williamson pointed out in the comments on my blog, there are also these things called wind tunnels that allow planes to fly "in place," by making all of the relative velocity between the plane and the air be the air's motion (in the frame of reference of an observer on the Earth's surface).
-Rob