Talk:Carroll's paradox
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This idea can be approximated experimentally. Arrange a greased hook in the ceiling to be cut at the 6:00 position. Screw an eye into the top of your rod. Catch the hook with the eye, and pull back to (nearly) horizontal. Release. Observe the rod as it flies off the hook. Measure the whirling and infer the angular momentum.
At the instant of release, the axis of rotation suddenly changes from the end of the rod to its center of gravity. At that moment, there will be a sudden change in the motion, like a bad manual gear shift. Nonethess, my casual experiments with a pencil seem to indicate that spinning continues.
Thus, one of the assumptions behind the "no angular momentum" assertion must be wrong, as it is empirically falsified.
Another experimental setup: Only a bottom guide. Find a large bowl, grease it, and align your pencil horizontally against the inside. Now, slowly release your finger from the eraser until friction with your finger approaches zero. Thud. Not much angular momentum, but it's not following the other rules either.
Which assumption is false? Probably the idea that there is no force perpendicular to the rod. The inside guide MUST exert such a force, or else the rod cannot "move like the hands of a clock". This force need not be supplied by friction, as demonstrated by the "greased hook" approach. It is, rather, supplied by supporting the inner end of the rod against gravity.
Hi
Well, something pretty odd is going on in the physical situation. The resolution of the paradox is that the constraints on the rod must supply an infinite force to maintain the rod in a radial position (more precisely: as the deviation from radiality tends to zero, the force exterted by the constraints tends to infinity).
There are two papers in the American Journal of Physics that discuss different resolutions. When I get a minute I'll type them up.
best wishes
Robinh 19:01, 28 Mar 2005 (UTC)
- Yes, it would be great if someone added information on the resolutions of this "paradox". "The resolution of this paradox is not clear." is at best unsatisfying, and at worse, false. dbenbenn | talk 19:33, 13 August 2005 (UTC)
To be honest, I fail to see the "paradox" in all of it. The problem with looking at the situation apparently lies in the frictionless guidance from the disks (which in turn would require infinite forces to keep the assembly from breaking apart.)
Consider a slightly altered arrangement: You have two concentric disks of different radii which can freely and independently from each other rotate around their centers. At the edges of the disks, there are two pivots. When the pivots are aligned, their distance is exactly the length of the rod in question. Now, attach the rod to the pivots.
You will notice the following:
- ) Although the individual disks can rotate independently, and although the rod could rotate around each single pivot at will, the current assembly makes the whole arrangement effectively rigid: Both disks and the rod must move in unison.
- ) From this point on, considering the disks to be massless, the whole situation is just that of the rod being attached by means of a massless lever to the common center of the disks.
Am I missing something? -- Syzygy 09:22, 27 March 2006 (UTC)
Hi. Sorry, to delete good-faith edits, but I removed the following paragraph:
However, even this resolution does not avoid the main cause of the seeming paradox. The reason why this system does not conserve angular momentum is because it simply does not have to. All the conservation laws are ony applicable to closed system, those which contain all the objects that exert forces on the system. By not including the Earth, which is tacitly assumed to produce a pull on the rod, we are not considering the angular momentum that will be gained as a result of the rod pulling on the Earth. This 'additional' angular momentum will add with that of the rod to result in sero gain.
the "system" under consideration is the rod, which (according to the second analysis) cannot change its angular momentum, because the constraints cannot exert a force lateral to the rod. I can't see how including the Earth is relevant. What additional angular momentum will be gained as a result of the rod pulling on the Earth? please explain this using Newton's Laws.
Robinh 21:51, 30 April 2006 (UTC)
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[edit] Gravity
- What has been neglected in the above is the ubiquitous force of gravity. The force of gravity is always directed downwards and provides a torque that imparts angular momentum to the rod. If you were to ignore gravity, the rod wouldn't slide in the first place.
I removed this, because it is not quite right. Gravity acts uniformly, effectively on the center of the rod, therefore imparts no torque. Or, if you like: if you ignore the walls, the rod wouldn't rotate in the first place. -Dan 17:42, 11 June 2006 (UTC)
[edit] Glaring omission
Who is the Carroll who noted this paradox? Jefferson Anderson 18:43, 19 December 2006 (UTC)
- Yikes! Added. 192.75.48.150 19:39, 19 December 2006 (UTC)
[edit] reversion
Sorry to revert a good-faith addition. The deleted material did not add to the article: the issue is the angular momentum of the rod per se. Robinh 08:14, 14 March 2007 (UTC)
[edit] reversion of a reversion
Dear Robinh, I kindly suggest that you read any physics book and try to understand the laws of conservation of angular and translational momentum properly. Momenta conserve only in systems that do contain all the bodies exerting forces on the system, i.e. if you consider a car on the surface of the Earth, that was initially stationary, then after it has accelerated it will have angular momentum. But the Earth will have precisely on opposite angular momentum because the car had to act by a friction force on the Earth and hence gave it a little additional spin. Think about this.
- Angular momenta conserve when all bodies exerting torque are considered. But see above: gravity acts on the centre of mass, therefore no torque. --192.75.48.150 18:01, 28 March 2007 (UTC)
