At 3:02 PM 3/9/5, Robin van Spaandonk wrote: >The above appears to describe the situation in the tank before the >plug is pulled. IOW the water is rotating, and the calculations >show the energy of any given chunk of water at any given radius. > >However, after the plug is pulled, the radius of any given chunk >of water is constantly shrinking. If no angular momentum is passed >out of the system, then for any given chunk of water > >mv1r1 = mv2r2, i.e. v2 = v1 x r1/r2. (m is the same before and >after, because we are dealing in both cases with the identical >chunk of water).
Yes, you are certainly right about this. Momentum is conserved. The speed of a chunck is thus not constant in a vortex, as I had assumed. In fact, I think in your imaginary tank the tangential speed is proportional to 1/r, and the vertical speed to 1/r as well. This is shown in Feynman's *Lectures on Physics*, Vol II, 40-10 ff. Fig. 40-12 is a great drawing of your tank, showing the surface countour of a sample vortex. The reason kinetic energy is increased is that work is done on the chunck as it moves inward. In the case of a cylinder of water the work is just the work of falling, m*g*h. The work is completely analogous to the work done by a skater pulling in her arms in order to spin faster. This F*d work supplies the needed kinetic energy to permit the reduction in radius R. In all cases the angular momentum L = I * w is conserved, so no torque upon the earth is required to gain the angular velocity. > >IOW the velocity has to increase inversely with the radius, which >leads back to the initial question, where does the energy come >from, or where is the fallacy in the argument? It comes from gravity. > >Possible answer: Gravity forms a link connecting the water to the >planet, and as gravity pulls the water into circular motion, the >water in turn pulls back on the planet, causing it to rotate in >the opposite direction. No, there need be no link to the earth for the major part of the kinetic energy gained. Angular momentum is conserved internal to the vortex, so any exchange of angular momentum with the earth would violate COAM. However, there is indeed a link to the earth for the gain in angular momentum that occurs due to the coreolis force, though this is small. >IOW angular momentum is passed to the >planet. Only due to coreolis effects, which are very small for ordinary tanks. >My problem with this explanation is that I'm not sure that >gravity can transfer more angular momentum than it would have been >able to do had the water initially been motionless. The water is not motionless though. Do the experiment if you doubt it. I did it in 1962, so I think it should still work, even in Australia. Water spins in the direction it spun when the tank was last affected. >(BTW I think that when the water *is* initially motionless, the >vortex starts at the centre and spreads out, with the velocity >dropping appropriately). I think it does that in all cases. The water starts to fall in the center first. Falling is how the kinetic energy is gained to shorten tha radius. The vortex cone is on an angle that slopes downward toward the hole. As a chunk slides downward the radius decreases and the kinetic energy increases, and momentum stays fixed. I hope I got it all right this time! 8^) Regards, Horace Heffner
