In reply to Horace Heffner's message of Mon, 07 Mar 2005 04:27:45 -0900: Hi, [snip] >Ignoring the coreolis force for a moment, the fallacy in the above >statement is the assumption that the instantaneous speed v of some small >chunk of the water changes as it approaches the drain. The speed of the >chunk remains constant at all times, except for the speed added by >converting gravitational potential energy PE to kinetic energy. Thus the >instantaneous linear kinetic energy KE = 1/2 m v^2 of the chunck remains >constant except for speed added by falling in the gravitational field. The >angular velocity w increases however, because w = v/r. Now, you might say >that for a rotational system KE = 1/2 I w^2, and w is increasing with >reduction in radius, so where does the free energy come from? Well, the >answer is that in a vortex the moment of inertia I of a chunk is not >constant. We have I = m R^2, and w = v/R, so when we substitute these into >KE = 1/2 I w^2 and we have: > > KE = 1/2 (m R^2) (v/R)^2 = 1/2 m v^2 > >which is constant, except for the PE converted to KE by falling down hill. >Since the KE of every chunk remains constant the energy of what remains in >the tank is the original gravitational potential energy PE plus KE less the >KE of what went down the drain. [snip]
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). 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? 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. IOW angular momentum is passed to the planet. 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. (BTW I think that when the water *is* initially motionless, the vortex starts at the centre and spreads out, with the velocity dropping appropriately). Regards, Robin van Spaandonk All SPAM goes in the trash unread.
