law of conserv.. of energy..??

Question

imagine theres a horizontal planeA bove which the effects of gravity are not felt. any object above the plane just floats and gravity has no influence on it.
now take a chain(for e.g:made of small steel ring) of length hundred metres above the planeA and arrange it so that its in a heap just above the planeA. but take care the chain is not entangled and should be easy to stretch along its length.

now pull the lower end of the thread below the planeA and the chain runs down like a single thread and hits the ground and as links pull down on the ones they are linked to , the whole length of the chain will run like a thread and hit the ground in some time. now take planeA is 20metres above the ground surface. the terminal velocity of links when they hit ground will be around 14 metres .
now consider hundred metres of the chain weighs 100 units. the potential energy of the chain just above the planeA will be m*g*h =100 *9.8*20
=19600 units.

but kinetic energy recovered would be 1/2 * m * v*v = 0.5 * 100* 14 *14 = 9800 units. so in this method only half the energy is recovered hence violating the law of conservation of energy ....

any views pls...?


here the source is the gravity force acting on 20 metres of chain thats below the plane. when 20 metres weigh exactly 20 units it can produce a momentum of 20 * 9.8 units per sec.... which is 14*14 and not 20*20......so rings will hit the ground with the velocity of 14 m/s and not 20m/s as in free fall...

Answer 1

I believe that all the variations on this problem (9 so far and counting) that you have posted fall into either of two basic scenarios:
1. The descending part of the chain accelerates the (finite) not-yet-falling part (e.g., the stretched-out chain that firctionlessly slides over the edge). In this case I claim that the not-falling part gets accelerated by the falling part so velocities become greater than free-fall impact velocity as the process continues, and that energy is conserved.
2. The descending part is essentially unconnected to the (potentially infinite) not-falling part, and a demon is sitting up there adding links to the top of the falling part as needed. In this situation I claim that the falling part is in a steady-state situation, falling at a speed greater than 0 and less than free-fall impact speed, so kinetic energy is indeed less than potential energy.. The missing energy goes into the inelastic collision involved in attaching a link at 0 velocity to the moving chain. This link is jerked up to speed with no rebound. Momentum, but not energy, is conserved, and all the energy is accounted for.

Answer 2

Lets consider a simple example. A chain with only two links. As the chain is falling, there is tension in the chain. Why is there tension in the chain? Because the lower link has fallen farther than the upper link, it has experienced v = gt for a longer period of time and "wants" to go faster. The tension is a force acting through a distance, that's work. So the lower link is doing work speeding up the upper link. At the same time there is an equal and opposite force and the upper link is working to slow down the lower link. The overall acceleration of the lower link is less than g, and the overall acceleration of the upper link is greater than g. Energy is conserved.

Answer 3

i'm just guessing here but it seems like in the space you describe, conservation of energy (and momentum) would not be expected because the space is not symmetrical with respect to the gravitational force.

http://math.ucr.edu/home/baez/noether.html

Answer 4

I'm sorry, my eyes just glazed over.

You seem very smart; hopefully someone on your level will come along and wish to discuss scientific matters with you :)