Just keep in mind. This model has 3 phases. Bodies at phase1 is different from
bodies at phase3. The law of momentum conservation works, but it gives different
result for bodies with different mass.

This is the comment from other forum.

"If the platform is completely free to move (say floating in outer space)
momentum conservation requires that it will end up with a positive forward
velocity V=(nm/(M+nm) )v. Kinetic energy is not conserved because as each link
slaps down on the surface some energy is converted to heat.

For a full ring rolling at constant velocity there's no horizontal force between
the bottom of the ring and the surface but that requires the ring to be balanced
(rotationally symmetric). As links become missing from the circle that's no
longer true so the succeeding links that hit the surface do have a forward pull
on them accelerating the platform forward."
Please read and look on knol site for details.
http://knol.google.com/k/alex-belov/paradox-of-classical-mechanics-2/1xmqm1l0s4y\
s/9#


--- In gravitationalpropulsionstevenson@yahoogroups.com, "abelov0927"
<abelov0927@...> wrote:
>
> link to problem site
>
http://knol.google.com/k/alex-belov/paradox-of-classical-mechanics-2/1xmqm1l0s4y\
s/9#
>
> The idea is very simple.
>
> If spit a rolling ring to small parts set of n elements (1,2,3,...,n) with
mass m then each of them conduct linear and circular movement on surface.
>
> Each piece has constant angular velocity. Each piece of ring has variable
linear velocity at surface point. Once per circle each element of ring stop on
surface. At surface point this element has linear velocity value equal to zero.
>
> The ring must be broken in one location and one element of chain which has a
zero value of linear velocity holding by the surface. This is not mean to stop
the whole ring at this time. This mean stop the red piece and cut the ring at
the same time. The surface holds just one element of ring and other elements of
chain is continuing movement by own trajectories.
>
> If calculate net linear momentum of these elements then this net should be
equal to this ring initial linear momentum. But one of these elements is stop
already and net momentum will be for n-1 elements. In this case one element has
been join to the surface and mass is M(suface) + m(element). Set of elements has
a mass equal to (n-1)*m now. It's change initial condition. The surface is still
keeping same momentum and increase own mass. But chain (set of elements n-1)
should hold same ring initial momentum.
>
>
> Is this net of linear momentums for set of elements n-1 with net mass (n-1)*m
is equal to the ring initial linear momentum?
>
> Would set of n-1 elements return whole ring momentum back to surface?
>
> Will this surface return to initial velocity?
>
>
> In problem complexity
>
> Each element of chain won't stop at the same time. Each element has different
momentum but same mass m. If join each stopped element to the surface, then
surface mass increase faster than chain return momentum. This means the surface
mass growth will help ground to keep own momentum.
>
>
>
> My suggestion:
>
> If ring has set of n elements then for surface point only ring's set of n-1
elements is always move. But one element with zero value of velocity stand down
and it will be always part to surface.
>
> For this particular case, this set of n-1 elements (broken thin ring or chain)
not equivalent to set of n elements (initial ring). Base on law of momentum
conservation net momentum for set n-1 elements would have same initial momentum,
but momentum density will change for each element of this set n-1. The surface
will take all elements momentums back when they'll stop. Base on law of momentum
conservation, from same momentum the body with higher mass will take lower
velocity then body with lower mass.
>
> Initially:
> The ring has mass n*m
> The surface has mass M
>
> After ring to chain conversion:
> The chain has mass (n-1)*m
> The surface has mass M+m
>
> P = m*V = const
>
>
> m1*V1 = m2*V2
>
> The surface with new mass will take a velocity V1 from set of elements n-1.
This velocity is different from initial surface velocity V0, because the surface
mass has been changed.
>
> Follow the law of momentum conservation, even all elements of broken ring will
stop on the surface, this surface will have linear velocity more than zero.
>
> The surface will return back to initial velocity V=0 if rest of the chain
could increase momentum on cut action. But it's nonsense. Base on law of
momentum conservation the chain should keep same initial ring's momentum.
>