## Would rolling body transformation can help find a clue?

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• link to problem site http://knol.google.com/k/alex-belov/paradox-of-classical-mechanics-2/1xmqm1l0s4ys/9# The idea is very simple. If spit a rolling ring to
Message 1 of 4 , Jun 26, 2009
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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?

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.
• mirror of this problem page is: http://mysite.verizon.net/vze27vxm/index.htm
Message 2 of 4 , Jun 27, 2009
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mirror of this problem page is:
http://mysite.verizon.net/vze27vxm/index.htm
• 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
Message 3 of 4 , Jul 1, 2009
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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."

--- In gravitationalpropulsionstevenson@yahoogroups.com, "abelov0927" <abelov0927@...> wrote:
>
>
> 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?
>
>
>
> 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.
>
• Curious, some of the properties of this design echo some of my concerns with a Space Elevator . While the size factor is many magnitudes different, some
Message 4 of 4 , Jul 1, 2009
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Curious, some of the properties of this design echo some of my concerns with a "Space Elevator". While the size factor is many magnitudes different, some factors are similar. When I calculated the length of a space elevator Ribbon or Cable due to orbital mechanics and the rotation of the Earth, I came up with a length close to 100,000 miles just to reach an altitude of about 200 miles, in a counter-rotational direction encircling the Earth many times before reaching that altitude, and as far as the theoretical "space anchor" well that would be even further by far. This would be needed just to counter shear factors, however even supposing great advancements in nano tech, there just is nothing on the drawing boards that could possibly withstand the stress of the weight/mass of the cable alone, let alone an elevator.
I know that this is a bit off topic from the original concept you mention, however, one of the factors that I contemplated using to alter the force factors was inclusion of CMGs (Giant Gyros) along the cable to alter attitude and hence certain shearing factors. Never really came up with a way to make the numbers doable, however factors did begin to improve. Have you ever considered the addition of Gyros in your design?CMGs can be altered as wished to influence attitude as needed.

Mystery
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