How to calculate this?

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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=92s 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?

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