Process of moving of the considered system it is possible to repeat.

For this purpose it is required to move mass , which gathers on distance from the center of mass , in the center of mass . Not breaking position of the center of mass of all system, in regular intervals to disperse system of mobile elements on a circle of radius . To give to system of mobile elements angular speed . After that, it is possible to repeat a running cycle (a Fig. 25,Fig. 26).

To exclude from calculations coordinate  and  (where — a corner of turn of all mechanical system), it is convenient to bring in the closed mechanical system one more system of the mobile elements making movement, mirror symmetric to the first system of mobile elements concerning an axis.

Fig. 13

Fig. 14

It is meant mirror symmetry of movement:

starting corner                

angular speed      

change of mass of mobile elements  

elements of mobile system stop in a point with coordinates  

Coordinates of the center of mass of the second mobile system

 

 

In a projection to axes of  coordinates system XOY a linear momentum :

 

 

Total momentum of systems of mobile elements in a projection to an axis it is equal to zero.

Total momentum of two systems of mobile elements in a projection to an axis it is equal to the double momentum of one of systems of mobile elements.

 

The sum of the moments of momentums of two systems of mobile elements is equal to zero

As it was already mentioned above (Fig. 3), to coordinates system XOY it is possible to compare moving of the center of mass of system of mobile elements to moving the center of mass of a pendulum of variable length and variable mass .  For a case of two systems of mobile elements, moving of their general center of mass can be presented as back and forth motion of a body of variable mass on one coordinate. (Fig. 15).

That is, the mechanical system consisting of two systems of mobile elements, possesses one degree of freedom.

Fig. 15

 

Animated image Fig. 15 :

During the first half-cycle there is a reduction of mass of two systems of mobile elements. Thus coordinate (the center of mass of systems of mobile elements) decreases.

During the second half-cycle , coordinate increases, but mass continues to decrease up to zero value. (see Fig. 8)