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  2.2.2. Calculation of a linear momentum of the closed mechanical system of bodies in absolute coordinates system for the working period

Here:

- a linear momentum of all closed system consisting of a body and systems of mobile elements a lump

- speed of the center of mass  of a body with the joined elements  .

-a linear momentum of system of mobile elements. Projections on an axis of coordinates are found above:(11), (12).

As is known, the geometrical point, radius-vector r  which refers to as the center of mass of material system is defined by equality

As for the working period to a body additional particles, coordinates of the center of mass of a body join pay off as follows:

where and coordinates of the center of mass of a body in coordinate system  XOY.

After substitution

and

,

let's receive:

Projections of speeds to corresponding axes of coordinates:

or, after simplification:

In a projection to axes of  coordinates system XOY a linear momentum (from(13)):

According to a principle of conservation of momentum of the closed system:

The decision of the given system of the differential equations concerning coordinates and     (coordinates of the center of mass of the case), in view of entry conditions:

yields following results:

, where:

Euler's constant:   

Cosine integral:       

Sine integral:                   

In a Fig. 4 the plot of change of coordinates is presented and    for the working period.

Fig. 4

Once again we shall remind, that and    Coordinates of the center of mass of a body in coordinates system XOY, i.e. in "absolute" coordinates system.

_and   coordinates of the center of mass of all closed mechanical system in coordinates system XOY.

values and it is possible to find as follows:

_and     coordinates of the center of mass of all closed mechanical system in coordinates system XOY.

With coordinate the center of mass of all system (16) coincides also.

Moving of the center of mass of all system of bodies looks like (Fig. 5):

Fig. 5

Dependence of moving of the center of mass of all closed system are presented in a Fig. 6 and a Fig. 7.

Fig. 6

Fig. 7

Two plots presented  on Fig.4 and Fig 5, collected in one plot:

Red color designates a trajectory of moving of the center of mass of all system,  dark blue - moving of the case (Mc).

Conclusions.

At observance condition of preservation of a momentum the given system ( Fig. 1) moves for the certain time interval on the certain distance ( Fig. 5).

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