Linear momentum of material system to equally mass of all system increased for speed of its center of inertia. In our case:

(2)

 

Here  Is a mass of all elements of the system moving on a circle. From conditions of a task follows, that during the initial moment of time the mass is distributed in regular intervals. From here it is possible to accept:

(3)

 

Angular density of distribution of elements of system of bodies on a circle (an arch of a circle).

Size  changes linearly in time :

(4)

 

whence:

(5)

 

or, as dependence on a corner of filling:

It is possible to write down also:

Size  defines a corner on a trajectory filled by moving elements.

In case of our task, the center of mass of system of mobile elements in coordinates system XOY it is calculated:

,

where x_i , y_i -coordinates i-th site of a body in mass .

Coordinates i-th mobile element in the chosen coordinates system:

From conditions of our task:

 

Coordinates of the center of mass of mobile elements in the chosen coordinates system are calculated by means of integrals with a variable bottom limit:

(6)

 

After integration we shall receive:

(7)

(8)

  

Expressions (7) and (8) define coordinates of the center of mass of mobile elements depending on a corner of an aperture . From a condition of a task the corner of an aperture changes linearly eventually:

The plot of moving center of mass of system of mobile elements is presented in a Fig. 2:

Fig. 2

In a Fig. 3 it is shown conditional  movings of the center of mass of mobile system.

Fig. 3: