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 xi
, yi -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:
Moving of the center of mass of system
of mobile elements to coordinates system
XOY it is possible to compare
to moving the center of mass of a pendulum of variable length
and variable mass
.
In an animated kind it looks as follows:
In a projection to axes of coordinates
system
XOY a linear momentum
:
|
 |
(9) |
|
 |
(10) |
In a projection to axes of coordinates
system XOY a linear momentum
:
|
 |
(11) |
|
 |
(12) |
, where
projections of speed of the
beginning of coordinates of system XOY on corresponding axes of system XOY.
Let's remind: the beginning of coordinates
of system XOY it is connected with the center of mass of a body
2. 
2.2.1. Calculation of
a linear momentum of system of mobile elements in relative coordinates
system.