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ON THE POSSIBILITY OF MOVING A CLOSED MECHANICAL SYSTEM
AT THE BASIS OF INTERNAL FORCES
The author of the article does not come up with new laws and in no case
not trying to break acting.
Is it possible to move a closed mechanical system that is not affected by external forces?
There is a momentum theorem:
The time derivative of the momentum vector of the system of material points is equal to the main vector of all external forces acting on the system.
Several consequences follow from this theorem:
- Internal forces do not directly affect the change in the momentum of the material system.
- If the main vector of all external forces acting on the system is zero, then the vector of momentum of the material system remains constant in magnitude and direction.
- If the projection of the main vector of all external forces applied to the system onto some fixed axis is equal to zero, then the projection of the momentum of the material system on this axis remains constant.
Corollaries (2.3), in fact, are called the law of conservation of momentum.
For a closed system, that is, a system that does not experience external influences, or in the case when the geometric sum of the external forces acting on the system is zero, the law of conservation of momentum holds.
In this case, the amount of motion of individual parts of the system (for example, under the action of internal forces) can vary, but so that the quantity remains constant.
The law of conservation of momentum cannot be violated. Impossible!
And it can not be bypassed.
But can it be used to solve a problem that, at first glance, contradicts this law?
I bring to your attention the following scheme of reasoning:
The body movement shown in Fig. 1 cannot be carried out without the participation of another body in this process.
Since the momentum of the body, in projection on the coordinate axis, is constantly changing and different from zero.
Bound together by an indissoluble and inextensible bond, bodies can move as shown in Fig. 2.
With this movement, bodies will move around circles whose radii are inversely proportional to their masses.
This is a consequence of the law of conservation of momentum.
In this case, the total momentum of the system is zero.
This is how it would look in the case of two “real” objects (Fig. 3).
We will call one of them a “body” and the other a “working fluid” or “working substance”.
If two working bodies (having the same masses and relative displacement speeds) take part in the relative displacement inside (around) the casing (Fig. 4), then in this case, the casing will not acquire rotational motion.
In this case, the total momentum of the two working bodies in the projection onto the X axis is zero.
Therefore, the momentum of the body, in projection onto the same axis, will also be zero.
The body will make reciprocating movements relative to the center of mass of the entire system of bodies.
The movement of the body will be carried out along one axis of the "absolute", fixed coordinate system.
In the coordinate system associated with the housing, the working bodies move around the circle.
In a fixed coordinate system, working bodies move along an elliptical trajectory.
It is possible to give kinetic energy to the components of a mechanical system by converting other types of energy included in a closed system.
That is, it is possible to organize the relative movements of system components due to internal forces.
In this case, the center of mass of the entire system will always remain in place, that is, do not change its coordinates (or move rectilinearly and evenly).
We modify the system shown in Fig. 1.
We use not a “point” body, but a working mass uniformly distributed over a portion of the trajectory (Fig. 5).
With uniform relative displacement, the “body” and “working mass” will move along circles whose radii are inversely proportional to their masses (Fig. 6). By the radius of the trajectory of the working mass, it is necessary to mean the radius of the trajectory of the center of mass (CM) of the working mass (red dot in the figure).
This is how it would look for a “real” body system. (Fig. 7)
Similarly (Fig. 4), several working masses can be used in the system to balance the angular momentum. (Fig. 8)
Again, the center of mass of the entire system is fixed.
But what happens if the working mass uniformly and inextricably fills the entire trajectory of movement? (Fig. 9) Nothing! The center of mass of the working mass coincides with the center of mass of the housing. The total momentum of the distributed working mass in the projection on all coordinate axes is zero. Therefore, the momentum of the case is equal to zero.
The centers of mass of the system components do not move relative to each other. The system is stably stationary. In this case, we can transfer significant kinetic energy to the moving working mass.
The "interesting" begins when we begin to stop the moving working mass at a certain point in the trajectory (Fig. 10)
We stop - this means, in this case, - a consistent "connection" of all elementary particles of the working mass with the housing.
That is, particles of the working mass at a certain point in the trajectory sequentially acquire the speed of the housing, losing the speed of relative displacement. In this case, the rest of the working mass continues to move until all its particles acquire the speed of the housing.
For example, moving some liquid along a closed path, we suddenly begin to collect it in a container that is fixedly mounted in the housing.
In the animated figure, the conditional CM of the working mass is indicated in red. Green - the trajectory of the movement of the CM in its relative movement relative to the "body".
The movement of the radius vector of the CM of the working mass can be considered as the movement of a pendulum of variable mass and variable length. (In the coordinate system associated with the body.)
Here an interesting task was determined:
- calculate how system components move?
The calculations led to the following result (Fig. 11).
After completion of the stopping cycle of the entire working mass, the CM of the entire system is shifted relative to its initial position. The system moves in full accordance with the law of conservation of momentum!
It can be said differently: In order for the law of conservation of momentum to be satisfied, the center of mass of a given mechanical system must move.
The first calculation is based on the law of conservation of momentum.
The second - with the help of Lagrange equations of the second kind.
The results are the same.
In solving this problem, the internal interactions of system components are not considered at all.
No inertia forces!
Only the kinetic and energy state of the system.
In Fig. 11, a final, additional stage is added - the reduction of the centers of mass of the housing and the working mass at one point. This is necessary for a more visual representation of the entire cycle of movement, with the possibility of repeating this cycle. The red line in the figure indicates the trajectory of the movement of the CM of the entire system, the blue - the CM of the case.
Mathematical calculations describing this alleged phenomenon, you can see at: http://varipend.narod.ru
Posted by: Sergey Butov
PS Material is protected.
Publication date 12/23/2006