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ABOUT POSSIBILITY OF MOVEMENT OF CLOSED MECHANICAL SYSTEM
At the expense of the internal forces
The author of the article does not invent new laws and in any case
not trying to break the acting.
Is it possible to move a closed mechanical system that is not affected by external forces?
There is a number of motion theorem:
The time derivative of the momentum vector of a system of material points is equal to the main vector of all external forces acting on the system.
From this theorem several consequences follow:
- Internal forces do not directly affect the change in the amount of movement of the material system.
- If the main vector of all external forces acting on the system is zero, then the vector of the 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 zero, then the projection of the momentum of the material system onto 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 external forces acting on the system is zero, the law of conservation of momentum holds.
The amount of movement of individual parts of the system (for example, under the action of internal forces) may vary, but so that the magnitude remains constant.
The law of conservation of momentum cannot be violated. Impossible!
And it can not be circumvented.
Is it possible to use it to solve a problem, at first glance, contrary to this law?
I bring to your attention the following scheme of reasoning:
The movement of the body shown in Fig. 1 cannot be carried out without the participation of another body in this process.
Since the amount of body motion, in the projection on the coordinate axes, is constantly changing and is different from zero.
Interconnected and inextricable bond of the body can move as shown in Fig.2.
With such a movement, the bodies will move in circles, the radii of which are inversely proportional to their masses.
This is a consequence of the law of conservation of momentum.
In this case, the total impulse of the system is zero.
Here is how it would look in the case of two "real" objects (Fig. 3).
Let's call one of them "the body", and the other - "working body", or "working substance".
If two working bodies (having the same masses and speeds of relative movement) (Fig.4) take part in the relative movement inside (around) the case, then the case will not acquire a rotational movement.
In this case, the total momentum of two working bodies projected on the X axis is zero.
Therefore, the momentum of the hull, in the projection on the same axis, will also be equal to zero.
The corps will reciprocate relative to the center of mass of the entire system of bodies.
The body will move along one axis of the "absolute", fixed coordinate system.
In the coordinate system associated with the body, the working bodies move in a circle.
In a fixed coordinate system, working bodies move along an elliptical path.
It is possible to impart kinetic energy to the components of a mechanical system by converting other types of energy that are included in a closed system.
That is, it is possible to organize relative movements of the system components at the expense of internal forces.
At the same time, the center of mass of the entire system will always remain in place, that is, not to change its coordinates (or move in a straight line and evenly).
Let's modify the system shown in Fig.1.
We use not the “point” body, but the working mass uniformly distributed on a certain part of the trajectory (Fig.5).
With a uniform relative displacement, the "body" and "working mass" will move in circles, the radii of which are inversely proportional to their masses (Fig.6). By the radius of the trajectory of the working mass it is necessary to imply 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 the “real” system of bodies. (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.
And what will happen if the working mass uniformly and inseparably 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 body. The total impulse of the distributed working mass in the projection on all coordinate axes is zero. Therefore, equal to zero and the momentum of the hull.
The centers of mass of the system components do not move relative to each other. The system is stably fixed. At the same time, we can transfer considerable kinetic energy to a moving working mass.
"Interesting" begins when we start to stop the moving working mass at a certain point in the trajectory. (Fig.10)
We stop - this means, in this case, the sequential "connection" of all the elementary particles of the working mass with the body.
That is, the particles of the working mass at a certain point of the trajectory successively acquire the velocity of the body, losing the speed of relative displacement. At the same time, the rest of the working mass continues to move until all its particles acquire the velocity of the body.
For example, moving some liquid along a closed path, we suddenly begin to collect it in a container fixedly mounted in the body.
In the animated figure, the conditional CM of the working mass is indicated in red. Green - the trajectory of the CMU in its relative movement relative to the "body".
Moving the radius vector of the CM of the working mass can be considered as moving the pendulum of variable mass and variable length. (In the coordinate system associated with the hull.)
Here the interesting task was defined:
- calculate how the system components move?
The calculations led to the following result (Fig.11).
After the completion of the cycle of stopping the entire working mass, the CM of the entire system is shifted relative to its initial position. The system moves in full compliance with the law of conservation of momentum!
It can be said otherwise: In order for the law of conservation of momentum to be fulfilled, the center of mass of this mechanical system must move.
The first calculation is made on the basis of the law of conservation of momentum.
The second is using the Lagrange II type equations.
The results are the same.
In solving this problem, the internal interactions of the system components are not considered at all.
No inertia forces!
Only the kinetic and energy state of the system.
A final, additional step was added in Fig.11 - the reduction of the centers of mass of the hull and the working mass into 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 CM of the entire system, the blue line - the CM of the body.
Mathematical calculations describing this alleged phenomenon can be found at: http://varipend.narod.ru
Author: Sergey Butov
PS Material is protected.
Publication date 12/23/2006