Analogy between rotatory and forward motions

Using an analogy method, it can be assumed that must exist the closed mechanical system, capable of changing its the location only with the aid of the internal forces, without the external action.

Analogy method. Method of investigating any process, by the replacement by its process, described by the same differential equation,
like the studied process.
(the explanatory dictionary of physical terms)

Analogy between rotatory and forward motions

" I value the skill to construct the analogies, which, if they are daring and reasonable, derive us beyond the limits of the fact that nature wished to us to open, making it possible to foresee facts even before we will see them. ".
Zh.L.D'Alamber

Between the motion of solid body around the fixed axis and the motion of separate material point (or by forward motion of body) there is a close and far reaching analogy. A similar value from the kinematics of the rotation of solid body corresponds to each linear value from the kinematics of point. To coordinate s corresponds the angle φ , the linear speed v - the angular velocity w , to linear (tangential) acceleration a - the angular acceleration ε .

Comparative parameters of the motion:

Forward motion		The rotary motion
Displacement	S	The angular displacement	φ
Linear speed		Angular velocity
Acceleration		The angular acceleration
Mass	m	Moment of the inertia	I
Pulse		Moment of momentum
Force	F	Moment of the force	M

Table can be continued and further.

Work:

Kinetic energy

Expressions for the rotary motion resemble the appropriate expressions of forward motion.
They are obtained from the latter by the formal replacement of m → I , v → w , p → L
The represented table cannot pretend to full weight of the scope of analogous values.

For rotatory and forward motions are formulated the analogous laws:

Law Of momentum conservation (LMC )

, with F_ext = 0

Law of conservation of momentum of pulse (LCMP )

, with M_ext = 0

These laws are formulated as follows:

"if vector sum of the external forces, which act on the system, is equal to zero, then the pulse of system remains, i.e., does not change in the course of time. In particular, this occurs, when system is locked "	"if the moment of external forces relative to fixed beginning o is equal to zero, then the moment of momentum of system relative to the same beginning remains time-constant"
[ D.V.Sivukhin. General course the physicists of t..I Mechanics ]

In the classical mechanics the law of momentum conservation usually is derived as the consequence of Newton's law. However, this law of conservation is accurate in cases when Newtonian mechanics is not applied (relativistic physics, quantum mechanics). It can be obtained as the consequence of intuitive- accurate assertion about the fact that the properties of our peace will not change, if all its objects (or reference point!) to move for the certain vector R. At present there does not exist any experimental facts, which testify about the non fulfillment of the law of momentum conservation.

The law of conservation of momentum of pulse is the consequence of assertion about the fact that the properties of the surrounding peace do not change with the turnings (or the turning of frame of reference) in the space. The moment of momentum of the system of the point bodies L is defined as the sum of the moments of each of the points and remains in the time with the condition of equality to zero moments of external forces.

Given laws relate to the global laws of conservation.

Answer to the natural question of why are valid the laws of conservation, in physics was found comparatively recently. It turned out that the laws of conservation appear in the systems when in them certain elements of symmetry are present. (element of the symmetry of system is called any conversion, that transfer system into itself, i.e. not changing it).

German mathematician By Emmi Noether in 1918 mathematically proved the connection between the laws of conservation and the symmetry, which the laws of nature possess in physics. In the simplified formulation the theorem Noether says, that if the properties of system do not change from any conversion of variables, then to this corresponds a certain law of conservation.

Theorem Noether - simplest and universal means, which makes it possible to find the laws of conservation in the classical mechanics, quantum mechanics, field theory, etc.
This is how appear proofs LMC and LCMP on the basis of the property of the symmetry of the space [ Of d.V.Sivukhin. General course the physicists of t..I Of Mechanic.(gl..Y, 38) ]:

...4. Turning to the proof of the law of momentum conservation. Let us assume that the mechanical system is locked. All forces F₁ , F₂ ..., the acting on the material points systems, are forces internal, there are no external forces. Let us transfer system from arbitrary position 1 to another arbitrary position 2 so that all its material points would undergo one and the same displacement r, and besides so that their speeds would remain before in the value and the direction. In view of the uniformity of space, to this displacement does not be required the expenditures of work. But this work is the scalar product (F₁ + F₂ +...)r . It means, it is equal to zero, whatever displacement r. Hence it follows that for the closed system F₁ +F₂ +... of =0. A this is the very thing condition, with fulfilling of which of second Newton's law is obtained the law of momentum conservation.

5. The law of conservation of momentum of pulse for the closed system proves exactly so. Using isotropy of space, it is possible to prove that the vector sum of the moments of the internal forces, which act in the system, is equal to zero: M₁ +M₂ +... =0. hence immediately follows the law in question.

As is evident, proofs are very similar.
With the entire similarity of progressive and rotary motion, they have differences.
One of them is examined in this article.

There is a phenomenon, which enters, it would seem, in the contradiction with LCMP.

The closed mechanical system can be turned to any angle
with the aid of some internal forces alone.

And not only to turn, but also to make it necessary to revolve.

How it is possible to turn mechanical system with the aid of some internal forces alone?
Let us examine an example.

Let us assume we have two balanced bodies, fixed coaxially. Bodies have the capability to revolve relative to each other. One of the bodies (green outline, a red radius) is capable of changing its geometric dimensions, changing, thus, its moment of inertia (I_v). Let us assume the maximum value of I_{v max} of this body corresponds to value of I_c of the second body (dark-blue outline, a purple radius). The minimum value of I_{v min} is considerably lower than I_{v max}.

Let us examine the cycle of motions of this mechanical system.

Fig.1

Body with by variable I_vacquires maximum geometric sizes. (due to the internal energy, it understands).

Due to the same internal energy let us create the internal torque, which acts equally on both bodies. Under the action of this moment, body they begin to be turned in opposite directions.

It is possible to stop motion of both bodies at any moment of time. Since the moments of inertia of both bodies are equal, angular positions both bodies will engage identical (in the absolute value). At the moment of stop, the angular position of entire system remained before. Will not change this angular position and when we will begin to change geometric dimensions of one of the bodies.

After decreasing the geometric dimension, and respectively the moment of inertia of one of the bodies, let us try to reduce both bodies to the previous angular positions of these bodies, with the aid of the power moment, directed to other side.

Here and occurs interesting.

Since the moments of inertia of both bodies are already different, both bodies it is turned to the different angles. It is possible to again increase dimensions of one of the bodies. System will again be such as before.
Almost...
In reality, entire system proves to be that turned to a certain angle. The value of this angle is proportional to the difference of the moments of the inertia of two bodies.

These cycle of motions can be repeated.
And it occurs that the system revolves!

Fig.2

Moreover, it revolves entirely well!

Let us assume the components of system change their geometric dimensions in such a way that total moment of the inertia of entire system remains without the changes.
Depicted on fig3 system can be examined as interaction of bodies with the variable torque of inertia, with the constant mass and constant moment of the inertia of entire system as a whole.

Fig.3

The system, as shown in animated fig3, accomplishes the complete revolution in 16 seconds.
It is possible even to calculate the "speed" of rotation of this system:

rad/s

But, in reality, it cannot be spoken about this value as about the "speed of rotation of system". More right it will sound: the "rate of the process of changing the angular position", since value w_φ it is not the measure of the kinetic state of this system, since kinetic energy of system does not change.

The motion examined is inertia-free. If we stop the relative displacements of components inside the closed system, motion w_φ immediately it ceases. To value w_φmost of all approaches name precession.

In the process of a similar "rotation" is not disrupted LCMP, since total moment of momentum of all components of system, at each moment of time, is equal to zero.

At the same time it is possible to determine the moment of force, equivalent to the external moment of force, which should have been applied to the system in order to cause a similar displacement.
Specifically, - equivalent, since for the situation in question, the external power moment is equal to zero.

The conclusions, which can be made on the basis of the given example:

within the framework of the law of conservation of momentum of pulse it is possible to change the angular coordinates of the closed mechanical system only due to the internal forces;
important role during the organization of angular displacement with the aid of the internal forces, plays a change in the moment of the inertia of the components of system;
the angular displacement of entire system - is inertia-free;
the rate of change in the angular coordinates, having a dimensionality the same as angular velocity, by speed can be counted only formally.

Let us return to the proofs LMC and LCMP on the basis of the property of the symmetry of space.

Proofs are constructed on the assumption of a certain displacement of system (turning, in the case LCMP).
But in the case of the revolving system, this displacement actually can be produced due to the internal energy.
I.e., displacement not is virtual, but it is completely actual.

In the large table of analogous expressions for the rotatory and progressive displacements, appears the large empty cell:

Displacement due to the internal forces

Turning due to the internal forces

φ≠constwith: ∑L_i=0_,, M_ext = 0,

Displacement due to the internal forces? In the case with the progressive displacement, such it seems inadmissibly. On this score even there is a "theorem about the motion of the center of masses", which, it is at first glance, does not allow the similar displacement:
"the centre of the masses of system move as the material point, whose mass is equal to the summary mass of entire system, and the acting force - to vector sum of all external forces, which act on the system".

It does mean, I do be empty cell in the table it must occupy this theorem?
But analogy method makes it possible to assume that and for the linearly moving system, must exist the possibilities, inherent in the revolving system.
This assumption is constructed on the analogy of the equation of motions of rotatory and forward motions.
Cell must be filled, approximately, with the following content:

Displacement due to the internal forces

s ≠constwith:∑ p_i=0, , F_ext = 0

This expression is obtained as a result of the formal replacement φ→S , M→F , L→p

The conclusions, which can be made with the aid of the analogy method:

is possible existence of mechanical system, which is capable of linearly being moved only due to the internal forces;
linear displacement can be achieved due to the change in the parameter, analogous to the moment of inertia, namely - mass;
the linear displacement of entire system must be inertia-free;
the "speed" of the displacement of a similar system not will be the measure of kinetic state, i.e., in the process of displacement the total impulse of the components of system will remain constant.
Since the "speed" during this motion is formal value, its appearance cannot indicate of the disturbance of the "theorems about the motion of the centre of masses".

Varipend

The properties of mechanical system, by the name "varipend", most fully satisfy the enumerated assumptions.

Fig.4

Name "varipend" is obtained from the confluence of two words: varipend =variable +pendulum .

In the process of displacing the working mass and its stoppage in the "housing" of system, the motion of the centre of the masses of work substance can be examined as the pendulum motion of variable length and variable mass. (in the coordinate system, connected with the "housing")

Fig.5

Or, in the projection on one coordinate:
(this case of displacing the centre of masses it is possible with the displacement of two working masses, which revolve in opposite directions.)

Fig.6

The changing size of components in the figure symbolizes a change in the mass of the elements of the system:
the work substance, which decreases its mass during the operating cycle,
and the housing of system, which increases its mass due to the connection of the particles of work substance.

With constant overall mass, entire system can be examined as interaction of the components of variable mass.

In one of the calculations of the displacement of "varipend" is used the equation of motion of the body of variable mass - equation of Meshcherski'y.

In Fig.4 is represented setting task "varipend".
Solution of this task appears as follows:

Blue track the movement of the body(case).
Red - moving center of mass of the all system

Fig.7

If we examine "varipend" in the absolute coordinate system, then with the repetition of the internal cycles of the displacement of work substance, "varipend" continuously changes its attitude:

Fig.8

It is possible to conduct analogy with figure 3.
System, depicted on fig.8, are moved up to the distance of 55 mm in 20 seconds.
It would seem, it is possible to calculate the speed of "varipend":

But, since in the process of displacement the total impulse of system remains constant, value v_s will be formal value, which determines only changes in the coordinates of the centre of masses. This value is in no way connected with the pulse of entire system.

The displacement of "varipend" possesses the non-inertia properties.
If we stop the relative displacement of working mass inside the closed mechanical system, entire system will instantly end its motion.

The mathematical calculations, which describe the assumed properties of mechanical system by the name "varipend", can be looked with the address:
https://varipend.narod.ru

Note

In many training publications on theoretical mechanics, are given examples of the turning of the closed mechanical system due to the internal forces.
For example, man is capable to turn himself on Zhukovskiy's bench, accomplishing some rotations by hands.
Turning massive bodies in the space vehicle, the angular orientation of these apparatuses is produced.
Cosmonaut, find in weightlessness, also can turn his housing, after completing several revolutions by hand.

However, in reality, in all given examples the turning of system is absent.
The given examples demonstrate, if one may put it that way, the "theorem about the retention of the angular position" of the closed mechanical system.
I will attempt to explain my words.
If the moments of the inertia of the components of system do not change and external moment is absent, then it is possible to write down:
I₁φ₁(t)+I₂φ₂(t)=0
This is easy to verify.
Let us assume we is pivoted two bodies relative to each other.
If we after stoppage turn components in the opposite side, system will compulsorily occupy accurately the same angular position, which was at the initial moment of time.

Fig.9

To Fig.9 two bodies first are turned to a certain angle, and then are turned in the opposite side to accurately the same angles. In this case the system occupies its previous angular position.
The illusion of the turning of entire system appears because the angular function is periodical. As soon as angular positions of components they exceed the angle of 2π appears sensation, that the system turned itself. But this is not so. Certainly, angles it is necessary to continue to count off, also, with exceeding of the angle of 2π.
The case of two-body interaction can serve as an analogous example of linear displacement:

Fig.10

Bodies can they will be removed from each other, due to the internal forces, approach, but the relationship will be always carried out for this system:

This is - corollary about the motion of the centre of masses....

But as the same expressions for the conditions of the system, which consists of the components with the variable torque of inertia (variable mass) will appear?:

comment on "question marks"...

S.Butov
7 July 2007

" ... analogy is the specific case of symmetry, the special form of the unity of retention and change. Consequently, to use in the analysis an analogy method, means to act in accordance with the principle of symmetry. Analogy not only is permitted, but also is necessary in the knowledge of nature of things...."[ Ovchinnikov N.F. Principles retention. M., 1966 ]

ButovSV 07/07/2007

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