Theorem on changing the number of point movement

Since the mass of the point is constant, and its acceleration is the equation expressing the basic law of dynamics, can be represented as

The equation expresses at the same time the theorem on the change in the number of motion of the point in the differential form: derivative in time the amount of point movement is equal to the geometric sum of the forces acting on the point.

Integrate this equation. Let the point of mass. m.moving under the action of force (Fig.15) has at the time t.\u003d 0 speed, and at the moment t. 1-Speed.

Fig.15

Multiply then both parts of equality on and take from them certain integrals. At the same time, to the right, where integration is in time, the limits of the integrals will be 0 and t. 1, and the left where the speed is integrated, the corresponding speed values \u200b\u200band . Since the integral from Raven , As a result, we get:

.

Standing on the right of the integrals represent the impulses of the current forces. Therefore, we will finally have:

.

The equation expresses the theorem about changing the number of point in the final form: changing the number of motion of the point for a certain period of time is equal to the geometric sum of the pulses of all the forces acting on the point over the same period of time (fig. fifteen).

When solving problems, instead of the vector equation, they often use equations in projections.

In the case of a straightforward movement occurring along the axis Ohtheorem is expressed by the first of these equations.

Example 9. Find the law of movement material point masses m.moving along the axis h. Under the action of constant for the module of force F.(Fig. 16) under the initial conditions:, when .

Fig.16.

Decision. Make up differential equation Movement point in the projection on the axis h.:. Integrating this equation, we find: . The constant is determined from the initial condition for speed and equal. Finally

.

Next, considering that V \u003d dX /dt., come to the differential equation: integrating which we get

Permanent determining from the initial condition for the coordinate point. It is equal. Consequently, the point of motion of the point has the form

Example 10.. Cargo weight R (Fig.17) begins to move from the state of rest along the smooth horizontal plane under the action of force F \u003d KT.. Find the law of traffic.

Fig.17.

Decision. Choose the start of the coordinate system ABOUT in initial position cargo and send the axis h. towards movement (Fig. 17). Then the initial conditions are: x.(t \u003d.0) \u003d 0, V ( t \u003d.0) \u003d 0. Forces act on the goods F,P. and reaction force plane N.. Projections of these forces on the axis h. matter F. X. = F. = kt., R X. = 0, N X. \u003d 0, so the corresponding movement equation can be written as follows :. Separating the variables in this differential equation and then integrating, we get: v \u003d g.kt. 2 /2P. + C. one . Substituting the initial data ( v.(0) \u003d 0), find that C. 1 \u003d 0, and we get the law of speed change .

The last expression, in turn, is a differential equation, which is integrating that we will find the law of motion of the material point: . Incoming here are constant determining from the second initial condition h.(0) \u003d 0. It is easy to make sure that. Finally

Example 11. On the cargo located at rest on the horizontal smooth plane (see Fig. 17) at a distance a. from the beginning of the coordinates, begins to act in the positive direction of the axis x. force F \u003d K. 2 (P./g.)x., where R -cargo weight. Find the law of traffic.

Decision. The equation of movement of the goods under consideration (material point) in the projection on the axis h.

The initial conditions of equation (1) are: x.(t \u003d.0) = a., V ( t \u003d.0) = 0.

Included in the equation (1) time derivative from speed will be submitted so

.

Substituting this expression to equation (1) and reducing on ( P./g.), we get

Separating variables in the last equation, find that. Integrating the last, we have :. Using the initial conditions , get, and, therefore,

, . (2)

Since the force acts on the cargo in the positive direction of the axis h., it is clear that in the same direction he should and move. Therefore, in the decision (2), select the plus sign. Replacing further in the second expression (2) on, we obtain a differential equation to determine the law of movement of the cargo. Where, separating variables, we have

.

Integrating the last, we find: . After finding constant finally get

Example 12. Ball M. masses m. (Fig.18) falls without initial speed under the action of gravity. When falling the ball is experiencing resistance, where – permanent resistance coefficient. Find the law of the ball.

Fig.18.

Decision. We introduce the coordinate system with the beginning at the point of the ball location when t \u003d.0, sending the axis w. vertically down (Fig. 18). Differential equation of ball movement in the projection on the axis w. Has then appearance

The initial conditions for the ball are written as: y.(t \u003d.0) \u003d 0, V ( t \u003d.0) = 0.

Separating variables in equation (1)

and integrating, finding:, where. Or after finding constant

or . (2)

It follows that the limit speed, i.e. Speed \u200b\u200bat is equal.

To find the law of movement, replace in equation (2) V on dY /dt.. Then, integrating the obtained equation, taking into account the initial condition, we finally find

.

Example 13.Research submarine of a spherical shape and mass m. \u003d \u003d 1.5 × 10 5 kg Begins to dive with off-off engines, having a horizontal velocity V H. 0 = 30 m / S. and negative buoyancy R 1 = 0.01mG.where - vector sum of the archimedean pushing force Q. And gravity strength mG.operating on the boat (Fig. 20). Water resistance force , kg / s. Determine the equations of motion of the boat and its trajectory.

Differential equation of motion of the material point under the action of force F. Can be represented in the next vector form:

As the mass of the point m. Adopted constant, it can be made under the sign of the derivative. Then

Formula (1) expresses the theorem about changing the number of point movement in differential form: the first time derivative on the amount of point movement is equal to the current force.

In the projections on the coordinate axis (1) can be represented as

If both parts (1) multiply on dt., I get another form of the same theorem - pulse theorem in differential form:

those. the differential from the amount of motion of the point is equal to the elementary impulse of the force acting on the point.

Projecting both parts (2) on the coordinate axes, we get

Integrating both parts (2) ranging from zero to T (Fig. 1), we have

where - the speed of the point at the moment t. ; - Speed \u200b\u200bat t. = 0;

S. - impetus for force during the time t..

The expression in the form (3) is often referred to as the pulse theorem in the final (or integral) form: changing the number of motion of the point for any period of time is equal to the force impulse for the same period of time.

In the projections on the coordinate axis, this theorem can be represented as follows:

For the material point of the theorem on the change in the amount of movement in any of the forms, essentially does not differ from the differential equations of the point of movement.

Theorem on the change in the number of system movement

The number of system movement will be called a vector magnitude Q.equal to the geometric amount (the main vector) of the movement of all points of the system.

Consider a system consisting of n. material points. We will make a differential equation equations for this system and lay them so far. Then we get:

The last amount by the property of the internal forces is zero. Moreover,

Finally find:

Equation (4) expresses the theorem about changing the number of system movement in differential form: The time derivative on the amount of system movement is equal to the geometric sum of all external forces acting on the system.

Find another expression of the theorem. Let at the moment t.= 0 The number of system movement is equal Q 0. , and at the time of time t 1. It becomes equal Q 1. Then, multiplying both parts of equality (4) on dt. And integrating, we get:

Or, where:

(S-Pulse force)

since the integrals standing on the right give pulses of external forces,

equation (5) expresses the theorem about changing the number of system movement in the integral form: the change in the amount of system movement over a certain period of time is equal to the sum of the pulses acting on the external strength system over the same period of time.

In the projections on the axis coordinates we will have:

The law of preserving the number of movement

From the theorem on changing the number of system movement, you can get the following important consequences:

1. Let the sum of all external forces acting on the system are zero:

Then from equation (4) it follows that Q \u003d Const.

In this way, if the sum of all external forces acting on the system is zero, then the system of the amount of motion of the system will be constant by 10module and direction.

2. 01 The external forces acting on the system are such that the sum of their projections on some axis (for example, oh) is zero:

Then from equations (4`) it follows that Q \u003d Const.

In this way, if the amount of projections of all current external forces on some axis is zero, then the projection of the number of system movement on this axis is permanent.

These results and express the law of preserving the number of system movement. It follows that the internal forces change the total number of system movement cannot.

Consider some examples:

· I am in l and e o t d a h and l and o t k a t a. If we consider the rifle and bullet as one system, then the pressure of the powder gases during the shot will be internal power. This force cannot change the total number of system movement. But since the powder gases, acting on the bullet, tell her a number of movement sent forward, they should simultaneously inform the rifle of the same amount of movement in the opposite direction. This will cause the movement of the rifle back, i.e. The so-called return. A similar phenomenon is obtained when shooting from gun (rollback).

· R A B O T A R E B N O G O V N T A (P R O P E L L E R A). The screw reports some mass of air (or water) movement along the screw axis, swinging this mass back. If we consider the discarded mass and aircraft (or vessel) as a single system, then the power of the screw and the medium as internal cannot change the total amount of the movement of this system. Therefore, when dropping the mass of air (water) back, the aircraft (or vessel) is obtained by the corresponding speed of forward, such that the total number of movement of the system under consideration remains equal to zero, as it was zero before the start of movement.

A similar effect is achieved by the action of cheerful or rowing wheels.

· R e a to and in n about e d in and f and e. In the reactive projectile (rocket) gaseous products The combustion of fuel at high speed is thrown out of the hole in the tail part of the rocket (from the nozzle of the jet engine). The pressure forces acting at the same time will be internal and they cannot change the total number of movement of the rocket-powder gases. But since the removable gases have a known amount of movement, directed back, then the rocket receives the corresponding speed of forward speed.

Theorem of moments relative to the axis.

Consider the material point of mass m.moving under the action of power F.. We will find for it the relationship between the moment of vectors mV and F.regarding any fixed axis Z.

m z (f) \u003d xf - uf (7)

Similar to magnitude m (MV)if reinstalled m. Behind the bracket will be

m. z (mv) \u003d m (xv - uv)(7`)

Taking from both parts of this equality of time derivatives in time, we find

In the right part of the obtained expression, the first bracket is 0, since dX / DT \u003d V and DU / DT \u003d V , the second bracket according to formula (7) is equal

m z (f)because according to the main law of the speakers:

Let us finally have (8)

The resulting equation expresses the theorem of moments relative to the axis: the time-derived from the moment of the number of motion of the point relative to any axis is equal to the moment of the current force relative to the same axis. A similar theorem takes place for moments relative to any Center O.

The number of system movement, as vector value, is determined by formulas (4.12) and (4.13).

Theorem. The derivative of the amount of the time system in time is equal to the geometric sum of all external forces acting on it.

In projections, Cartesian axes we obtain scalar equations.

You can write vector

(4.28)

and scalar equations

Which express the theorem about changing the number of system movement in the integrated form: Changing the amount of system movement over a certain period of time is equal to the sum of pulses for the same period of time. When solving tasks, equations (4.27) are more often used

The law of preserving the number of movement

Change theorem kinetic moment

The theorem on changing the moment of the number of point movement relative to the center: the time derivative from the moment of the number of motion of the point relative to the fixed center is equal to the vector torque acting on the point of force relative to the same center.

Or (4.30)

Comparing (4.23) and (4.30), we see that the moments of the vectors are associated with the same dependence that the vectors themselves are connected (Fig. 4.1). If we design equality on the axis passing through the center of Oh, then we get

(4.31)

This equality expresses the moment the moment of the number of motion of the point relative to the axis.

Fig. 4.1.

The theorem on the change in the main moment of the number of movement or the kinetic moment of the mechanical system relative to the center: the time derivative from the kinetic moment of the system relative to a certain fixed center equal to the sum of the moments of all external forces on the same center.

(4.32)

If we design an expression (4.32) on the axis passing through the center of O, we obtain equality that characterizes the theorem about the change in the kinetic moment relative to the axis.

(4.33)

Substituting (4.10) into equality (4.33), you can record the differential equation of a rotating solid (wheels, axes, shafts, rotors, etc.) in three forms.

(4.34)

(4.35)

(4.36)

Thus, the theorem on the change in the kinetic moment is advisable to use for the study of a very common in the technique of a solid body, its rotation around the stationary axis.

The law of preserving the kinetic moment of the system

1. Let in the expression (4.32).

Then from equation (4.32) it follows that, i.e. If the sum of the moments of all applied to the system of the wound force relative to of this center Equally zero, the kinetic moment of the system relative to this center will be numerically and in the direction will be constant.

2. If, then. Thus, if the sum of the moments of the external forces acting on the system relative to some axis is zero, then the kinetic moment of the system relative to this axis will be the magnitude of constant.

These results are expressed by the law of preserving the kinetic moment.

In the case of a rotating solid from equality (4.34), it follows that if, then. From here we come to the following conclusions:

If the system is immutable (absolutely solid), then, therefore, the solid is rotated around the stationary axis with a constant angular velocity.

If the system is changeable, then. With an increase (then the individual elements of the system are removed from the axis of rotation) the angular speed decreases, because , and when decreasing, it increases, thus, in the case of a variable system with the help of internal forces, you can change the angular speed.

Second task d2. test work Deals the theorem on the change in the kinetic moment of the system relative to the axis.

Task d2.

The homogeneous horizontal platform (the round radius R or rectangular with the sides R and 2R, where R \u003d 1,2m) is rotated with an angular velocity around the vertical axis z, separated from the center of mass from the platform at a distance OC \u003d B (Fig. D2.0 - d2.9, table. D2); Dimensions for all rectangular platforms are shown in Fig. D2,0A (top view).

At the time of time on the chute platform, the cargo D is started (under the influence of the domestic forces), the weight of kg by law, where S is expressed in meters, T - in seconds. At the same time, a pair of forces begins to operate on the platform with the moment M (set in Newtonometers; at m< 0 его направление противоположно показанному на рисунках).

Determine, neglecting the mass of the shaft, dependence i.e. Corner speed platform as a function of time.

In all drawings, the load D is shown in the position at which S\u003e 0 (when s< 0, груз находится по другую сторону от точки А). Изображая чертеж решаемой задачи, провести ось z на заданном расстоянии OC = b от центра C.

Instructions.The task of D2 - to apply the theorem on the change in the kinetic moment of the system. When applying the theorem to a system consisting of a platform and cargo, the kinetic moment of the system relative to the z axis is defined as the sum of the moments of the platform and the cargo. It should be noted that the absolute speed of cargo is consisted of relative and portable velocities, i.e. . Therefore, the amount of movement of this cargo . Then you can use the Varignon Theorem (Static), according to which; These moments are calculated the same as the moments of forces. Read more The solution is clarified in the example of D2.

When solving the problem, it is useful to portray on the auxiliary drawing a view of the platform from above (from the end of Z), as is done in Fig. D2,0, a - d2.9, a.

The moment of inertia plates with a mass M relative to the Cz axis, perpendicular plate and passing through its center of mass, is: for a rectangular plate with sides and

;

For a round radius plate R

Condition number	B.	S \u003d F (T)	M.
	R R / 2 R R / 2 R R / 2 R R / 2 R R / 2	-0.4 0.6 0.8 10 T 0.4 -0.5T -0.6T 0.8T 0.4 0.5	4T -6 -8T -9 6 -10 12

	Fig. D2.0.		Fig. D2.0a

	Fig. D2.1.		Fig. D2.1A

	Fig. D2.2.		Fig. D2.2A.

	Fig. D2.3.		Fig. D2.3A.

	Fig. D2.4.
		Fig. D2.4A.

		Fig. D2.5A.
	Fig. D2.5.

	Fig. D2.6.		Fig. D2.6A.

	Fig. D2.7		Fig. D2.7A.

	Fig. D2.8.		Fig. D2.8A.

	Fig. D2.9.		Fig. D2.9A.

	Fig. D 2

EXAMPLE D2.. The homogeneous horizontal platform (rectangular with the sides 2L and L), having a mass is rigidly bonded with a vertical shaft and rotates with it around the axis z.with angular speed (Fig. D2A ). At the time of time, the torque torque is started to act the oppositely ; at the same time the cargo D.mass found in the gutter AUat point FROM,begins to move on the groove (under the action of the internal forces) by law s \u003d Cd \u003d F (T).

Danched: M 1 \u003d 16 kg, t 2.\u003d 10 kg, l.\u003d 0.5 m, \u003d 2, s \u003d 0.4t 2 (S - in meters, t - in seconds), M.= ktwhere k.\u003d 6 nm / s. Determine: - the law of change angular speed platforms.

Decision. Consider mechanical systemconsisting of platform and cargo D.To determine W, we apply the theorem about changing the kinetic moment of the system relative to the axis z:

(1)

We will depict the external force acting on the system: the gravity of the reaction and the torque M. Since the strength and parallel to the axis Z, and the reactions and this axis are crossed, their moments relative to the z axis are zero. Then, counting for the moment a positive direction (i.e., against the course of the clockwise), we get And equation (1) will take this species.

As a system, which is discussed in the theorem, any mechanical system consisting of any bodies can act.

The wording of the theorem

The number of motion (impulse) of the mechanical system is called a value equal to the amount of the amount of movement (pulses) of all bodies that are in the system. The impulse of the external forces acting on the body of the system is the sum of the pulses of all external forces acting on the body of the system.

( kg · m / s)

The theorem on the change in the number of the system is approved

The change in the amount of system movement for a certain period of time is equal to the pulse of the external forces acting on the system during the same period of time.

The law of preserving the number of system movement

If the sum of all external forces acting on the system is zero, then the number of motion (impulse) of the system is the value constant.

, we obtain the expression of the theorem on changing the number of system movement in differential form:

Integrating both parts of the equality obtained on an arbitrarily taken period of time between some and, we obtain the expression of the theorem on changing the number of system movement in the integral form:

Law of preserving impulse (The law of preserving the number of movement) It claims that the vector sum of the pulses of all bodies of the system is the value constant if the vector sum of the external forces acting on the system is zero.

(the moment of the number of motion m 2 · kg · s -1)

Theorem on changing the moment of the number of movement relative to the center

the time derivative from the moment of the amount of movement (kinetic torque) of the material point relative to a fixed center is equal to the moment of the current force on the same center.

dk 0 /dT \u003d M. 0 (F. ) .

Theorem on changing the moment of the amount of movement relative to the axis

the time-derived from the moment of the amount of movement (kinetic moment) of the material point relative to any fixed axis is equal to the moment of force acting on this point relative to the same axis.

dk x. /dT \u003d M. x. (F. ); dk y. /dT \u003d M. y. (F. ); dk z. /dT \u003d M. z. (F. ) .

Consider a material point M. Mass m. moving under the action of power F. (Figure 3.1). We write and build the moment of the moment of movement (kinetic moment) M. 0 material point relative to the center O. :

Differentiating the expression of the moment of the amount of movement (kinetic moment k. 0) by time:

As dr. /dt. = V. , then vector work V. ⊗ m. ⋅ V. (Collinear vectors V. and m. ⋅ V. ) Equally zero. In the same time d (M. ⋅ V) /dT \u003d F. According to the theorem on the number of motion of the material point. So we get that

dk 0 /dt. = r. ⊗F. , (3.3)

where r. ⊗F. = M. 0 (F. ) - vector moment of power F. Regarding a fixed center O. . Vector k. 0 ⊥ planes ( r. , m. ⊗V. ), and vector M. 0 (F. ) ⊥ planes ( r. ,F. ), finally we have

dk 0 /dT \u003d M. 0 (F. ) . (3.4)

Equation (3.4) expresses the theorem about changing the moment of the amount of movement (kinetic torque) of the material point relative to the center: the time derivative from the moment of the amount of movement (kinetic torque) of the material point relative to a fixed center is equal to the moment of the current force on the same center.

Projecting equality (3.4) on the axis of the Cartesian coordinates, we get

dk x. /dT \u003d M. x. (F. ); dk y. /dT \u003d M. y. (F. ); dk z. /dT \u003d M. z. (F. ) . (3.5)

Equality (3.5) express the theorem of changing the moment of the amount of movement (kinetic moment) of the material point relative to the axis: the time-derived from the moment of the amount of movement (kinetic moment) of the material point relative to any fixed axis is equal to the moment of force acting on this point relative to the same axis.

Consider the consequences arising from Theorems (3.4) and (3.5).

Corollary 1. Consider the case when the power F. At all the time the movement point passes through a fixed center O. (Case of Central Force), i.e. when M. 0 (F. ) \u003d 0. Then, from Theorem (3.4) it follows that k. 0 = const. ,

those. In the case of the central force, the moment of the amount of movement (kinetic moment) of the material point relative to the center of this force remains constant by the module and direction (Figure 3.2).

Figure 3.2

From condition k. 0 = const. It follows that the trajectory of the moving point is a flat curve, the plane of which passes through the center of this force.

Corollary 2. Let be M. z. (F. ) \u003d 0, i.e. Power crosses the axis z. or her parallel. In this case, as can be seen from the third of equations (3.5), k. z. = const. ,

those. If the moment of the current force is always equal to the point of force, then the moment of the amount of movement (kinetic moment) points relative to this axis remains constant.

Proof of the theorem Ob i with the amount of movement

Let the system consist of material points with masses and accelerations. All forces acting on the body of the system, divide into two types:

External forces - the forces acting on the part of the bodies that are not included in the system under consideration. Equality of external forces acting on the material point with the number i. Denote.

Internal forces - the forces with which they interact with each other of the body itself. Strength with which to the point with the number i. acts dot with number k., we will designate, and the power of exposure i.Point on k.- Point -. Obviously, when, then

Using the introduced notation, write Newton's second law for each of the material points under consideration in the form of

Considering that and summing up all the equations of Newton's second law, we get:

The expression is the sum of all internal forces operating in the system. According to the third law of Newton in this amount, each force corresponds to the force such as, it means Since the entire amount consists of such pairs, the amount itself is zero. So you can record

Using the designation of the system to move the system, we get

Entering into consideration a change in the impulse of external forces , We obtain the expression of the theorem on changing the number of system movement in differential form:

Thus, each of the last obtained equations allows you to assert: a change in the amount of system movement occurs only as a result of the action of external forces, and the internal forces can not have any influence on this magnitude.

Integrating both parts of the equality obtained according to an arbitrarily taken period of time between some and, we obtain the expression of the theorem on changing the number of system movement in the integral form:

where and is the values \u200b\u200bof the amount of system movement at the moments of time and, accordingly, a - pulse of external forces over the time interval. In accordance with those who have said earlier and the designations are performed.

Let the material point move under force F.. It is required to determine the movement of this point in relation to the mobile system. Oxyz. (see complex motion of the material point), which moves a known manner with respect to the fixed system O. 1 x. 1 y. 1 z. 1 .

The main equation of speakers in the fixed system

We write the absolute acceleration of the point by the Coriolis theorem

where a. abs - absolute acceleration;

a. relative - relative acceleration;

a. per - portable acceleration;

a. corner - Coriolis acceleration.

Remember (25), taking into account (26)

We introduce notation
- portable inertia force,
- Coriolis is the power of inertia. Then equation (27) acquires the view

The main equation of the dynamics to study the relative movement (28) is recorded as as for the absolute movement, only a portable and Coriolis for the power of inertia should be added to the forces.

General Material Dynamics Theorems

When solving many tasks, you can use the pre-preparations made on the basis of Newton's second law. Such methods for solving problems are combined in this section.

Theorem on changing the amount of material point

We introduce the following dynamic characteristics:

1. The amount of motion of the material point - vector magnitude equal to the product of the point of point on the vector of its speed

. (29)

2. Power pulse

Elementary power impulse - vector magnitude equal to the work of the strength vector on an elementary period of time

(30).

Then full impulse

. (31)

For F.\u003d const S.=Ft..

A complete pulse for a finite period of time can be calculated only in two cases, when the power is permanent or dependent on the point. In other cases, it is necessary to express force as a function of time.

The equality of the dimensions of the impulse (29) and the amount of movement (30) allows you to establish a quantitative relationship between them.

Consider the movement of the material point M under the action of arbitrary strength F. According to an arbitrary trajectory.

ABOUT UD:
. (32)

We divide into (32) variables and integrate

. (33)

As a result, taking into account (31), we get

. (34)

Equation (34) expresses the following theorem.

Theorem: Changing the amount of material movement for a certain period of time is equal to the pulse of force acting on the point, during the same time interval.

When solving problems, equation (34) must be designed on the axis of coordinates

It is convenient to use this theorem when among the specified and unknown values \u200b\u200bthere are plenty of point, its initial and final speed, strength and time of movement are present.

Theorem on changing the moment of material point of motion

M.
omiment of the amount of motion of the material point Regarding the center is equal to the product of the module of the movement of the point on the shoulder, i.e. The shortest distance (perpendicular) from the center to the line coinciding with the speed vector

, (36)

. (37)

The relationship between the moment of force (cause) and the moment of the amount of movement (consequence) establishes the following theorem.

Let the point M of a given mass m. moving under the action of power F..

,
,

, (38)

. (39)

Calculate the derivative from (39)

. (40)

Combining (40) and (38), finally get

. (41)

Equation (41) expresses the following theorem.

Theorem: The time derivative from the moment of the moment of the amount of material of the material point relative to some center is equal to the moment the point of force on the same center.

When solving problems, equation (41) must be designed on the coordinate axes

In equations (42), the moments of the amount of movement and force are calculated relative to the coordinate axes.

From (41) follows the law of preserving the moment of the number of movement (the law of Kepler).

If the moment of force acting on the material point relative to any center is zero, then the moment of the number of motion of the point relative to this center retains its size and direction.

If a
T.
.

The theorem and the law of conservation are used in problems on curvilinear movement, especially under the action of central forces.