Quantity of movement

measure mechanical movement, equal for a material point to the product of its mass m for speed v. K. l. mv- a vector quantity, directed in the same way as the speed of a point. Sometimes CD is also called an impulse. Under the action of a force, the efficiency of a point changes in the general case both numerically and in direction; this change is determined by the second (fundamental) law of dynamics (see Newton's laws of mechanics).

K.d.Q mechanical system equals geometric sum The efficiency of all its points or the product of the mass M the entire system for speed vc its center of mass: Q= ∑m k v k =Mv s. A change in the efficiency of the system occurs under the influence only external forces, that is, forces acting on the system from bodies not included in this system. According to the theorem on the change in efficiency, Q 1 -Q 0 = ∑S k e. where Q 0 and Q 1 are the efficiency of the system at the beginning and at the end of a certain period of time, S k e - impulses of external forces F k e (see Force impulse) for this period of time (in differential form the theorem is expressed by the Dynamics equation) , in particular in the theory of Impact a.

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 equal to zero, the law of conservation of efficiency holds. In this case, the efficiency of individual parts of the system (for example, under the influence internal forces) can change, but in such a way that the value Q = ∑m to v k remains constant. This law explains such phenomena as jet motion, recoil (or rollback) when fired, the operation of a propeller or oars, etc. For example, if we consider a gun and a bullet as one system, then the pressure of the powder gases when fired will be an internal force for this system and cannot change the efficiency of the system, which is equal to zero before the shot. Therefore, telling the bullet K. d. m 1 v 1 , directed towards the muzzle, the powder gases will simultaneously impart to the gun a numerically the same, but oppositely directed K. d. m 2 v 2, what will cause recoil; from equality m 1 v 1 = m 2 v 2(where v 1, v 2 are the numerical values ​​of the speeds) it is possible, knowing the speed v 1; bullets when leaving the barrel, find the highest speed v 2 recoil (and for a gun - recoil).

At speeds close to the speed of light, the c.d., or momentum, of a free particle is determined by the formula p = mv/β=v/c; when vc, this formula becomes the usual one: p = mv(see Relativity theory).

K. d. possess and physical fields (electromagnetic, gravitational, etc.). The efficiency of fields is characterized by the density of the field (the ratio of the efficiency of an elementary volume to this volume) and is expressed in terms of the field strength or its potential, etc.

S. M. Targ.

Big Soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what “Quantity of Movement” is in other dictionaries:

A measure of mechanical motion, equal for a material point to the product of its mass m and speed v. The amount of motion mv is a vector quantity, directed in the same way as the speed of the point. The amount of motion is also called impulse... Big Encyclopedic Dictionary

- (impulse), measure of mechanical motion, equal for a material point to the product of its mass m and speed v. K. d. mv is a vector quantity, directed in the same way as the speed of a point. Under the influence of a force, the efficiency of a point changes in the general case both numerically and... ... Physical encyclopedia

See Impulse. Philosophical encyclopedic dictionary. 2010 … Philosophical Encyclopedia

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A measure of mechanical motion, equal for a material point to the product of its mass m and speed v. The amount of motion mv is a vector quantity coinciding in direction with the velocity vector v. The quantity of motion is also called impulse. * * *… … Encyclopedic Dictionary

Impulse (quantity of motion) additive integral of motion of a mechanical system; the corresponding conservation law is associated with the fundamental symmetry and homogeneity of space. Contents 1 History of the term 2 “School” definition... ... Wikipedia

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momentum- judesio kiekis statusas T sritis fizika atitikmenys: engl. kinetic momentum momentum; quantity of motion vok. Bewegungsgröße, f; Impuls, m rus. impulse, m; momentum, n pranc. impulse, f; quantité de mouvement, f … Fizikos terminų žodynas

Quantity of movement- the same as impulse, a measure of mechanical motion equal to the product of the mass of a body m by its speed v. The momentum vector coincides in direction with the velocity vector... The beginnings of modern natural science

Mechanical measure motion, equal for a material point to the product of its mass from and speed v. K. d. mv is a vector quantity coinciding in direction with the velocity vector v. K. d. called. also an impulse... Natural science. Encyclopedic Dictionary

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14. The amount of motion of a material point and a mechanical system.

Number of doors mat/points is called a vector quantity equal to the product of mass and its speed (directed as well as tangentially).

Number of motors we will call a vector quantity equal to the geometric sum (principal vector) of the number of points of all points:

Number of motors is equal to the product of the mass of the entire object by the speed of its center of mass:

15. Elementary impulse of force and impulse of force for a finite period of time.

Elem imp of strength is called a vector quantity equal to the product of force and elemental time interval dt: (directed along the line of action of the force)

Impulse force over a certain period of time t 1 is equal to definite integral from the element pulse taken within the range from 0

16. Theorems on the change in momentum of a material point in differential and finite forms.

T-ma about changing the number of moving parts/points in diff/form: the time derivative of the number of moving points is equal to the sum of the forces acting on the point:

At t=0 speed, at t 1 speed

T-ma about changing the number of moving parts/points (in con/form): quantity change

The movement of a point over a certain period of time is equal to the sum of the impulses of all forces acting on the point over the same period of time.

17. Theorem on the change in momentum of a mechanical system. Law of conservation of momentum.

T-ma about changing the number of motors in diff/form: the time derivative of the number of motors is equal to the geometric sum of all forces acting on

s-mu external forces. On

At t=0 number of doors, at t 1 number/door:

T-ma about changing the number of motors in integral form: the change in the number/dv of s-we over a certain period of time is equal to the sum of the impulses acting on s-th of external forces over the same period of time.

Drying number of motors:

1) Let , then = const. If the sum of external forces acting on the c-mu is equal to 0, then the vector of quantity/movement of the c-mu will be constant in magnitude and direction.

2) Let , then=const. If the sum of the projections of all acting external forces onto any axis is equal to 0, then the projection of the quantity/movement on this axis is a constant value.

18. Moment of momentum of a material point relative to the center and relative to the axis.

Moment of number of points relative to some center O is called a vector quantity defined by the equality (directed perpendicularly

plane passing through and center O)

Moment of the number of points relative to the Oz axis passing through the center O:

19. Kinetic moment of a mechanical system relative to the center and relative to the axis. The kinetic moment of a rigid body relative to the axis of rotation.

The main moment of the number of movements (or the kinetic moment) is relative to this center This is a quantity equal to the geometric sum of the moments of the number of movements of all points relative to this center:

Axis projection:

At any point of the body located at a distance from the axis of rotation, the speed is, therefore:

The kinetic moment of rotation of the body relative to the axis of rotation equal to the product of the moment of inertia of the body relative to this axis

to the angular velocity of the body:

20. number of double mat. points - vectormυ dimension [kg*m\s]=[N*s]

Theorem: The time differential from the number of two mating points is equal to the geometric sum of the forces acting on them.

Multiply bydt, : d(mυ) . Full impulseS=multiply bydtwe obtain the integral final form of writing the theorem:m . – The change in the number of double mathematical points over a certain period of time is equal to the geometric sum of the force impulses acting on the point over the same period of time. Analytical record form:m mm

(21). Theorem on the change in the kinetic moment of a mechanical system. Law of conservation of angular momentum.

T-ma moments for s-we: the time derivative of the main moment of the number of movements with respect to some fixed center is equal to the sum of the moment of all external forces with respect to the same center. Axis projection:

Law of conservation of momentum:

By definition, the amount of motion of a system is a vector

Therefore, according to Newton's second law

and by virtue of relation (5)

This statement is called the theorem on the change in momentum (momentum) of the system:

The time derivative of the momentum of the system is equal to the main vector of all external forces acting on the system.

Projecting equality (7) to any fixed axis, we get

where is the projection onto the vector axis, and is the projection of the vector onto it.

If the system is closed, then, by definition, external forces do not act on its points, i.e.

(9)

This establishes the law of conservation of momentum: When a closed system moves, the momentum (momentum) of the system does not change.

This statement is also true, of course, for a system that is acted upon by external forces, if .

From equality (8) it follows that if , then , i.e., that for any system the projection of the momentum onto a certain axis does not change during movement if the main vector of the external forces of the system is perpendicular to this axis.

The theorem on the change in momentum and the law of conservation of momentum can be given a different form if we introduce the concept of the center of inertia of the system.

The center of inertia of the system is called a geometric point

From space, determined by the radius vector

The quantity is called the mass of the system.

During the movement of the points of the system, , and therefore changes, i.e., when the points of the system move, its center of inertia also moves. The trajectory of the center of inertia is locus(hodograph) of the ends of the vectors, and the speed of point C is directed tangentially to this hodograph and is determined by the equality

which is obtained by differentiating equality (10) with respect to .

From equality (11) it follows that

i.e., that the momentum of the system is equal to the mass of the system multiplied by the speed of its center of inertia.

From the theorem on the change in momentum it follows then

But equality (13) expresses Newton’s second law for a material point placed at the center of inertia and moving with it, if the mass of this point is equal to M and if a force is applied to it. It follows that the theorem on the change in momentum can be formulated as follows:

When the system moves material points its center of inertia moves in the same way as a material point placed at the center of inertia would move if the masses of all points of the system were concentrated in it and all external forces acting on the points of the system were applied to it.

In this formulation, the theorem on the change in momentum is called the theorem on the movement of the center of inertia.

In closed systems and

(14)

Therefore, the law of conservation of momentum can be formulated as follows: the center of inertia of a closed system moves with constant speed(perhaps equal to zero).

Of course, this statement is also true for projections of the corresponding vectors. If the projection of the main vector of external forces onto a certain axis is identically equal to zero, then the center of inertia moves in such a way that the projection of the velocity of the center of inertia on this axis remains constant.

Further, it will sometimes be convenient to introduce into consideration an auxiliary reference system that moves translationally and whose origin is placed at the center of inertia of the system. We will further call such a reference system central. In the case where the speed of the center of inertia is constant, central system is inertial.

To solve many dynamics problems, especially in system dynamics, instead of direct integration differential equations motion, it turns out to be more effective to use the so-called general theorems, which are consequences of the basic law of dynamics.

The significance of general theorems is that they establish visual relationships between the corresponding dynamic characteristics of the motion of material bodies and thereby open up new opportunities for studying the motion of mechanical systems, widely used in engineering practice. In addition, the use of general theorems eliminates the need to perform for each problem those integration operations that are performed once and for all when deriving these theorems; This simplifies the solution process.

Let us move on to consider general theorems on the dynamics of a point.

§ 83. AMOUNT OF MOTION OF A POINT. POWER IMPULSE

One of the main dynamic characteristics of the movement of a point is the amount of movement

The momentum of a material point is a vector quantity equal to the product of the mass of the point and its speed. The direction of the vector is the same as the speed of the point, i.e., tangent to its trajectory.

The unit of measurement of momentum is in SI - and in the MKGSS system - .

Impulse of force. To characterize the action exerted on a body by a force over a certain period of time, the concept of force impulse is introduced. First, let us introduce the concept of an elementary impulse, i.e., an impulse over an elementary period of time

An elementary impulse of force is a vector quantity equal to the product of force F and an elementary period of time

The elementary impulse is directed along the line of action of the force.

The impulse S of any force F for a finite period of time is calculated as the limit of the integral sum of the corresponding elemental impulses, i.e.

Consequently, the impulse of a force over a certain period of time is equal to a certain integral of the elementary impulse, taken in the range from zero to

and mechanical system

The momentum of a material point is a vector measure of mechanical motion, equal to the product of the mass of the point and its speed, . The unit of measurement of momentum in the SI system is
. The amount of motion of a mechanical system is equal to the sum of the amounts of motion of all material points forming the system:

. (5.2)

Let's transform the resulting formula

.

According to formula (4.2)
, That's why

.

Thus, the momentum of a mechanical system is equal to the product of its mass and the speed of the center of mass:

. (5.3)

Since the amount of motion of a system is determined by the motion of only one of its points (the center of mass), it cannot be a complete characteristic of the motion of the system. Indeed, for any motion of the system, when its center of mass remains stationary, the momentum of the system is zero. For example, this occurs when rotating solid around a fixed axis passing through its center of mass.

Let's introduce a reference system Cxyz, having its origin at the center of mass of the mechanical system WITH and moving translationally relative to the inertial system
(Fig. 5.1). Then the movement of each point
can be considered as complex: portable movement together with axes Cxyz and movement relative to these axes. Due to the progressive movement of the axes Cxyz the portable speed of each point is equal to the speed of the center of mass of the system, and the amount of motion of the system, determined by formula (5.3), characterizes only its translational portable motion.

5.3. Impulse force

To characterize the action of a force over a certain period of time, a quantity called impulse of force . An elementary impulse of a force is a vector measure of the action of a force, equal to the product of the force by the elementary time interval of its action:

. (5.4)

The SI unit of force impulse is
, i.e. The dimensions of force impulse and momentum are the same.

Force impulse over a finite period of time
is equal to a certain integral of the elementary momentum:

. (5.5)

The impulse of a constant force is equal to the product of the force and the time of its action:

. (5.6)

In general, the force impulse can be determined by its projections onto the coordinate axes:

. (5.7)

5.4. Momentum change theorem

material point

In the basic equation of dynamics (1.2), the mass of a material point is a constant quantity, its acceleration
, which makes it possible to write this equation in the form:

. (5.8)

The resulting relationship allows us to formulate theorem on the change in momentum of a material point in differential form: The time derivative of the momentum of a material point is equal to the geometric sum (principal vector) of the forces acting on the point.

Now we obtain the integral form of this theorem. From relation (5.8) it follows that

.

Let's integrate both sides of the equality within the limits corresponding to the moments of time And ,

. (5.9)

The integrals on the right side represent the impulses of the forces acting on the point, so after integrating the left side we get

. (5.10)

Thus it is proven theorem on the change in momentum of a material point in integral form: The change in the momentum of a material point over a certain period of time is equal to the geometric sum of the impulses of the forces acting on the point over the same period of time.

The vector equation (5.10) corresponds to a system of three equations in projections onto the coordinate axes:

;

; (5.11)

.

Example 1. The body moves translationally along an inclined plane forming an angle α with the horizon. At the initial moment of time it had a speed , directed upward along an inclined plane (Fig. 5.2).

After what time does the speed of the body become equal to zero if the coefficient of friction is equal to f ?

Let us take a translationally moving body as a material point and consider the forces acting on it. It's gravity
, normal plane reaction and friction force . Let's direct the axis x along the inclined plane upward and write the 1st equation of the system (5.11)

where are the projections of quantities of motion, and are the projections of impulses of constant forces
,And are equal to the products of the projections of forces and the time of movement:

Since the acceleration of the body is directed along the inclined plane, the sum of the projections onto the axis y of all forces acting on the body is equal to zero:
, from which it follows that
. Let's find the friction force

and from equation (5.12) we obtain

from where we determine the time of movement of the body

.