Similarly, for how for one material point, we derive the theorem about changing the amount of movement for the system in various forms.

We convert the equation (the theorem on the movement of the cents of the mechanical system)

in the following way:

;

The resulting equation expresses the theorem on the change in the number of movement of the mechanical system in differential form: a derivative of the amount of movement of the mechanical system in time is equal to the main vector of external forces operating on the system .

In projections on Cartesian coordinate axes:

; ; .

Taking integrals from both parts of the last equations in time, we obtain the theorem of changing the number of mechanical system in an integrated form: a change in the amount of movement of the mechanical system is equal to the impulse of the main vector of external forces operating on the system .

.

Or in the projections on the Cartesian coordinate axes:

; ; .

Corollary from Theorem (the laws of maintaining the amount of movement)

The law of preserving the amount of movement is obtained as special cases of the theorem on the change in the amount of movement for the system, depending on the characteristics of the external strength system. Internal forces can be anyhow, as they do not affect changes in the amount of movement.

Two cases are possible:

1. If the vector sum of all external forces attached to the system is zero, the number of system movement is constantly large and direction

2. If zero is equal to the projection of the main vector of external forces on which or the coordinate axis and / or / or / or, then the projection of the amount of movement on the same axes is the value of constant, i.e. and / or / or respectively.

Similar entries can be made for the material point and for the material point.

The task. From gun, the mass of which M., flies in the horizontal direction m. with speed v.. Find speed V. guns after a shot.

Decision. All external forces acting on the mechanical system of an instrument-projectile, vertical. Therefore, on the basis of the investigation from the theorem on the change in the number of system movement, we have :.

The number of mechanical system movement to the shot:

The number of movement of the mechanical system after a shot:

.

Equating the right parts of expressions, we get that

.

The sign "-" in the resulting formula indicates that after the shot, the instrument rolls back in the direction opposite to the axis OX..

Example 2. A stream of fluid density flows at a velocity V from the pipe with a cross-sectional area F and hits an angle of the vertical wall. Determine the pressure of the fluid on the wall.

DECISION. Apply the theorem about changing the amount of movement in the integral form to the volume of fluid mass m. hitting the wall for a certain period of time t..

Meshchersky equation

(Basic equation of variable mass speakers)

In modern technology, there are cases when the mass of the point and the system does not remain constant during the movement, but changes. For example, under flight of space missiles, due to the throwing of combustion products and individual unnecessary parts of missiles, the mass change reaches 90-95% of the total initial value. But not only the space technology can be an example of the dynamics of the movement of the variable mass. In the textile industry there are significant changes in the mass of various spindles, bumps, rolls at modern speeds of machines and machines.

Consider the main features associated with the change in mass, on the example of the progressive movement of the body of the variable mass. To the body of variable mass can not be directly applying the main law of speakers. Therefore, we obtain differential equations of motion of the variable mass point by applying the theorem about changing the number of system movement.

Let the point mass m + DM.moves at speed. Then there is a separant from the point of some particle mass. dM. moving at speed.

The amount of body movement to the separation of the particle:

The number of motion of the system consisting of a body and a broken particle after its separation:

Then change the amount of movement:

Based on the theorem on the change in the number of system movement:

Denote by the value - the relative speed of the particle:

Denote

Magnitude R. Called reactive power. The reactive force is the engine of the engine, due to the emission of gas from the nozzle.

Finally get

-

This formula expresses the main equation of the dynamics of the body of the variable mass (Meshchersky formula). From the last formula, it follows that the differential equations of motion of the variable mass point have the same form as for a constant mass point, except for an additional reactive force applied to the point, due to a change in mass.

The basic equation of the dynamics of the body variable mass indicates that the acceleration of this body is formed not only due to external forces, but also due to the reactive force.

The reactive force is the power, a relative of the one that the shooting person feels - when shooting from a pistol, it feels hands with a brush; When shooting from the rifle is perceived by the shoulder.

The first formula of Tsiolkovsky (for a single-stage rocket)

Let the variable mass or rocket be moved straightforwardly under the action of only one reactive force. As for many modern jet engines , where - the most allowed jet power allowed by the engine (motor); - The strength of gravity acting on the engine located on the earth's surface. Those. The above allows the component in the Meshchers' equation to neglect and to further adopt this equation in the form:,

Denote:

Fuel supply (with liquid jet engines - dry mass of the rocket (the remaining mass after the burnout of all fuel);

The mass of particles separated from the rocket; It is considered as a variable value varying from before.

We write the equation of the rectilinear movement of the variable mass point in the following form

.

Since the formula for determining the mass rocket mass

Consequently, the point of motion equations Taking integrals from both parts

where - characteristic speed - This is the speed that the rocket acquires under the action of thrust after eruptions from the rocket of all particles (with liquid jet engines - after the burnout of the entire fuel).

Announced for the integral sign (which can be done on the basis of the medium theorem known from the highest mathematics) is the average speed of the particles sponsored from the rocket.

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 integrated 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.

Consisting of n. material points. We highlight some point from this system M J. With mass m J.. At this point, as is well known, external and internal forces act.

We applied to the point M J. Equality of all internal forces F J I. and equal all external forces F J E. (Figure 2.2). For highlighted material point M J. (as for a free point) Write the theorem about changing the amount of movement in differential form (2.3):

We write similar equations for all points of the mechanical system (j \u003d 1,2,3, ..., n).

Figure 2.2.

Moving so far everything n.equations:

Σd (m j × v j) / dt \u003d σf j e + σf j i, (2.9)

dς (m j × v j) / dt \u003d σf j e + σf j i. (2.10)

Here Σm j × v j \u003d q - the number of mechanical system movement;
ΣF j e \u003d r e - the main vector of all external forces operating on the mechanical system;
ΣF j i \u003d r i \u003d 0 - The main vector of the internal forces of the system (by the property of the internal forces it is zero).

Finally for the mechanical system we get

dQ / DT \u003d R E. (2.11)

The expression (2.11) is a theorem that changes in the number of motion of the mechanical system in differential form (in vector terms): the time-derived from the number of the number of mechanical system movement is equal to the main vector of all external forces operating on the system.

Projecting vector equality (2.11) on the Cartesian coordinate axes, we obtain expressions for the theorem on changing the number of movement of the mechanical system in the coordinate (scalar) expression:

dQ X / DT \u003d R x E;

dQ Y / DT \u003d R Y E;

dQ z / dt \u003d r z e, (2.12)

those. the time-derived from the projection of the number of mechanical system movement on any axis is equal to the projection on this axis of the main vector of all external forces acting on this mechanical system.

Multiplying both parts of equality (2.12) on dt., I get theorem in another differential form:

dQ \u003d R E × DT \u003d ΔS E, (2.13)

those. the differential of the amount of movement of the mechanical system is equal to the elementary impulse of the main vector (amount of elementary pulses) of all external forces operating on the system.

Integrating equality (2.13) within the change in time from 0 to t., We obtain the theorem about changing the amount of movement of the mechanical system in the final (integral) form (in vector terms):

Q - Q 0 \u003d s e,

those. changing the amount of movement of the mechanical system for the final period of time is equal to the complete impulse of the main vector (summation of full pulses) of all external forces operating on the system in the same time.

Projecting vector equality (2.14) on the Cartesian axis of the coordinates, we obtain expressions for theorem in projections (in the scalar expression):

those. changes in the projection of the number of movement of the mechanical system on any axis for the final period of time equal to the projection on the same axis of the total pulse of the main vector (the amount of complete pulses) of all external forces operating on the mechanical system in the same time.

From the considered theorem (2.11) - (2.15), the investigations flow:

1. If a R E \u003d ΣF j e \u003d 0T. Q \u003d Const. - We have a law of saving the vector of the number of mechanical system: if the main vector R E. All external forces acting on the mechanical system are zero, the vector of the amount of movement of this system remains constant in size and direction and equal to its initial value. Q 0. Q \u003d Q 0.
2. If a R x E \u003d Σx j e \u003d 0 (R e ≠ 0)T. Q x \u003d const - We have the law of preserving the projection on the axis of the number of mechanical system movement: if the projection of the main vector of all the forces acting on the mechanical system on any axis is zero, then the projection on the same axis of the number of the number of this system will be the magnitude of the constant and equal projection on this axis Initial vector of the amount of movement, i.e. Q x \u003d Q 0x.

The differential form of the theorem on the change in the amount of motion of the material system has important and interesting applications in the mechanics of a solid medium. From (2.11) you can get the Euler theorem.

View:this article read 14066 times

PDF Select a tongue ... Russian Ukrainian English

Short review

Fully the material is downloaded above, after selecting the language

Number of traffic

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.

The unit of measurement of the amount of movement is (kg m / s).

The number of movement of the mechanical system - The vector value equal to the geometric sum (the main vector) of the amount of movement of the mechanical system is equal to the mass of the entire system on the speed of its center mass.

When the body (or system) moves so that its center of the masses is immobile, the amount of body movement is zero (for example, the rotation of the body around the stationary axis passing through the center of mass body).

In the case of a complex movement, the number of system movement will not characterize the rotational part of the movement during rotation around the mass center. Those., The number of movement characterizes only the translational movement of the system (together with the center of the masses).

Power pulse

The power pulse characterizes the effect of force for a certain period of time.

Pulse force for the final period of time Determined as the integral sum of the corresponding elementary pulses.

Theorem on changing the amount of material point

(in differential forms e. ):

The time derivative on the amount of motion of the material point is equal to the geometric sum of the strength acting on the point.

(in integral form ):

The change in the amount of motion of the material point over a certain period of time is equal to the geometric sum of the pulses of the forces applied to the point over this time period.

Theorem on changing the number of mechanical system movement

(in differential form ):

The time derivative on the amount of system movement is equal to the geometric sum of all external forces operating on the system.

(in integrated form ):

The change in the amount of system movement over a certain period of time is equal to the geometric sum of the pulses of the external forces acting on the system over this period of time.

Theorem makes it possible to exclude unknown internal forces from consideration.

The theorem on the change in the amount of movement of the mechanical system and the theorem on the movement of the center of mass is two different forms of one theorem.

The law of preserving the number of system movement

1. If the sum of all external forces acting on the system is zero, then the system of the amount of system movement will be constant in the direction and module.
2. If the amount of projections of all existing external forces on any arbitrary axis is zero, then the projection of the amount of movement on this axis is a permanent value.

conclusions:

1. The laws of conservation indicate that the internal forces cannot change the total number of system movement.
2. The theorem on the change in the number of movement of the mechanical system does not characterize the rotational motion of the mechanical system, but only the translational.

An example is: determine the amount of movement of the disk of a certain mass, if its angular speed and size is known.

An example of calculating the strait of cylindrical transmission
An example of calculating the strait of cylindrical transmission. The choice of material, the calculation of the allowable stresses, the calculation on the contact and flexural strength.

Example Solving Beam Beam Tasks
In the example, the line of transverse forces and bending moments are built, a dangerous cross-section has been found and a mellover is selected. The task is analyzed by the construction of an EPUR using differential dependencies, a comparative analysis of various transverse sections of the beam was conducted.

Example Solving Task Task
The task is to check the strength of the steel shaft at a given diameter, material and allowable voltages. During the solution, the plots of torque, tangent stresses and spinning angles are being built. Own weight of the shaft is not taken into account

Example Solving Tensile Testing-Compression Rod
The task is to check the strength of the steel rod at given to the allowed voltages. During the solution, the supports of the longitudinal forces, normal stresses and movements are being built. Own weight rod is not taken into account

Application of the theorem on the preservation of kinetic energy
An example of solving the task of applying the theorem on the preservation of the kinetic energy of the mechanical system

Determination of the speed and acceleration of the point according to the specified motion equations
Example Solution of the task for determining the speed and acceleration of the point according to the given movement equations

Determination of speeds and accelerations of solid points with plane-parallel motion
An example of solving the problem of determining the speeds and accelerations of solid points with plane-parallel motion

Definition of efforts in the rods of a flat farm
An example of solving the problem on the definition of efforts in the rods of a flat farm by the ritter method and the method of cutting nodes

Application of a kinetic moment theorem
An example of solving the task of applying the theorem on the change in the kinetic moment to determine the angular velocity of the body performing the rotation around the stationary axis.