Change in angular momentum. Moments of momentum of a point relative to the center and axis Moment of momentum of a mechanical system

Consider a material point M mass m, moving under the influence of force F(Figure 3.1). Let's write down and construct the vector of angular momentum (kinetic momentum) M0 material point relative to the center O:

Figure 3.1

Let us differentiate the expression for the angular momentum (kinetic moment k 0) by time:

Because dr/dt=V, then the vector product V × m∙V(collinear vectors V And m∙V) is equal to zero. At the same time d(m∙V)/dt=F according to the theorem on the momentum of a material point. Therefore we get that

dk 0 /dt = r×F, (3.3)

Where r×F = M 0 (F)– vector-moment of force F relative to a fixed center O. Vector k 0⊥ plane ( r, m×V), and the vector M0(F)⊥ plane ( r, F), we finally have

dk 0 /dt = M 0 (F). (3.4)

Equation (3.4) expresses the theorem about the change in angular momentum (angular momentum) of a material point relative to the center: the time derivative of the moment of momentum (kinetic moment) of a material point relative to any fixed center is equal to the moment of force acting on the point relative to the same center.

Projecting equality (3.4) onto the axes of Cartesian coordinates, we obtain

dk x /dt = M x (F);

dk y /dt = M y (F);

dk z /dt = M z (F). (3.5)

Equalities (3.5) express the theorem about the change in angular momentum (kinetic momentum) of a material point relative to the axis: the time derivative of the moment of momentum (kinetic moment) of a material point relative to any fixed axis is equal to the moment of the force acting on this point relative to the same axis.

Let us consider the consequences following from Theorems (3.4) and (3.5).

Corollary 1

Consider the case when the force F during the entire movement of the point passes through the stationary center O(case of central force), i.e. When M 0 (F) = 0. Then from Theorem (3.4) it follows that k 0 = const, those. in the case of a central force, the angular momentum (kinetic moment) of a material point relative to the center of this force remains constant in magnitude and direction(Figure 3.2).

Figure 3.2

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

Corollary 2

Let M z (F) = 0, i.e. force crosses the axis z or parallel to it.

In this case, as can be seen from the third of equations (3.5), k z = const, those. if the moment of force acting on a point relative to any fixed axis is always zero, then the angular momentum (kinetic moment) of the point relative to this axis remains constant.

• 1. Algebraic angular momentum about the center. Algebraic ABOUT-- scalar quantity taken with the sign (+) or (-) and equal to the product of the modulus of momentum m to a distance h(perpendicular) from this center to the line along which the vector is directed m:
• 2. Vector moment of momentum relative to the center.

Vector moment of momentum of a material point relative to some center ABOUT -- vector applied at this center and directed perpendicular to the plane of vectors m And in the direction from which the point’s movement is visible counterclockwise. This definition satisfies the vector equality

Momentum material point relative to some axis z is a scalar quantity taken with the sign (+) or (-) and equal to the product of the modulus projection vector momentum per plane perpendicular to this axis perpendicular h, lowered from the point of intersection of the axis with the plane to the line along which the indicated projection is directed:

Kinetic moment of a mechanical system relative to the center and axis

1. Momentum relative to the center.

Kinetic moment or the main moment of the quantities of motion of a mechanical system relative to some center is called the geometric sum of the moments of momentum of all material points of the system relative to the same center.

2. Kinetic moment about the axis.

The kinetic moment or principal moment of the quantities of motion of a mechanical system relative to a certain axis is the algebraic sum of the moments of the quantities of motion of all material points of the system relative to the same axis.

3. Kinetic moment of a rigid body rotating around a fixed axis z with angular velocity.

Theorem on the change in angular momentum of a material point relative to the center and axis

1. Theorem of moments about the center.

Derivative in time from the moment of momentum of a material point relative to some fixed center is equal to the moment of force acting on the point relative to the same center

2. Theorem of moments about an axis.

Derivative in time from the moment of momentum of a material point relative to a certain axis is equal to the moment of force acting on the point relative to the same axis

Theorem on the change in the angular momentum of a mechanical system relative to the center and axis

Theorem of moments about the center.

Derivative in time from the kinetic moment of a mechanical system relative to some fixed center is equal to the geometric sum of the moments of all external forces acting on the system relative to the same center;

Consequence. If the main moment of external forces relative to some center is equal to zero, then the kinetic moment of the system relative to this center does not change (the law of conservation of kinetic moment).

2. Theorem of moments about an axis.

Derivative in time from the kinetic moment of a mechanical system relative to some fixed axis is equal to the sum of the moments of all external forces acting on the system relative to this axis

Consequence. If the main moment of external forces relative to a certain axis is zero, then the kinetic moment of the system relative to this axis does not change.

For example, = 0, then L z = const.

Work and power of forces

Work of force-- scalar measure of the action of force.

1. Elementary work of force.

Elementary the work of a force is an infinitesimal scalar quantity equal to the scalar product of the force vector and the vector of infinite small displacement of the point of application of the force: ; - radius vector increment the point of application of force, the hodograph of which is the trajectory of this point. Elementary movement points along the trajectory coincides with due to their small size. That's why

if then dA > 0;if, then dA = 0;if , That dA< 0.

2. Analytical expression of elementary work.

Let's imagine the vectors And d through their projections on the Cartesian coordinate axes:

, . We get (4.40)

3. The work of a force on a final displacement is equal to the integral sum of the elementary works on this displacement

If the force is constant and the point of its application moves linearly,

4. Work of gravity. We use the formula: Fx = Fy = 0; Fz = -G = -mg;

Where h- moving the point of application of force vertically downwards (height).

When moving the point of application of gravity upward A 12 = -mgh(dot M 1 -- down, M 2 - at the top).

So, . The work done by gravity does not depend on the shape of the trajectory. When moving along a closed path ( M 2 matches M 1 ) work is zero.

5. The work of the elastic force of the spring.

The spring stretches only along its axis X:

F y = F z = ABOUT, F x = = -сх;

where is the magnitude of the spring deformation.

When the point of application of force moves from the lower position to the upper position, the direction of force and the direction of movement coincide, then

Therefore, the work of the elastic force

Work of forces on final displacement; If = const, then

where is the final angle of rotation; , Where p -- the number of revolutions of a body around an axis.

Kinetic energy of a material point and a mechanical system. Koenig's theorem

Kinetic energy- scalar measure of mechanical motion.

Kinetic energy of a material point - a scalar positive quantity equal to half the product of the mass of a point and the square of its speed,

Kinetic energy of a mechanical system -- the arithmetic sum of the kinetic energies of all material points of this system:

Kinetic energy of a system consisting of n interconnected bodies is equal to the arithmetic sum of the kinetic energies of all bodies of this system:

Koenig's theorem

Kinetic energy of a mechanical system in the general case of its motion is equal to the sum of the kinetic energy of motion of the system together with the center of mass and the kinetic energy of the system when it moves relative to the center of mass:

Where Vkc -- speed k- th points of the system relative to the center of mass.

Kinetic energy of a rigid body under various motions

Forward movement.

Rotation of a body around a fixed axis . ,Where -- moment of inertia of a body relative to the axis of rotation.

3. Plane-parallel motion. , where is the moment of inertia of a flat figure relative to an axis passing through the center of mass.

When moving flat body kinetic energy consists of the kinetic energy of the translational motion of the body with the speed of the center of mass and kinetic energy of rotational motion around an axis passing through the center of mass, ;

Theorem on the change in kinetic energy of a material point

The theorem in differential form.

Differential from the kinetic energy of a material point is equal to the elementary work of the force acting on the point,

The theorem in integral (finite) form.

Change kinetic energy of a material point at a certain displacement is equal to the work of the force acting on the point at the same displacement.

Theorem on the change in kinetic energy of a mechanical system

The theorem in differential form.

Differential from the kinetic energy of a mechanical system is equal to the sum of the elementary works of external and internal forces acting on the system.

The theorem in integral (finite) form.

Change kinetic energy of a mechanical system at a certain displacement is equal to the sum of the work of external and internal forces applied to the system at the same displacement. ; For a system of solid bodies = 0 (according to the property of internal forces). Then

Law of conservation of mechanical energy of a material point and mechanical system

If for material point or mechanical system, only conservative forces act, then in any position of the point or system the sum of kinetic and potential energies remains constant.

For a material point

For mechanical system T+ P= const

Where T+ P -- total mechanical energy of the system.

Rigid body dynamics

Differential equations of motion of a rigid body

These equations can be obtained from general theorems of the dynamics of a mechanical system.

1. Equations of translational motion of a body - from the theorem on the movement of the center of mass of a mechanical system In projections on the axes of Cartesian coordinates

2. The equation for the rotation of a rigid body around a fixed axis - from the theorem on the change in the kinetic moment of a mechanical system relative to an axis, for example, relative to an axis

Since the kinetic moment L z rigid body relative to the axis, then if

Since or, the equation can be written as or, the form of writing the equation depends on what needs to be determined in a particular problem.

Differential equations of plane-parallel the motions of a rigid body are a set of equations progressive movement of a flat figure together with the center of mass and rotational movement relative to an axis passing through the center of mass:

Physical pendulum

Physical pendulum is a rigid body that rotates around a horizontal axis that does not pass through the center of mass of the body and moves under the influence of gravity.

Differential equation of rotation

In case of small fluctuations.

Then where

Solution of this homogeneous equation.

Let at t=0 Then

-- equation of harmonic vibrations.

Period of oscillation of a pendulum

Given length of a physical pendulum is the length of a mathematical pendulum whose period of oscillation is equal to the period of oscillation of a physical pendulum.

angular momentum

MOMENTUM OF MOTION (kinetic momentum, angular momentum, angular momentum) is a measure of the mechanical motion of a body or system of bodies relative to some center (point) or axis. To calculate the moment of momentum K of a material point (body), the same formulas are valid as for calculating the moment of force, if you replace the force vector in them with the vector of momentum mv, in particular K0 = . The sum of the angular momentum of all points of the system relative to the center (axis) is called the principal angular momentum of the system (kinetic moment) relative to this center (axis). In the rotational motion of a rigid body, the main angular momentum relative to the axis of rotation z of the body is expressed by the product of the moment of inertia Iz and the angular velocity? bodies, i.e. КZ = Iz?.

Momentum

kinetic moment, one of the measures of mechanical motion of a material point or system. Mechanical motion plays a particularly important role in the study of rotational motion. As with the moment of force, a distinction is made between mechanical action relative to the center (point) and relative to the axis.

To calculate the mechanical efficiency k of a material point relative to the center O or the z axis, all the formulas given for calculating the moment of force are valid if the vector F is replaced by the momentum vector mv. Thus, ko = , where r ≈ radius vector of the moving point drawn from the center O, and kz is equal to the projection of the vector ko onto the z axis passing through the point O. The change in the M. efficiency of the point occurs under the influence of the moment mo (F) of the applied force and is determined by the theorem on the change in M. efficiency, expressed by the equation dko/dt = mo(F). When mo(F) = 0, which, for example, is the case for central forces, the motion of the point obeys the area law. This result is important for celestial mechanics, the theory of motion of artificial Earth satellites, spacecraft, etc.

The main mechanical coefficient (or kinetic moment) of a mechanical system relative to the center O or the z axis is equal, respectively, to the geometric or algebraic sum of the mechanical coefficient of all points of the system relative to the same center or axis, i.e. Ko = Skoi, Kz = Skzi. Vector Ko can be determined by its projections Kx, Ky, Kz onto the coordinate axes. For a body rotating around a stationary axis z with angular velocity w, Kx = ≈ Ixzw, Ky = ≈Iyzw, Kz = Izw, where lz ≈ axial, and Ixz, lyz ≈ centrifugal moments of inertia. If the z axis is the main axis of inertia for the origin O, then Ko = Izw.

A change in the main M. efficiency of the system occurs under the influence of only external forces and depends on their main moment Moe. This dependence is determined by the theorem on the change in the main M. efficiency of the system, expressed by the equation dKo/dt = Moe. A similar equation relates the moments Kz and Mze. If Moe = 0 or Mze = 0, then, respectively, Ko or Kz will be constant quantities, i.e., the law of conservation of mechanical efficiency holds (see Conservation laws). Thus, internal forces cannot change the efficiency of the system, but the efficiency of individual parts of the system or angular velocities under the influence of these forces can change. For example, for a figure skater (or ballerina) rotating around the vertical axis z, the value Kz = Izw will be constant, since practically Mze = 0. But by changing the value of the moment of inertia lz with the movement of his arms or legs, he can change the angular velocity w. Dr. An example of the fulfillment of the law of conservation of mechanical efficiency is the appearance of a reactive torque in an engine with a rotating shaft (rotor). The concept of mechanical dynamics is widely used in rigid body dynamics, especially in the theory of the gyroscope.

Dimension of M. k.d. ≈ L2MT-1, units of measurement ≈ kg×m2/sec, g×cm2/sec. MKDs also have electromagnetic, gravitational, and other physical fields. Most elementary particles have their own, internal magnetodynamic efficiency ≈ spin. MQD is of great importance in quantum mechanics.

Lit. see under art. Mechanics.

Kinetic moment of a point and a mechanical system

Rice. 3.14

One of the dynamic characteristics of the motion of a material point and a mechanical system is the kinetic moment or angular momentum.

For a material point, the angular momentum relative to any center O is the angular momentum of the point relative to this center (Fig. 3.14),

The kinetic moment of a material point relative to an axis is the projection onto this axis of the kinetic moment of the point relative to any center on this axis:

The kinetic moment of a mechanical system relative to the center O is the geometric sum of the kinetic moments of all points of the system relative to the same center (Fig. 3.15):

(3.20)

The kinetic moment is applied to the point ABOUT, relative to which it is calculated.

If we project (3.20) onto the axes of the Cartesian coordinate system, we obtain projections of the kinetic moment onto these axes, or kinetic moments relative to the coordinate axes:

Let us determine the kinetic moment of the body relative to its fixed axis of rotation z(Fig. 3.16).

According to formulas (3.21), we have

But when the body rotates with angular velocity w, the speed and the amount of motion of the point perpendicular to the segment dk and lies in the plane perpendicular to the axis of rotation Oz, hence,

Rice. 3.15

Rice. 3.16

For the whole body:

Where Jz– moment of inertia relative to the axis of rotation.

Consequently, the angular momentum of a rigid body relative to the axis of rotation is equal to the product of the moment of inertia of the body relative to a given axis and the angular velocity of the body.

2. Theorem on the change in angular momentum
mechanical system

Kinetic moment of the system relative to the stationary center O(Fig. 3.15)

Let us take the derivative with respect to time from the left and right sides of this equality:

(3.22)

Let's take into account that then expression (3.22) will take the form

Or, given that

– the sum of the moments of external forces relative to the center O, we finally have:

(3.23)

Equality (3.23) expresses the theorem about the change in angular momentum.

Theorem on the change in angular momentum. The time derivative of the kinetic moment of a mechanical system relative to a fixed center is equal to the principal moment of the external forces of the system relative to the same center.

Having projected equality (3.23) onto the fixed axes of Cartesian coordinates, we obtain a representation of the theorem in projections onto these axes:

From (3.23) it follows that if the main moment of external forces relative to any fixed center is zero, then the kinetic moment relative to this center remains constant, i.e. If

(3.24)

If the sum of the moments of the external forces of the system relative to any fixed axis is zero, then the corresponding projection of the kinetic moment remains constant,

(3.25)

Statements (3.24) and (3.25) represent the law of conservation of angular momentum of the system.

Let us obtain a theorem about the change in the kinetic moment of the system by choosing the point as a point when calculating the kinetic moment A, moving relative to the inertial reference frame with speed

Kinetic moment of the system relative to the point A(Fig. 3.17)

Rice. 3.17

because That

Considering that where is the speed of the center of mass of the system, we obtain

Let's calculate the time derivative of the angular momentum

In the resulting expression:

Combining the second and third terms, and considering that

finally we get

If the point coincides with the center of mass of the system C, That and the theorem takes the form

those. it has the same shape as for a fixed point ABOUT.

3. Differential equation of rotation of a rigid body
around a fixed axis

Let a rigid body rotate around a fixed axis Az(Fig. 3.18) under the influence of a system of external forces
Let us write the equation of the theorem on the change in the angular momentum of the system in projection onto the axis of rotation:

Rice. 3.18

For the case of rotation of a rigid body around a fixed axis:

Where Jz– constant moment of inertia relative to the axis of rotation; w – angular velocity.

Taking this into account, we get:

If we introduce the angle of rotation of the body j, then, taking into account the equality we have

(3.26)

Expression (3.26) is the differential equation for the rotation of a rigid body around a fixed axis.

4. Theorem on the change in the angular momentum of the system
in relative motion relative to the center of mass

To study a mechanical system, we choose a fixed coordinate system Ox 1 y 1 z 1 and movable Cxyz with origin at the center of mass C, moving forward (Fig. 3.19).

From a vector triangle:

Rice. 3.19

Differentiating this equality with respect to time, we obtain

or

where is the absolute speed of the point Mk, - absolute speed of the center of mass WITH,
- relative speed of the point Mk, because

Momentum about a point ABOUT

Substituting the values ​​and , we get

In this expression: – mass of the system; ;

– kinetic moment of the system relative to the center of mass for relative motion in the coordinate system Сxyz.

The kinetic moment takes the form

Theorem on the change in angular momentum relative to a point ABOUT looks like

Let's substitute the values ​​and we get

Let us transform this expression taking into account that

or

This formula expresses the theorem on the change in the angular momentum of a system relative to the center of mass for the relative motion of the system with respect to a coordinate system moving translationally with the center of mass. It is formulated in the same way as if the center of mass were a fixed point.

Momentum of a material point(kinetic moment) relative to a selected point in space is the result of the vector product of a vector drawn from the selected point to any point on the line of action of the force by the vector of momentum of the material point:

Momentum of a mechanical system(kinetic moment of the system) relative to a selected point in space is the sum of the angular momentum of all material points of the system relative to the same point:

Let us limit ourselves to considering only plane problems. In this case, similar to the moment of force, we can assume that the moment of momentum of a point is a scalar quantity and is equal to:

Where v i– module of the point velocity vector;

h i-shoulder.

The sign of the moment of momentum is chosen in the same way as the sign of the moment of force.

Theorem: The angular momentum of a translationally moving body is equal to the product of the mass of the body and the speed of any point on the body and the leverage of the velocity of the center of mass relative to the selected point:

Where h c– arm of the velocity of the center of mass of the system relative to the selected point.

Theorem: The moment of momentum of a rotating body is equal to the product of the moment of inertia of the body relative to the axis of rotation and the angular velocity:

where is the distance from the point in question to the axis of rotation.

Theorem: the angular momentum of a body moving plane-parallel is equal to the sum of the angular momentum of the center of mass of the body relative to the selected point and the product of the body’s own moment of inertia and the angular velocity:

Elementary impulse– is the product of the moment of force and the elementary time interval of the force action

1.3.11. The principle of possible movements

Possible relocation- this is any infinitesimal movement of an arbitrary point of the body, which is allowed by the connections imposed on the body without changing the connection itself.

Perfect connection is a connection in which the sum of the possible work of all its reactions on all possible movements of the system is equal to zero.

All connections that were considered before, excluding the rough surface, are ideal.

Active power– any force acting in a system, excluding reaction forces. From the definition of ideal connections it follows that the work of reactive forces in the case of a system with ideal connections is always equal to zero.

Number of degrees of freedom of the system is the number of linearly independent possible generalized movements of the system. Independent movements can be selected arbitrarily. So a flat body resting on a plane (Fig. 1.52) has many possible movements (right, left, up at an angle), but linearly independent

Only three (for example, horizontal offset dx, vertical displacement up dy and the angle of rotation around the point A - dj).

It is customary to denote possible movements with the symbol “ δ ” before moving. It is necessary to distinguish possible movements from actual ones. There may be many possible ones, but only one actual one. Actual movement is necessarily included among the possible ones.