The moment of momentum of a material point relative to the center and axis. What does moment of momentum mean?

In some problems, instead of the momentum itself, its moment relative to some center or axis is considered as a dynamic characteristic of a moving point. These moments are defined in the same way as the moments of force.

Moment of momentum material point with respect to some center O is called a vector defined by the equality

The angular momentum of a point is also called angular momentum .

Moment of momentum relative to any axis, passing through the center O, is equal to the projection of the momentum vector on this axis.

If the momentum is given by its projections on the coordinate axis and the coordinates of a point in space are given, then the moment of momentum relative to the origin is calculated as follows:

The projections of the angular momentum on the coordinate axes are:

The SI unit of momentum is -.

End of work -

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Dynamics

Lecture.. summary introduction to the dynamics of the axioms of classical mechanics.. introduction..

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Unit systems
CGS Si Technical [L] cm m m [M]

Differential equations of motion of a point
The basic equation of dynamics can be written as

Basic tasks of dynamics
The first or direct task: The mass of the point and the law of its motion are known, it is necessary to find the force acting on the point. m

The most important cases
1. Strength is constant.

Number of point movement
The amount of motion of a material point is a vector equal to the product m

Elemental and full force impulse
The action of a force on a material point over time

Theorem on the change in momentum of a point
Theorem. The time derivative of the momentum of a point is equal to the force acting on the point. Let's write down the basic law of dynamics

Theorem on the change in the angular momentum of a point
Theorem. The time derivative of the angular momentum of a point, taken with respect to some center, is equal to the moment of the force acting on the point with respect to the same

Force work. Power
One of the main characteristics of a force that evaluates the effect of a force on a body during some movement.

Theorem on the change in the kinetic energy of a point
Theorem. Differential kinetic energy point is equal to the elementary work of the force acting on the point.

d'Alembert's principle for a material point
The equation of motion of a material point relative to an inertial frame of reference under the action of applied active forces and reaction forces of constraints has the form:

Dynamics of a non-free material point
A non-free material point is a point whose freedom of movement is limited. The bodies that limit the freedom of movement of a point are called bonds.

Relative motion of a material point
In many problems of dynamics, the motion of a material point is considered relative to a frame of reference moving relative to an inertial frame of reference.

Special cases of relative motion
1. Relative motion by inertia If a material point moves relative to a moving frame of reference in a straight line and uniformly, then such a movement is called relative

Mass geometry
Consider a mechanical system that consists of a finite number of material points with masses

Moments of inertia
To characterize the distribution of masses in bodies when considering rotational motions, it is necessary to introduce the concepts of moments of inertia. Moment of inertia about a point

Moments of inertia of the simplest bodies
1. Uniform rod 2. Rectangular plate 3. Uniform round disc

Quantity of movement system
The amount of motion of a system of material points is the vector sum of quantities

Theorem on the change in the momentum of the system
This theorem exists in three different forms. Theorem. The time derivative of the momentum of the system is equal to the vector sum of all external forces acting n

Laws of conservation of momentum
1. If the main vector of all external forces of the system is zero (), then the momentum of the system is constant

Theorem on the motion of the center of mass
Theorem The center of mass of the system moves in the same way as a material point, the mass of which is equal to the mass of the entire system, if all external forces applied to the considered point act on the point.

moment of momentum of the system
The moment of momentum of the system of material points relative to some

The angular momentum of a rigid body relative to the axis of rotation during rotational motion of a rigid body
Let us calculate the angular momentum of a rigid body relative to the axis of rotation.

Theorem on the change in the angular momentum of the system
Theorem. The time derivative of the angular momentum of the system, taken relative to some center, is equal to the vector sum of the moments of external forces acting on

Laws of conservation of angular momentum
1. If the main moment of the external forces of the system relative to the point is equal to zero (

Kinetic energy of the system
The kinetic energy of a system is the sum of the kinetic energies of all points in the system.

Kinetic energy of a rigid body
1. Translational movement of the body. The kinetic energy of a rigid body during translational motion is calculated in the same way as for a single point whose mass is equal to the mass of this body.

Theorem on the change in the kinetic energy of the system
This theorem exists in two forms. Theorem. The differential of the kinetic energy of the system is equal to the sum of the elementary works of all external and internal forces acting on the system

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

Vector angular momentum of a material point relative to some center O -- a vector applied at this center and directed perpendicular to the plane of the vectors m and in the direction from which the movement of the point can be seen counterclockwise. This definition satisfies the vector equality

moment of momentum material point about some axis z is called a scalar value taken with a sign (+) or (-) and equal to the product of the modulus vector projections amount of motion to a plane perpendicular to this axis, to a perpendicular h, lowered from the point of intersection of the axis with the plane to the line along which the indicated projection is directed:

momentum mechanical system relative to the center and axis

1. Kinetic moment relative to the center.

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

2. Kinetic moment about the axis.

The angular momentum or the main moment of the quantities of motion of a mechanical system relative to some 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. Momentum of a rigid body rotating around a fixed axis z with angular velocity.

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

1. Theorem of moments with respect to 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. The theorem of moments about the axis.

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

Theorem on the change in the kinetic moment of a mechanical system relative to the center and axis

Theorem of moments about the center.

Derivative in time from the angular momentum of the mechanical system relative to some fixed center is equal to geometric sum 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 angular momentum of the system relative to this center does not change (the law of conservation of angular momentum).

2. The theorem of moments about the axis.

Derivative in time from the angular momentum 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 about some axis is equal to zero, then the kinetic moment of the system about this axis does not change.

For example = 0, then L z = const.

Work and power of forces

Force work is a scalar measure of the action of a force.

1. Elementary work of force.

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

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

2. Analytic expression for elementary work.

Imagine vectors and d through their projections on the axes Cartesian coordinates:

, . Get (4.40)

3. The work of the force on the 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 in a straight line,

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

where h- moving the point of application of force vertically down (height).

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

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

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

The spring stretches only along the axis X:

F y = F z = O, F x = = -sh;

where is the value of the spring deformation.

When moving the point of application of force from the lower position to the upper position, the direction of force and the direction of movement are the same, then

Therefore, the work of the elastic force

The work of forces on the final displacement; If = const, then

where is the final angle of rotation; , where P -- the number of revolutions of the body around the axis.

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

Kinetic energy- scalar measure mechanical movement.

Kinetic energy of a material point - a scalar positive value 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:

The kinetic energy of a system consisting of P interconnected bodies is equal to arithmetic sum kinetic energies of all bodies of this system:

König'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 the system motion together with the center of mass and the kinetic energy of the system as 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 in various motions

Progressive movement.

Rotation of a body around a fixed axis . ,where -- the moment of inertia of the body about the axis of rotation.

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

With flat motion body kinetic energy is the sum of the kinetic energy forward movement bodies with the speed of the center of mass and kinetic energy rotary motion around an axis passing through the center of mass, ;

Theorem on the change in the kinetic energy of a material point

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,

Theorem in integral (finite) form.

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

Theorem on the change in the kinetic energy of a mechanical system

Theorem in differential form.

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

Theorem in integral (finite) form.

Change The kinetic energy of a mechanical system at some 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 rigid bodies = 0 (according to the property of internal forces). Then

moment of momentum

MOMENTUM AMOUNT (kinetic moment, angular momentum, angular momentum) is a measure of the mechanical movement of a body or system of bodies relative to any 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 we replace the force vector in them with the vector of momentum mv, in particular, K0 = . The sum of the moments of the momentum of all points of the system about the center (axis) is called the main moment of the momentum of the system (kinetic moment) about this center (axis). During the rotational motion of a rigid body, the main moment of momentum about the axis of rotation z of the body is expressed by the product of the moment of inertia Iz by angular velocity? body, i.e. KZ = Iz?.

Moment of momentum

kinetic moment, one of the measures of the mechanical motion of a material point or system. A particularly important role is played by M. k. d. in the study of rotational motion. As for the moment of force, M. c.d. is distinguished relative to the center (point) and relative to the axis.

To calculate the M. f. 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 we replace the vector F in them with the momentum vector mv. Thus, ko = , where r ≈ the 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. c. f. point occurs under the action of the moment mo (F) of the applied force and is determined by the theorem on the change of M. c. d., expressed by the equation dko / dt = mo (F). When mo(F) = 0, which, for example, takes place for central forces, the motion of a point obeys the area law. This result is important for celestial mechanics, the theory of motion artificial satellites Earth, space aircraft and etc.

The main M. c. d. (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 M. c. d. of all points of the system with respect to the same center or axis, i.e. Ko = Skoi, Kz = Skzi. The vector Ko can be defined by its projections Kx, Ky, Kz onto the coordinate axes. For a body rotating around a fixed axis z with an 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 principal axis of inertia for the origin O, then Ko = Izw.

The change in the main magnetic force of the system occurs under the action of only external forces and depends on their main moment Moe. This dependence is determined by the theorem on the change of the main M. c. d. of the system, expressed by the equation dKo / dt = Moe. The moments Kz and Mze are related by a similar equation. If Moe = 0 or Mze = 0, then respectively Ko or Kz will be constant values, i.e., the conservation law of M. c.d. holds (see Conservation laws). That., internal forces They cannot change the MCF of the system, but the MCF of individual parts of the system or the angular velocities can change under the action of these forces. 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 by moving his arms or legs, he can change the angular velocity w. Dr. An example of the fulfillment of the law of conservation of M. c.d. is the appearance of a reactive torque in an engine with a rotating shaft (rotor). The concept of M. to. d. is widely used in dynamics solid body, especially in gyroscope theory.

Dimension M. c. d. ≈ L2MT-1, units of measurement ≈ kg × m2 / s, g × cm2 / s. Electromagnetic, gravitational, and other physical fields also have magnetic fields. Most elementary particles own, internal M. c. d. ≈ spin is inherent. Great importance M. to. d. has in quantum mechanics.

Lit. see at Art. Mechanics.

Problems with solutions

28.1 railway train moves horizontally and straight section way. When braking, a resistance force equal to 0.1 of the train weight develops. At the start of braking, the speed of the train is 20 m/s. Find the braking time and stopping distance.
SOLUTION

28.2 On a rough inclined plane, making an angle α = 30 ° with the horizon, descends heavy body without initial speed. Determine during what time T the body will pass the way length l=39.2 m, if the coefficient of friction f=0.2.
SOLUTION

28.3 A train of mass 4*10^5 kg enters the rise i=tg α=0.006 (where α is the angle of rise) at a speed of 15 m/s. The coefficient of friction (coefficient of total resistance) when the train is moving is 0.005. 50 s after the train enters the rise, its speed drops to 12.5 m/s. Find the traction force of the locomotive.
SOLUTION

28.4 A weight M is tied to the end of an inextensible string MOA, part of which OA is passed through a vertical tube; the weight moves around the axis of the tube along a circle of radius MC=R, making 120 rpm. Slowly pulling the thread OA into the tube, shorten the outer part of the thread to the length OM1, at which the weight describes a circle with a radius R/2. How many revolutions per minute does the weight make along this circle?
SOLUTION

28.5 To determine the mass of a loaded train, a dynamometer was installed between the diesel locomotives and wagons. The average reading of the dynamometer for 2 minutes turned out to be 10 ^ 6 N. During the same time, the train picked up a speed of 16 m / s (at first the train stood still). Find the mass of the composition if the coefficient of friction f=0.02.
SOLUTION

28.6 What should be the coefficient of friction f of the wheels of a braked car on the road if, at a driving speed v = 20 m/s, it stops 6 s after the start of braking.
SOLUTION

28.7 A bullet of mass 20 g flies out of the barrel of a rifle with a speed of v=650 m/s, running through the bore in the time t=0.00095 s. Determine the average pressure of the gases ejecting the bullet if the cross-sectional area of ​​the channel is σ=150 mm^2.
SOLUTION

28.8 The point M moves around a fixed center under the influence of the force of attraction to this center. Find the velocity v2 at the point of the trajectory furthest from the center, if the velocity of the point at the position closest to it is v1=30 cm/s, and r2 is five times greater than r1.
SOLUTION

28.9 Find the momentum of the resultant of all forces acting on the projectile during the time when the projectile moves from the initial position O to the highest position M. Given: v0=500 m/s; α0=60°; v1=200 m/s; projectile weight 100 kg.
SOLUTION

28.10 Two asteroids M1 and M2 describe the same ellipse, in the focus of which S is the Sun. The distance between them is so small that the arc M1M2 of the ellipse can be considered a straight line segment. It is known that the length of the M1M2 arc was a when its middle was at perihelion P. Assuming that the asteroids move with equal sectorial velocities, determine the length of the M1M2 arc when its middle passes through aphelion A, if it is known that SP=R1 and SA =R2.
SOLUTION

28.11 A boy of mass 40 kg stands on the runners of a sports sleigh, the mass of which is 20 kg, and every second he pushes with an impulse of 20 N * s. Find the speed acquired by the sleigh in 15 s if the coefficient of friction f=0.01.
SOLUTION

28.12 The point makes a uniform movement along a circle with a speed v=0.2 m/s, making a complete revolution in time T=4 s. Find the momentum S of the forces acting on the point during one half-cycle if the mass of the point is m=5 kg. Determine the average value of the force F.
SOLUTION

28.13 Two mathematical pendulums suspended on threads of lengths l1 and l2 (l1>l2) oscillate with the same amplitude. Both pendulums simultaneously began to move in the same direction from their extreme deflected positions. Find the condition that the lengths l1 and l2 must satisfy in order for the pendulums to simultaneously return to the equilibrium position after a certain period of time. Determine the smallest time interval T.
SOLUTION

28.14 A ball of mass m, tied to an inextensible thread, slides along a smooth horizontal plane; the other end of the thread is pulled at a constant speed a into a hole made on a plane. Determine the motion of the ball and the tension of the thread T, if it is known that at the initial moment the thread is located in a straight line, the distance between the ball and the hole is R, and the projection of the initial velocity of the ball onto the perpendicular to the direction of the thread is v0.
SOLUTION

28.15 Determine the mass M of the Sun, having the following data: the radius of the Earth R=6.37*106 m, the average density is 5.5 t/m3, the semi-major axis of the earth's orbit a=1.49*10^11 m, the time of the Earth's revolution around the Sun T=365.25 days Strength gravity between two masses equal to 1 kg, at a distance of 1 m, we consider equal to gR2/m H, where m is the mass of the Earth; It follows from Kepler's laws that the force of attraction of the Earth by the Sun is equal to 4π2a3m/(T2r2), where r is the distance of the Earth from the Sun.
SOLUTION

28.16 Mass point m affected by central force F, describes the lemniscate r2=a cos 2φ, where a is a constant value, r is the distance of the point from the center of force; at the initial moment r=r0, the speed of the point is equal to v0 and makes an angle α with the straight line connecting the point with the center of force. Determine the magnitude of the force F, knowing that it depends only on the distance r. According to Binet's formula, F =-(mc2/r2)(d2(1/r)/dφ2+1/r), where c is twice the sector velocity of the point.
SOLUTION

28.17 A point M, whose mass is m, moves near a fixed center O under the influence of a force F emanating from this center and depending only on the distance MO=r. Knowing that the speed of the point is v=a/r, where a is a constant value, find the magnitude of the force F and the trajectory of the point.
SOLUTION

28.18 Determine the movement of a point whose mass is 1 kg, under the action of a central force of attraction, inversely proportional to the cube of the distance of the point from the center of attraction, with the following data: at a distance of 1 m, the force is 1 N. At the initial moment, the distance of the point from the center of attraction is 2 m, velocity v0=0.5 m/s and makes an angle of 45° with the direction of the straight line drawn from the center to the point.
SOLUTION

28.19 A particle M of mass 1 kg is attracted to a fixed center O by a force inversely proportional to the fifth power of the distance. This force is equal to 8 N at a distance of 1 m. At the initial moment, the particle is at a distance of OM0=2 m and has a velocity perpendicular to OM0 and equal to 0.5 m/s. Determine the trajectory of the particle.
SOLUTION

28.20 A point of mass 0.2 kg, moving under the influence of the force of attraction to a fixed center according to Newton's law of gravitation, describes a complete ellipse with semi-axes 0.1 m and 0.08 m for 50 s. Determine the largest and smallest values ​​of the attractive force F during this movement.
SOLUTION

28.21 A mathematical pendulum, each swing of which lasts one second, is called a second pendulum and is used to measure time. Find the length l of this pendulum, assuming the acceleration due to gravity to be 981 cm/s2. What time will this pendulum on the Moon show, where the acceleration of gravity is 6 times less than the earth? What length l1 should a lunar second pendulum have?
SOLUTION

28.22 At some point on the Earth, the second pendulum counts time correctly. Being moved to another location, it lags behind by T seconds per day. Determine the acceleration due to gravity in the new position of the second pendulum.

MOMENTUM TORQUE(kinetic momentum, angular momentum, orbital momentum, angular momentum) - one of the dynamic. movement or mechanical characteristics. systems; plays a particularly important role in the study of rotation. movement. As for, they distinguish between M. c. d. relative to the center (point) and relative to the axis.

M. c. d. of a material point relative to the center O equals vector product radius-vector r point drawn from the center O, on her number of movements mv, i.e. k 0 = [r mu] or in other notation k 0 = r mu. M. k. d. kz material point about the z-axis passing through the center O, is equal to the projection of the vector k 0 for this axis. To calculate the M. c.d. of a point, all the f-ly given for the calculation are valid moment of force, if we replace the vector F (or its projections) vector mu(or its projections). The change in M. c. d. point occurs under the action of a moment m 0 (F ) applied force. The nature of this change is determined by the equation dk/dt = m 0 (F ), which is a consequence of the main law. When m 0 (F ) = 0, which, for example, takes place for the center. forces, M. c. d. points relative to the center O remains constant; the point moves along a flat curve and its radius vector describes equal areas in any equal time intervals. This result is important for celestial mechanics (cf. Kepler's laws), as well as for the theory of cosmic motion. fly. devices, satellites, etc.

For mechanical of the system, the concept of the main M. c.d. (or kinetic moment) of the system relative to the center is introduced O, equal to geom. the sum of M. c. d. of all points of the system relative to the same center:

Vector K 0 can be determined by its projections on mutually perpendicular axes Oxyz. Quantities K x , K y , K z, are at the same time the main M. c.d. of the system with respect to the corresponding axes. For a body rotating about a fixed axis z from ang. speed w, these quantities are: K x = -I xz w, K y \u003d -I yz w, Kz = Iz w, where Iz- axial, a I xz and I yz- centrifugal. If the body moves around a fixed point O, then for it in projections on the main axes of inertia drawn at the point O, will be K x =- I x w x , K y = 1 y w y, Kz = Iz w z, where I x , 1 y, I z- moments of inertia relative to Ch. axes; w x, w y, w z- projection of the instantaneous angle. speed w on these axes. From f-l is visible that the direction of the vector K 0 is the same direction w only when the body rotates around one of its chapters. (for point O) axes of inertia. In this case K 0 = Iw, where I- the moment of inertia of the body relative to this Ch. axes.

The change in the main M. to. d. of the system occurs only as a result of external. influences and depends on Ch. moment M e 0 external forces; this dependence is determined by the equation d K 0 /dt= M e 0 (equation of moments). In contrast to the case of the movement of a single point, the equation of moments for the system is not a consequence of the equation of the number of movements, and both of these equations can be used to study the movement of the system at the same time. With the help of the equation of moments alone, the motion of a system (body) can be completely determined only in the case of a purely rotation. movement (around a fixed axis or point). If Ch. moment ext. forces relative to - n. center or axis is equal to zero, then the main M. c.d. of the system with respect to this center or axis remains constant, i.e., the law of conservation of M. c.d. takes place (see.