Basic equation for the dynamics of rotating motion. Equation for the dynamics of rotational motion of a rigid body around a fixed axis

A moment of power F relative to a fixed point O is a physical quantity determined by the vector product of the radius vector r drawn from point O to point A of application of force and forceF (Fig. 25):

M = [ rF ].

HereM - pseudo-vector, its direction coincides with the direction of translational movement of the right propeller when it rotates fromG ToF .

Modulus of moment of force

M = Frsin = Fl, (18.1)

Where - angle betweenG AndF ; rsin = l- the shortest distance between the line of action of the force and point O -shoulder of strength.

Moment of force about a fixed axis zcalled the scalar quantity M z , equal to the projection onto this axis of the vector aM moment of force determined relative to an arbitrary point O of a given axis 2 (Fig. 26). Moment value M z does not depend on the choice of the position of point O on the axisz.

Equation (18.3) isequation of dynamics of rotational motion of a rigid body relative to a fixed axis.

14. Center of mass of a system of material points.

In Galileo-Newtonian mechanics, due to the independence of mass from velocity, the momentum of a system can be expressed in terms of the velocity of its center of mass.Center of mass (orcenter of inertia) system of material points is called an imaginary point C, the position of which characterizes the distribution of the mass of this system. Its radius vector is equal to

Wherem i Andr i - mass and radius vector, respectivelyith material point;n- number of material points in the system;

- mass of the system.

Center of mass speed

Considering thatp i = m i v i , A

there is momentumR systems, you can write

p = mv c , (9.2)

that is, the momentum of the system is equal to the product of the mass of the system and the speed of its center of mass.

Substituting expression (9.2) into equation (9.1), we obtain

mdv c / dt= F 1 + F 2 +...+ F n , (9.3)

that is, the center of mass of the system moves as a material point in which the mass of the entire system is concentrated and on which a force acts equal to the geometric sum of all external forces acting on the system. Expression (9.3) islaw of motion of the center of mass.

In accordance with (9.2), it follows from the law of conservation of momentum that the center of mass of a closed system either moves rectilinearly and uniformly or remains motionless

2) Trajectory of movement. Distance traveled. Kinematic law of motion.

Trajectory movement of a material point - a line described by this point in space. Depending on the shape of the trajectory, the movement can be rectilinear or curved.

Let's consider the movement of a material point along an arbitrary trajectory (Fig. 2). We will start counting time from the moment when the point was in position A. The length of the section of trajectory AB traversed by the material point since the start of counting time is calledpath length Asand is a scalar function of time: s =  s(t). Vector r= r- r 0 , drawn from the initial position of the moving point to its position in. a given point in time (increment of the radius vector of a point over the period of time under consideration) is calledmoving.

During rectilinear movement, the displacement vector coincides with the corresponding section of the trajectory and the displacement module | r| equal to the distance traveled s.

Questions for the exam in physics (I semester)

1. Movement. Types of movements. Description of the movement. Reference system.

2. Trajectory of movement. Distance traveled. Kinematic law of motion.

3. Speed. Average speed. Velocity projections.

4. Acceleration. The concept of normal and tangential acceleration.

5. Rotational movement. Angular velocity and angular acceleration.

6. Centripetal acceleration.

7. Inertial reference systems. Newton's first law.

8. Strength. Newton's second law.

9. Newton's third law.

10.Types of interactions. Interaction carrier particles.

11.Field concept of interactions.

12. Gravitational forces. Gravity. Body weight.

13. Friction forces and elastic forces.

14. Center of mass of a system of material points.

15. Law of conservation of momentum.

16. Moment of force relative to a point and an axis.

17. Moment of inertia of a rigid body. Steiner's theorem.

18. Basic equation for the dynamics of rotational motion.

19. Momentum. Law of conservation of angular momentum.

20. Work. Calculation of work. Work of elastic forces.

21. Power. Power calculation.

22. Potential field of forces. Conservative and non-conservative forces.

23. The work of conservative forces.

24. Energy. Types of energy.

25. Kinetic energy of the body.

26. Potential energy of the body.

27. Total mechanical energy of a system of bodies.

28. Relationship between potential energy and force.

29. Conditions for equilibrium of a mechanical system.

30. Collision of bodies. Types of collisions.

31. Conservation laws for various types of collisions.

32. Current lines and tubes. Continuity of the stream. 3 3. Bernoulli's equation.

34. Forces of internal friction. Viscosity.

35. Oscillatory motion. Types of vibrations.

36. Harmonic vibrations. Definition, equation, examples.

37. Self-oscillations. Definition, examples.

38. Forced vibrations. Definition, examples. Resonance.

39. Internal energy of the system.

40. The first law of thermodynamics. Work done by a body when volume changes.

41. Temperature. Equation of state of an ideal gas.

42. Internal energy and heat capacity of an ideal gas.

43. Adiabatic equation for an ideal gas.

44. Polytropic processes.

45. Van der Waals gas.

46. ​​Gas pressure on the wall. Average energy of molecules.

47.Maxwell distribution.

48. Boltzmann distribution.

Topic 3. Elements of solid body mechanics.

Lecture No. 5.

Kinematic relations

Determination of moment of force.

Moment of inertia, moment of momentum of a rigid body.

Kinematic relations.

A solid body can be considered as a system of material points rigidly fastened to each other. The nature of its movement may be different.

Mainly distinguish translational and rotational movements .

At progressive In motion, all points of the body move along parallel trajectories, so to describe the motion of the body as a whole, it is enough to know the law of motion of one point. In particular, the center of mass of a rigid body can serve as such a point

At rotational(more complex!) In motion, all points of the body describe concentric circles, the centers of which lie on the same axis. The velocities of points on any circle are related to the radii of these circles and the angular velocity
rotation: . Since a rigid body retains its shape during rotation, the radii of rotation remain constant and the linear acceleration will be equal to:

. (1)

Determination of moment of force.

To describe the dynamics of the rotational motion of a rigid body, it is necessary to introduce the concepts of moments of force.

Definition 1.

moment - strength – , applied to a material point T. A, relative to an arbitrary point T. ABOUT , drawn from the point T. ABOUT to the point T. A:

Note.

The modulus of the vector product, that is, the actual magnitude of the moment, is determined by the product - , and the direction of the moment is given by the definition of the right triple of vectors.

Definition 2.

moment– strength – , applied at point t.A, relative to an arbitrary axis is called the cross product of the radius vector and force component , lying in a plane perpendicular to the axis and passing through the point T. A:

.

Basic equation for the dynamics of rotational motion.

Let there be a rigid body of arbitrary shape that can rotate around an axis OO. Breaking the body into small elements, you can see that they all rotate around an axis OO in planes perpendicular to the axis of rotation with the same angular velocity w.

The movement of each of the individual elements of small mass m i described by Newton's second law.

For i th element we have:

Where f ik (k = 1,2, ...N) represent the internal forces of interaction of all

Items with selected and F i- the resultant of all external forces acting on i- element.

Speed v i each element, generally speaking, can change as desired, but since the body is solid, the displacement of points in the direction of the radii of rotation need not be considered. Therefore, we project equation (1) onto the direction of the tangent to the circle of rotation and multiply both sides of the equation by r i:

On the right side of the resulting equation, products of the type represent the moments of internal forces relative to the axis of rotation, since r i And f it mutually perpendicular. Similarly, the products are the moments of external forces acting on i-element.

Let us sum up in the equation of motion over all the elements into which the body was divided.

The sum of the moments of internal forces can be divided into pairs of terms, which owe their origin to the interaction of two symmetrical elements of the body with each other. Their moments are equal and oppositely directed. Based on this, we can conclude that when adding up all the moments of internal forces, they will be destroyed in pairs. Let us denote the total moment of all external forces S M i, Where M i = [ r i × F i ].

The left side of equation (2), taking into account relation (1) in the previous section, is presented as follows:

= = , (3)

where is the moment of inertia.

Equation (3) is basic equation of rotational motion.

4.Moment of inertia of a rigid body.

Definition 1.

Magnitude is called the moment of inertia of a rigid body about a given axis.

This article describes an important section of physics - “Kinematics and dynamics of rotational motion”.

Basic concepts of kinematics of rotational motion

Rotational motion of a material point around a fixed axis is called such motion, the trajectory of which is a circle located in a plane perpendicular to the axis, and its center lies on the axis of rotation.

Rotational motion of a rigid body is a motion in which all points of the body move along concentric (the centers of which lie on the same axis) circles in accordance with the rule for the rotational motion of a material point.

Let an arbitrary rigid body T rotate around the O axis, which is perpendicular to the plane of the drawing. Let us select point M on this body. When rotated, this point will describe a circle with radius around the O axis r.

After some time, the radius will rotate relative to its original position by an angle Δφ.

The direction of the right screw (clockwise) is taken as the positive direction of rotation. The change in the angle of rotation over time is called the equation of rotational motion of a rigid body:

φ = φ(t).

If φ is measured in radians (1 rad is the angle corresponding to an arc of length equal to its radius), then the length of the circular arc ΔS, which the material point M will pass in time Δt, is equal to:

ΔS = Δφr.

Basic elements of the kinematics of uniform rotational motion

A measure of the movement of a material point over a short period of time dt serves as an elementary rotation vector dφ.

The angular velocity of a material point or body is a physical quantity that is determined by the ratio of the vector of an elementary rotation to the duration of this rotation. The direction of the vector can be determined by the rule of the right screw along the O axis. In scalar form:

ω = dφ/dt.

If ω = dφ/dt = const, then such motion is called uniform rotational motion. With it, the angular velocity is determined by the formula

ω = φ/t.

According to the preliminary formula, the dimension of angular velocity

[ω] = 1 rad/s.

The uniform rotational motion of a body can be described by the period of rotation. The period of rotation T is a physical quantity that determines the time during which a body makes one full revolution around the axis of rotation ([T] = 1 s). If in the formula for angular velocity we take t = T, φ = 2 π (one full revolution of radius r), then

ω = 2π/T,

Therefore, we define the rotation period as follows:

T = 2π/ω.

The number of revolutions that a body makes per unit time is called the rotation frequency ν, which is equal to:

ν = 1/T.

Frequency units: [ν]= 1/s = 1 s -1 = 1 Hz.

Comparing the formulas for angular velocity and rotation frequency, we obtain an expression connecting these quantities:

ω = 2πν.

Basic elements of the kinematics of uneven rotational motion

The uneven rotational motion of a rigid body or material point around a fixed axis is characterized by its angular velocity, which changes with time.

Vector ε , characterizing the rate of change of angular velocity, is called the angular acceleration vector:

ε = dω/dt.

If a body rotates, accelerating, that is dω/dt > 0, the vector has a direction along the axis in the same direction as ω.

If the rotational movement is slow - dω/dt< 0 , then the vectors ε and ω are oppositely directed.

Comment. When uneven rotational motion occurs, the vector ω can change not only in magnitude, but also in direction (when the axis of rotation is rotated).

Relationship between quantities characterizing translational and rotational motion

It is known that the arc length with the angle of rotation of the radius and its value are related by the relation

ΔS = Δφ r.

Then the linear speed of a material point performing rotational motion

υ = ΔS/Δt = Δφr/Δt = ωr.

The normal acceleration of a material point that performs rotational translational motion is determined as follows:

a = υ 2 /r = ω 2 r 2 /r.

So, in scalar form

a = ω 2 r.

Tangential accelerated material point that performs rotational motion

a = ε r.

Momentum of a material point

The vector product of the radius vector of the trajectory of a material point of mass m i and its momentum is called the angular momentum of this point about the axis of rotation. The direction of the vector can be determined using the right screw rule.

Momentum of a material point ( L i) is directed perpendicular to the plane drawn through r i and υ i, and forms a right-hand triple of vectors with them (that is, when moving from the end of the vector r i To υ i the right screw will show the direction of the vector L i).

In scalar form

L = m i υ i r i sin(υ i , r i).

Considering that when moving in a circle, the radius vector and the linear velocity vector for the i-th material point are mutually perpendicular,

sin(υ i , r i) = 1.

So the angular momentum of a material point for rotational motion will take the form

L = m i υ i r i .

The moment of force that acts on the i-th material point

The vector product of the radius vector, which is drawn to the point of application of the force, and this force is called the moment of force acting on the i-th material point relative to the axis of rotation.

In scalar form

M i = r i F i sin(r i , F i).

Considering that r i sinα = l i ,M i = l i F i .

Magnitude l i, equal to the length of the perpendicular lowered from the point of rotation to the direction of action of the force, is called the arm of the force F i.

Dynamics of rotational motion

The equation for the dynamics of rotational motion is written as follows:

M = dL/dt.

The formulation of the law is as follows: the rate of change of angular momentum of a body that rotates around a fixed axis is equal to the resulting moment relative to this axis of all external forces applied to the body.

Moment of impulse and moment of inertia

It is known that for the i-th material point the angular momentum in scalar form is given by the formula

L i = m i υ i r i .

If instead of linear speed we substitute its expression through angular speed:

υ i = ωr i ,

then the expression for the angular momentum will take the form

L i = m i r i 2 ω.

Magnitude I i = m i r i 2 is called the moment of inertia relative to the axis of the i-th material point of an absolutely rigid body passing through its center of mass. Then we write the angular momentum of the material point:

L i = I i ω.

We write the angular momentum of an absolutely rigid body as the sum of the angular momentum of the material points that make up this body:

L = Iω.

Moment of force and moment of inertia

The law of rotational motion states:

M = dL/dt.

It is known that the angular momentum of a body can be represented through the moment of inertia:

L = Iω.

M = Idω/dt.

Considering that the angular acceleration is determined by the expression

ε = dω/dt,

we obtain a formula for the moment of force, represented through the moment of inertia:

M = Iε.

Comment. A moment of force is considered positive if the angular acceleration that causes it is greater than zero, and vice versa.

Steiner's theorem. Law of addition of moments of inertia

If the axis of rotation of a body does not pass through its center of mass, then relative to this axis one can find its moment of inertia using Steiner’s theorem:
I = I 0 + ma 2,

Where I 0- initial moment of inertia of the body; m- body mass; a- distance between axles.

If a system that rotates around a fixed axis consists of n bodies, then the total moment of inertia of this type of system will be equal to the sum of the moments of its components (the law of addition of moments of inertia).

The basic equation of the dynamics of rotational motion - section Mechanics, An unproven and unrefuted hypothesis is called an open problem According to Equation (5.8) Newton's Second Law of Rotational Motion...

This expression is called the basic equation of the dynamics of rotational motion and is formulated as follows: the change in the angular momentum of a rigid body is equal to the angular momentum of all external forces acting on this body.

Angular momentum (kinetic momentum, angular momentum, orbital momentum, angular momentum) characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed relative to the axis of rotation, and at what speed the rotation occurs.

Comment: angular momentum about a point is a pseudovector, and angular momentum about an axis is a scalar quantity.

It should be noted that rotation here is understood in a broad sense, not only as regular rotation around an axis. For example, even when a body moves in a straight line past an arbitrary imaginary point, it also has angular momentum. The angular momentum plays the greatest role in describing the actual rotational motion.

The angular momentum of a closed-loop system is conserved.

Law of conservation of angular momentum(law of conservation of angular momentum) - the vector sum of all angular momentum relative to any axis for a closed system remains constant in the case of equilibrium of the system. In accordance with this, the angular momentum of a closed system relative to any fixed point does not change with time.

The law of conservation of angular momentum is a manifestation of the isotropy of space.

Where does the law of conservation of angular momentum apply? Who among us does not admire the beauty of the movements of figure skaters on the ice, their rapid rotations and equally rapid transitions to slow sliding, the most complex somersaults of gymnasts or trampoline jumpers! This amazing skill is based on the same effect, which is a consequence of the law of conservation of angular momentum. By spreading his arms to the sides and moving his free leg, the skater imparts a slow rotation around the vertical axis (see Fig. 1). By sharply “grouping”, it reduces the moment of inertia and receives an increase in angular velocity.

If the axis of rotation of a body is free (for example, if the body is freely falling), then conservation of angular momentum does not mean that the direction of angular velocity is conserved in the inertial reference frame. With rare exceptions, the instantaneous axis of rotation is said to precess around the direction of the body's angular momentum. This manifests itself in the body tumbling when falling. However, bodies have so-called main axes of inertia, which coincide with the axes of symmetry of these bodies. The rotation around them is stable, the vectors of angular velocity and angular momentum coincide in direction, and no tumbling occurs.

If you carefully observe the work of a juggler, you will notice that when he throws objects, he gives them rotation. Only in this case the clubs, plates, hats are returned to his hands in the same position that was given to them. Rifled weapons provide better aiming and greater range than smooth-bore weapons. An artillery shell fired from a gun rotates around its longitudinal axis, and therefore its flight is stable.

Fig.2. Fig.3.

The well-known top, or gyroscope, behaves in the same way (Fig. 2). In mechanics, a gyroscope is any massive homogeneous body rotating around an axis of symmetry with a high angular velocity. Typically, the axis of rotation is chosen so that the moment of inertia about this axis is maximum. Then the rotation is most stable.

To create a free gyroscope in technology, a gimbal gimbal is used (Fig. 3). It consists of two annular cages that fit into one another and can rotate relative to each other. The point of intersection of all three axes 00, O"O" and O"0" coincides with the position of the center of mass of the gyroscope WITH. In such a suspension, the gyroscope can rotate around any of three mutually perpendicular axes, while the center of mass relative to the suspension will be at rest.

While the gyroscope is motionless, it can be rotated around any axis without much effort. If the gyroscope is brought into rapid rotation relative to the axis 00 and then try to rotate the gimbal, the gyroscope axis tends to keep its direction unchanged. The reason for such stability of rotation is associated with the law of conservation of angular momentum. Since the moment of external forces is small, it is not able to significantly change the angular momentum of the gyroscope. The axis of rotation of the gyroscope, with the direction of which the angular momentum vector almost coincides, does not deviate far from its position, but only trembles, remaining in place.

This property of the gyroscope has wide practical applications. A pilot, for example, always needs to know the position of the earth's true vertical in relation to the position of the aircraft at a given moment. An ordinary plumb line is not suitable for this purpose: with accelerated movement, it deviates from the vertical. Fast rotating gyroscopes on a gimbal are used. If the axis of rotation of the gyroscope is set so that it coincides with the earth's vertical, then no matter how the plane changes its position in space, the axis will maintain the vertical direction. This device is called a gyro horizon.

If the gyroscope is located in a rotating system, then its axis is set parallel to the axis of rotation of the system. In terrestrial conditions, this manifests itself in the fact that the axis of the gyroscope is eventually set parallel to the axis of rotation of the Earth, indicating the north-south direction. In marine navigation, such a gyroscopic compass is an absolutely indispensable device.

This seemingly strange behavior of the gyroscope is also in full agreement with the equation of moments and the law of conservation of angular momentum.

The law of conservation of angular momentum is, along with the laws of conservation of energy and momentum, one of the most important fundamental laws of nature and, generally speaking, is not derived from Newton’s laws. Only in the special case when we consider the circular motion of particles or material points, the totality of which forms a rigid body, is such an approach possible. Like other conservation laws, it, according to Noether's theorem, is associated with a certain type of symmetry.

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« Physics - 10th grade"

Angular acceleration.

Previously, we obtained a formula connecting the linear velocity υ, angular velocity ω and the radius R of the circle along which the selected element (material point) of an absolutely rigid body moves, which rotates about a fixed axis:

We know that linear the velocities and accelerations of the points of a rigid body are different. In the same time angular velocity is the same for all points of a rigid body.

Angular velocity is a vector quantity. The direction of angular velocity is determined by the gimlet rule. If the direction of rotation of the gimlet handle coincides with the direction of rotation of the body, then the translational movement of the gimlet indicates the direction of the angular velocity vector (Fig. 6.1).

However, uniform rotational motion is quite rare. Much more often we are dealing with movement in which the angular velocity changes, obviously this happens at the beginning and end of the movement.

The reason for the change in the angular velocity of rotation is the action of forces on the body. The change in angular velocity over time determines angular acceleration.

The angular velocity vector is a sliding vector. Regardless of the point of application, its direction indicates the direction of rotation of the body, and the module determines the speed of rotation,

The average angular acceleration is equal to the ratio of the change in angular velocity to the period of time during which this change occurred:

With uniformly accelerated motion, the angular acceleration is constant and with a stationary axis of rotation it characterizes the change in angular velocity in absolute value. When the angular velocity of rotation of a body increases, the angular acceleration is directed in the same direction as the angular velocity (Fig. 6.2, a), and when it decreases, in the opposite direction (Fig. 6.2, b).

Since the angular velocity is related to the linear velocity by the relation υ = ωR, the change in linear velocity over a certain period of time Δt is equal to Δυ =ΔωR. Dividing the left and right sides of the equation by Δt, we have either a = εR, where a - tangent(linear) acceleration, directed tangentially to the trajectory of motion (circle).

If time is measured in seconds and angular velocity is measured in radians per second, then one unit of angular acceleration is equal to 1 rad/s 2 , i.e., angular acceleration is expressed in radians per second squared.

Any rotating bodies, for example, a rotor in an electric motor, a lathe disk, a car wheel during acceleration, etc., move unevenly when starting and stopping.

Moment of power.

To create a rotational movement, not only the magnitude of the force is important, but also the point of its application. It is very difficult to open the door by applying pressure near the hinges, but at the same time you can easily open it by pressing on the door as far as possible from the axis of rotation, for example on the handle. Consequently, for rotational motion, not only the value of the force is important, but also the distance from the axis of rotation to the point of application of the force. In addition, the direction of the applied force is also important. You can pull the wheel with very great force, but still not cause it to rotate.

The moment of force is a physical quantity equal to the product of force per arm:

M = Fd,
where d is the force arm equal to the shortest distance from the axis of rotation to the line of action of the force (Fig. 6.3).

Obviously, the moment of force is maximum if the force is perpendicular to the radius vector drawn from the axis of rotation to the point of application of this force.

If several forces act on a body, then the total moment is equal to the algebraic sum of the moments of each force relative to a given axis of rotation.

In this case, the moments of forces causing the rotation of the body counterclockwise will be considered positive(force 2), and the moments of forces causing clockwise rotation are negative(forces 1 and 3) (Fig. 6.4).

Basic equation for the dynamics of rotational motion. Just as it was experimentally shown that the acceleration of a body is directly proportional to the force acting on it, it was found that angular acceleration is directly proportional to the moment of force:

Let a force act on a material point moving in a circle (Fig. 6.5). According to Newton’s second law, in projection onto the tangent direction we have ma k = F k. Multiplying the left and right sides of the equation by r, we get ma k r = F k r, or

mr 2 ε = M. (6.1)

Note that in this case, r is the shortest distance from the axis of rotation to the material point and, accordingly, the point of application of the force.

The product of the mass of a material point by the square of the distance to the axis of rotation is called moment of inertia of a material point and is designated by the letter I.

Thus, equation (6.1) can be written in the form I ε = M, whence

Equation (6.2) is called the basic equation of the dynamics of rotational motion.

Equation (6.2) is also valid for rotational motion solid, having a fixed axis of rotation, where I is the moment of inertia of the solid body, and M is the total moment of forces acting on the body. In this chapter, when calculating the total moment of forces, we consider only forces or their projections belonging to a plane perpendicular to the axis of rotation.

The angular acceleration with which a body rotates is directly proportional to the sum of the moments of forces acting on it, and inversely proportional to the moment of inertia of the body relative to a given axis of rotation.

If the system consists of a set of material points (Fig. 6.6), then the moment of inertia of this system relative to a given axis of rotation OO" is equal to the sum of the moments of inertia of each material point relative to this axis of rotation: I = m 1 r 2 1 + m 2 r 2 2 + ... .

The moment of inertia of a rigid body can be calculated by dividing the body into small volumes, which can be considered material points, and summing up their moments of inertia relative to the axis of rotation. Obviously, the moment of inertia depends on the position of the axis of rotation.

From the definition of the moment of inertia it follows that the moment of inertia characterizes the distribution of mass relative to the axis of rotation.

Let us present the values ​​of the moments of inertia for some absolutely rigid homogeneous bodies of mass m.

1. Moment of inertia of thin straight rod length l relative to the axis perpendicular to the rod and passing through its middle (Fig. 6.7) is equal to:

2. Moment of inertia straight cylinder(Fig. 6.8), or the disk relative to the axis OO", coinciding with the geometric axis of the cylinder or disk:

3. Moment of inertia ball

4. Moment of inertia thin hoop radius R relative to the axis passing through its center:

In its physical sense, the moment of inertia in rotational motion plays the role of mass, i.e., it characterizes the inertia of the body in relation to rotational motion. The greater the moment of inertia, the more difficult it is to make a body rotate or, conversely, to stop a rotating body.