Absolutely solid in terms of the resistance of materials is called a body. Material point

The easiest way to describe the movement of a body is that the relative positions of its parts do not change. Such a body is called absolutely solid.

When studying kinematics, we said that to describe the movement of a body means to describe the movement of all its points. In other words, you need to be able to find the coordinates, speed, acceleration, trajectories of all points of the body. In general, this is a difficult problem, and we will not attempt to solve it. It is especially difficult when bodies are noticeably deformed during movement.

In fact, there are no such bodies. This is a physical model. In cases where the deformations are small, real bodies can be considered as absolutely solid. However, the movement solid in general it is difficult. We will focus on the two simplest types of motion of a rigid body: translational and rotational.

Forward movement

A rigid body moves translationally if any segment of a straight line rigidly connected to the body constantly moves parallel to itself.

During translational motion, all points of the body make the same movements, describe the same trajectories, travel the same paths, and have equal speeds and accelerations. Let's show it.

Let the body move forward. Let's connect two arbitrary points A and B of the body with a straight line segment (Fig. 7.1). Line segment AB must remain parallel to itself. The distance AB does not change, since the body is absolutely rigid.

Rice. 7.1

During translational motion, the vector does not change, i.e. its magnitude and direction remain constant. As a result, the trajectories of points A and B are identical, since they can be completely combined by parallel translation to .

It is easy to see that the movements of points A and B are the same and occur in the same time. Therefore, points A and B have the same speeds. Their accelerations are also the same.

It is quite obvious that to describe the translational motion of a body it is enough to describe the movement of any one of its points, since all points move the same way. Only in this movement can we talk about the speed of the body and the acceleration of the body. With any other movement of a body, its points have different speeds and accelerations, and the terms “body speed” or “body acceleration” lose their meaning.

A desk drawer moves approximately progressively, the pistons of a car engine relative to the cylinders, cars on a straight section railway, lathe cutter relative to the bed (Fig. 7.2), etc.

Rice. 7.2

Rice. 7.3

Rotational movement

Rotational motion around a fixed axis is another type of motion of a rigid body.

Rotation of a rigid body around a fixed axis is a movement in which all points of the body describe circles, the centers of which are on the same straight line perpendicular to the planes of these circles. This straight line itself is the axis of rotation (MN in Figure 7.4).

Rice. 7.4

In technology, this type of motion occurs extremely often: rotation of the shafts of engines and generators, wheels of modern high-speed electric trains and village carts, turbines and airplane propellers, etc. The Earth rotates around its axis.

For a long time it was believed that there were no devices similar to a rotating wheel in living organisms: “nature did not create the wheel.” But research recent years showed that this is not so. Many bacteria, such as E. coli, have a “motor” that rotates flagella. With the help of these flagella, the bacterium moves in the environment (Fig. 7.5, a). The base of the flagellum is attached to a ring-shaped wheel (rotor) (Fig. 7.5, b). The plane of the rotor is parallel to another ring fixed in the cell membrane. The rotor rotates, making up to eight revolutions per second. The mechanism that causes the rotor to rotate remains largely unclear.

Rice. 7.5

Kinematic description of the rotational motion of a rigid body

When a body rotates, the radius r A of the circle described by point A of this body (see Fig. 7.4) will rotate during the time interval Δt by a certain angle φ. It is easy to see that due to the immutability relative position points of the body through the same angle φ, the radii of the circles described by any other points of the body will rotate in the same time (see Fig. 7.4). Consequently, this angle φ can be considered a quantity that characterizes the movement of not only an individual point of the body, but also the rotational movement of the entire body as a whole. Therefore, to describe the rotation of a rigid body around a fixed axis, only one quantity is sufficient - the variable φ(t).

This single quantity (coordinate) can be the angle φ through which the body rotates around an axis relative to some of its position, taken as zero. This position is specified by the O 1 X axis in Figure 7.4 (the segments O 2 B, O 3 C are parallel to O 1 X).

In § 1.28, the motion of a point along a circle was considered. The concepts of angular velocity ω and angular acceleration β were introduced. Since when a rigid body rotates, all its points rotate through the same angles at equal time intervals, all formulas that describe the motion of a point along a circle turn out to be applicable to describe the rotation of a rigid body. The definitions of angular velocity (1.28.2) and angular acceleration (1.28.6) can be related to the rotation of a rigid body. In the same way, formulas (1.28.7) and (1.28.8) are valid for describing the motion of a rigid body with constant angular acceleration.

The relationship between linear and angular velocities (see § 1.28) for each point of a rigid body is given by the formula

where R is the distance of the point from the axis of rotation, i.e., the radius of the circle described by the point of the rotating body. The linear velocity is directed tangentially to this circle. Various points rigid bodies have different linear velocities at the same angular velocity.

Various points of a rigid body have normal and tangential accelerations, determined by formulas (1.28.10) and (1.28.11):

Plane-parallel motion

Plane-parallel (or simply plane) motion of a rigid body is a motion in which each point of the body moves all the time in the same plane. Moreover, all the planes in which the points move are parallel to each other. A typical example of plane-parallel motion is the rolling of a cylinder along a plane. The movement of a wheel on a straight rail is also plane-parallel.

Let us remind you (once again!) that we can talk about the nature of the movement of a particular body only in relation to a certain frame of reference. So, in the above examples, in the reference system associated with the rail (ground), the movement of the cylinder or wheel is plane-parallel, and in the reference system associated with the axis of the wheel (or cylinder), it is rotational. Consequently, the speed of each point of the wheel in the reference system associated with the ground (absolute speed), according to the law of addition of speeds, is equal to the vector sum of the linear speed of rotational movement (relative speed) and the speed of translational movement of the axle (transferable speed) (Fig. 7.6):

Rice. 7.6

Instantaneous center of rotation

Let a thin disk roll along a plane (Fig. 7.7). A circle can be considered as a regular polygon with an arbitrarily large number of sides.

Therefore, the circle shown in Figure 7.7 can be mentally replaced by a polygon (Figure 7.8). But the movement of the latter consists of a series of small rotations: first around point C, then around points C 1, C 2, etc. Therefore, the movement of the disk can also be considered as a sequence of very small (infinitesimal) rotations around points C, C 1 C 2 etc. (2). Thus, at each moment of time the disk rotates around its lower point C. This point is called the instantaneous center of rotation of the disk. In the case of a disk rolling along a plane, we can talk about an instantaneous axis of rotation. This axis is the line of contact of the disk with the plane at a given time.

Rice. 7.7 and 7.8

The introduction of the concept of an instantaneous center (instantaneous axis) of rotation simplifies the solution of a number of problems. For example, knowing that the center of the disk has speed and, you can find the speed of point A (see Fig. 7.7). Indeed, since the disk rotates around the instantaneous center C, the radius of rotation of point A is equal to AC, and the radius of rotation of point O is equal to OC. But since AC = 20C, then

Similarly, you can find the speed of any point on this disk.

We met the most simple types motion of a rigid body: translational, rotational, plane-parallel. In the future we will have to deal with the dynamics of a rigid body.

(1) In what follows, for brevity, we will simply talk about a solid body.

(2) Of course, it is impossible to depict a polygon with an infinite number of sides.

Absolutely solid body (solid body) – a body the distance between its parts does not change when forces act on it, i.e. the shape and dimensions of a solid body do not change when any force acts on it. Of course, such bodies do not exist in nature. This is a physical model. In cases where the deformations are small, real bodies can be considered as absolutely solid. The motion of a rigid body is generally very complex. We will consider only two types of body movement:

1. Forward movement:

Body movement counts progressive , if any straight line segment rigidly connected to the body constantly moves parallel to itself. During translational motion, all points of the body make the same movements, travel the same paths, have equal speeds and accelerations, and describe the same trajectories.

2. Rotational movement:

When a body rotates, the radix of the circle described by a point of this body will rotate through a certain angle over a time interval. Due to the invariance of the relative position of the points of the body, the radii of the circles described by any other points of the body will rotate through the same angle during the same time.à This angle is a value that characterizes the rotational motion of the entire body as a whole. From this we can conclude that to describe the rotational motion of an absolutely rigid body around a fixed axis, you need to know only one variable - the angle through which the body will rotate in a certain time.

The relationship between linear and angular velocities for each point of a rigid body is given by the formula V = ώR

Also, points of a solid body have normal and tangential accelerations, which can be specified by the formulas:

a n = ώ 2 R a τ = βR

3. Plane-parallel movement:

Plane-parallel movement is a movement in which each point of the body moves constantly in one plane, while all planes are parallel to each other.

Now let's figure out what the instantaneous center of rotation is. Let us assume that the wheel is rolling along some plane. the movement of this wheel can be considered as a sequence of infinitesimal rotations around points. From this we can conclude that at every moment the wheel rotates around its lowest point. This point is called instantaneous center of rotation .

Instantaneous rotation axis – line of contact of the disk with the plane at a given time.

Statics is the branch of mechanics that deals with general doctrine about forces and study the conditions of equilibrium of material bodies under the influence of forces.

By equilibrium we mean the state of rest of a body in relation to other bodies, for example in relation to the Earth. The equilibrium conditions of a body depend significantly on whether the body is solid, liquid or gaseous. The equilibrium of liquid and gaseous bodies is studied in hydrostatics or aerostatics courses. In a general mechanics course, only problems on the equilibrium of rigid bodies are usually considered.

All solid bodies found in nature, under the influence of external influences, change their shape (deform) to one degree or another. The magnitude of these deformations depends on the material of the bodies, their geometric shape and sizes and from existing loads. To ensure the strength of various engineering structures and structures, the material and dimensions of their parts are selected so that the deformations under existing loads are sufficiently small. As a result, when studying equilibrium conditions, it is quite acceptable to neglect small deformations of the corresponding solid bodies and consider them as non-deformable or absolutely solid. An absolutely rigid body is a body whose distance between every two points always remains constant. In the future, when solving statics problems, all bodies are considered as absolutely rigid, although often for brevity they are simply called rigid bodies.

The state of equilibrium or movement of a given body depends on the nature of its mechanical interactions with other bodies, i.e., on the pressures, attractions or repulsions that the body experiences as a result of these interactions. The quantity, which is the main measure of the mechanical interaction of material bodies, is called force in mechanics.

The quantities considered in mechanics can be divided into scalar ones, i.e. those that are completely characterized by their numerical value, and vector, i.e. those that, in addition to their numerical value, are also characterized by direction in space.

Force is a vector quantity. Its action on the body is determined by: 1) the numerical value or modulus of the force, 2) the direction of the force, 3) the point of application of the force.

The force modulus is found by comparing it with the force taken as unity. The basic unit of force in the International System of Units (SI) that we will use (for more details, see § 75) is 1 newton (1 N); A larger unit of 1 kilonewton is also used. For static measurement of force, devices known from physics, called dynamometers, are used.

The force, like all other vector quantities, will be denoted by a letter with a bar over it (for example, F), and the force module will be denoted by a symbol or the same letter, but without a bar above it (F). Graphically, force, like other vectors, is represented by a directed segment (Fig. 1). The length of this segment expresses the modulus of the force on the selected scale, the direction of the segment corresponds to the direction of the force, point A in Fig. 1 is the point of application of the force (the force can also be depicted in such a way that the point of application is the end of the force, as in Fig. A, c). The straight line DE along which the force is directed is called the line of action of the force. Let us also agree on the following definitions.

1. We will call a system of forces the set of forces acting on the body (or bodies) under consideration. If the lines of action of all forces lie in the same plane, the system of forces is called flat, and if these lines of action do not lie in the same plane, it is called spatial. In addition, forces whose lines of action intersect at one point are called converging, and forces whose lines of action are parallel to each other are called parallel.

2. A body to which any movement in space can be imparted from a given position is called free.

3. If one system of forces acting on a free rigid body can be replaced by another system without changing the state of rest or motion in which the body is located, then such two systems of forces are called equivalent.

4. A system of forces under the influence of which a free rigid body can be at rest is called balanced or equivalent to zero.

5. If a given system of forces is equivalent to one force, then this force is called the resultant of this system of forces.

A force equal to the resultant in magnitude, directly opposite to it in direction and acting along the same straight line is called a balancing force.

6. The forces acting on a given body (or system of bodies) can be divided into external and internal. External are the forces that act on this body (or the bodies of the system) from other bodies, and internal are the forces with which the parts of a given body (or the bodies of a given system) act on each other.

7. A force applied to a body at any one point is called concentrated. Forces acting on all points of a given volume or a given part of the surface of a body are called distributed.

The concept of concentrated force is conditional, since it is practically impossible to apply force to a body at one point. Forces, which in mechanics are considered concentrated, are essentially the resultants of certain systems of distributed forces.

In particular, the gravitational force acting on a given solid body, considered in mechanics, is the resultant of the gravitational forces acting on its particles. The line of action of this resultant passes through a point called the center of gravity of the body.

The tasks of statics are: 1) transformation of systems of forces acting on a solid body into systems equivalent to them, in particular, bringing a given system of forces to its simplest form; 2) determination of the equilibrium conditions for systems of forces acting on a solid body.

Statics problems can be solved either by appropriate geometric constructions (geometric and graphical methods) or by numerical calculations (analytical method). The course will mainly use the analytical method, but it should be borne in mind that visual geometric constructions play an extremely important role in solving problems in mechanics.

In the section on the question what is an absolutely rigid body asked by the author European the best answer is An absolutely rigid body is the second supporting object of mechanics along with a material point. The mechanics of an absolutely rigid body is completely reducible to the mechanics of material points (with imposed constraints), but has its own content (useful concepts and relationships that can be formulated within the framework of the model of an absolutely rigid body), which is of great theoretical and practical interest.
There are several definitions:
An absolutely rigid body is a model concept of classical mechanics, denoting a set of material points, the distances between which are maintained during any movements performed by this body. In other words, an absolutely solid body not only does not change its shape, but also maintains the distribution of mass inside unchanged.
Absolutely rigid body - mechanical system, having only translational and rotational degrees of freedom. “Hardness” means that the body cannot be deformed, that is, no other energy can be transferred to the body except kinetic energy translational or rotational motion.
An absolutely rigid body is a body (system), the relative position of any points of which does not change, no matter what processes it participates in.
Thus, the position of an absolutely rigid body is completely determined, for example, by the position of the Cartesian coordinate system rigidly attached to it (usually its origin is made to coincide with the center of mass of the rigid body).
In three-dimensional space and in the absence of (other) connections, an absolutely rigid body has 6 degrees of freedom: three translational and three rotational. The exception is a diatomic molecule or, in the language of classical mechanics, a solid rod of zero thickness. Such a system has only two rotational degrees of freedom.
Absolutely rigid bodies do not exist in nature, however, in very many cases, when the deformation of the body is small and can be neglected, a real body can (approximately) be considered as an absolutely rigid body without prejudice to the problem.
Within the framework of relativistic mechanics, the concept of an absolutely rigid body is internally contradictory, as shown, in particular, by the Ehrenfest paradox. In other words, the model of an absolutely rigid body is generally speaking completely inapplicable to the case of fast movements (comparable in speed to the speed of light), as well as to the case of very strong gravitational fields

An absolutely rigid body is a body whose deformations can be neglected in this problem and under all conditions the distance between two points of this body remains constant.

The inertia of bodies during rotational motion is characterized by a quantity called the moment of inertia. The moment of inertia of a system (body) relative to a given axis is a physical quantity equal to the sum of the products of the masses and material points of the system by the square of their distances to the axis in question:

I=m i r i 2 (3.1)

In the case of a continuous mass distribution, this sum is reduced to the integral:

I=∫r 2 dm (3.2), where integration is carried out over the entire volume.

For a homogeneous solid disk (cylinder):

I=0.5 mR 2 (3.3), if the axis of rotation passes through the center of gravity (mass).

The moment of inertia about an arbitrary axis is determined by Steiner’s theorem:

I=I c +ma 2 (3.4), where a is the distance between the axes.

The ability of a force to rotate a body is characterized by a physical quantity called the moment of force:

O – axis of rotation
l – force arm
α – angle between vector F and radius vector r

Moment modulus: M=F r sinα=F l (3.6)

r sinα - shortest distance between the line of action of the force and point O is the shoulder of the force.

The moment of force is a physical quantity determined by the product of the force and its arm.

By analogy with translational motion, we can write the equation for the dynamics of rotational motion:

An analogue of the momentum of a body during rotational motion is the angular momentum relative to the axis. Vector quantity.

Momentum module:

L=r P sinα=m υ r sinα=Pl (3.9)
L z =I ω (3.10)

(3.12)

dL z /dt=M z (3.13)

This expression is another form of the equation for the dynamics of the rotational motion of a rigid body relative to a fixed axis: the derivative of the angular momentum relative to the axis is equal to the moment of force relative to the same axis. It can be shown that there is a vector equality:

In a closed system, the moment of external forces is M=0; dL/dt=0, whence L=const (3.15) represents the law of conservation of angular momentum: the angular momentum of a closed-loop system is conserved, i.e. does not change over time. The law of conservation of momentum is a fundamental law of nature. It is associated with the property of symmetry of space - its isotropy, i.e. invariance of physical laws with respect to the choice of direction of the coordinate axes of the reference system (relative to the rotation of a closed system in space at any angle).

Rotary operation:

dA=M z dφ (3.16)

Kinetic energy:

T=Iω 2 /2 (3.17)

The total energy of a system moving translationally and rotating is equal to:

E=+ (3.18)

You can make a table similar to the dynamics of translational and rotational motion.

Forward movement