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Basic concepts of statics entered science as a result of centuries practical activities person. They confirmed numerous experiments and observations of nature.

One of these basic concepts is the concept material point.

Body can be seen as material point, that is, it can be represented geometric the point at which all weight body, in the event that body size do not matter in the problem under consideration.

For example, when studying movement planets and satellites they are considered material points, because dimensions planets and satellites negligible compared with orbit sizes... On the other hand, studying traffic planets (for example, the Earth) around the axis, it is already it is forbidden be considered a material point.

Body can to be considered a material point in all cases when all its points perform the same traffic. For example, a piston in an internal combustion engine can be viewed as a material point where the entire mass of this piston is concentrated.

System called aggregate of material points, movements and positions of which interdependent... It follows from the above definition that any physical body can be considered as a system of material points.

When studying the balance of bodies, they are considered absolutely solid(or absolutely rigid), i.e., it is assumed that no external influences do not cause changes in their size and shape and what distance between any two points on the body always remains unchanged.

In fact all bodies under the influence of force from other bodies change its size and shape. So, if the rod, for example, made of steel or wood, squeeze, its length decrease, and at stretching she accordingly will increase(rice. a ).

Changes also the form a rod lying on two supports under the action of a load perpendicular to its axis (Fig. b ). At the same time, the rod bends.

Overwhelmingly cases deformations bodies (parts) that make up machines, apparatus and structures, very small, and when studying movement and balance these objects deformations are negligible.

Thus, the concept of an absolutely rigid body is conditional(abstraction). This concept is introduced for the purpose simplifying the study of the laws of equilibrium and motion of bodies.

Only by studying rigid body mechanics, you can start studying balance and movement deformable bodies, liquids, etc. When calculating the strength, it is necessary to take into account the deformations of bodies... In these calculations, the deformations play essential role and they cannot be neglected.

A section of mechanics is called statics, in which the general doctrine of forces is presented and the conditions of equilibrium of material bodies under the influence of forces are studied.

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

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

The state of balance or movement of a given body depends on the nature of its mechanical interactions with other bodies. The quantity, which is the main measure of the mechanical interaction of material bodies, is called force in mechanics.

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

The modulus of force is found by comparing it with the force taken as a unit. The basic unit of measure for force in the International System of Units (SI) is 1 Newton (1 N).

The length of this segment expresses the modulus of force in 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 will be the end of the force). 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. A system of forces will be called a set of forces acting on the considered body (or bodies). 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, the system of forces 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. The system of forces under the action of which a free rigid body can be at rest is called balanced or equivalent to zero.
5. If this system forces is equivalent to one force, then this force is called the resultant of the given system of forces.

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

6. Forces acting on a given body (or system of bodies) can be divided into external and internal. External forces are called forces that act on this body (or the bodies of the system) from other bodies, and internal forces are the forces with which the parts of this body (or the body of this system) act on each other.
7. The force applied to the body at any one of its points is called concentrated. Forces acting on all points of a given volume or a given part of the body surface are called distributed.

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

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

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

1. Translational movement:

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

2. Rotational movement:

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 one straight line, perpendicular to the planes these circles. This line itself is the axis of rotation.

When the body rotates, the radix of the circle described by the point of this body will rotate by a certain angle during the time interval. Due to the invariability 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 in the same time by the same angle.à This angle is a quantity that characterizes the rotational movement of the whole body as a whole. Hence, we can conclude that to describe the rotational motion of an absolutely rigid body around a fixed axis, one needs to know only one variable - the angle by which the body will rotate in a certain time.

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

Also, points of a rigid 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. Suppose the wheel is rolling on some plane. the movement of this wheel can be viewed as a sequence of infinitesimal turns 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 - the line of contact of the disc with the plane at a given time.

A section of mechanics is called statics, in which the general doctrine of forces is presented and the conditions of equilibrium of material bodies under the influence of forces are studied.

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

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

The state of equilibrium or motion of a given body depends on the nature of its mechanical interactions with other bodies, that is, on those 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, that is, those that are fully characterized by their numerical value, and vector, that is, those that, in addition to the numerical value, are also characterized by the direction in space.

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

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

Force, like all other vector quantities, will be denoted by a letter with a bar above it (for example, F), and the modulus of force by a symbol or the same letter, but without a bar above it (F). Graphically, the force, like other vectors, is depicted by a directed segment (Fig. 1). The length of this segment expresses the modulus of force in 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 will be 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. A system of forces will be called a set of forces acting on the considered body (or bodies). 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, the system of forces 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. The system of forces under the action 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 absolute value, directly opposite to it in direction and acting along the same straight line, is called a balancing force.

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

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

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

Statics problems can be solved either by means of appropriate geometric constructions (geometric and graphic methods), or by means of 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 of mechanics.

Newton's laws.

Newton's first law. Inertial frames of reference

Galileo, and then Newton, first came to the conclusion about the existence of the phenomenon of inertia. This conclusion is formulated as Newton's first law (law of inertia ): there are such reference systems relative to which the body (material point), in the absence of external influences on it (or with their mutual compensation), maintains a state of rest or uniform rectilinear motion.

Newton's first law postulates the presence of such a phenomenon as the inertia of bodies. Therefore, it is also known as the Law of Inertia. Inertia- this is the phenomenon of conservation of the speed of motion by the body (both in magnitude and in direction), when no forces act on the body. To change the speed of movement, it is necessary to act on the body with some force. Naturally, the result of the action of forces of the same magnitude on different bodies will be different. Thus, bodies are said to be inert. Inertia- this is the property of bodies to resist changing their current state. The amount of inertness is characterized by body weight.

Newton's second law.

Formula (1) expresses Newton's second law , which is formulated as follows: the force acting on a body is equal to the product of the body's mass by the acceleration imparted to this body by the force.

Force is a vector quantity that characterizes such an action on a given body of other bodies (or fields), which can cause acceleration and deformation of the body (here we mean an arbitrary rigid body, not a material point).

Newton's third law.

In all cases, when a body acts on another, there is no one-sided action, but the interaction of bodies. The forces of such interaction between bodies are of the same nature, appear and disappear simultaneously. When two bodies interact, both bodies receive accelerations directed along one straight line in opposite directions. Since a1 / a2 = m2 / m1, then m1a1 = m2a2, or in vector form

m1a1 = -m2a2. (1)

According to Newton's second law, m1a1 = F1 and m2a2 = F2. Then it follows from formula (2.7) that

Equality (2) expresses Newton's third law : bodies interact with each other by forces equal in magnitude and opposite in direction.

Absolutely solid. Moment of inertia. A moment of power.

Absolutely solid.

An absolutely solid body is the second reference 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 superimposed constraints), but has its own content ( useful concepts and relations that can be formulated within the framework of the absolutely rigid body model), which is of great theoretical and practical interest.

There are several definitions:

An absolutely rigid body is a mechanical system with 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 the kinetic energy of translational or rotational motion.

An absolutely rigid body is a body (system), the relative position of any points of which does not change, in whatever processes it participates.

Thus, the position of an absolutely rigid body is completely determined, for example, by the position of a Cartesian coordinate system rigidly attached to it (usually its origin is made to coincide with the center of mass of a rigid body).

Absolutely rigid bodies do not exist in nature, however, in very many cases, when the deformation of a body is small and can be neglected, a real body can (approximately) be considered as an absolutely rigid body without prejudice to the problem.

Moment of inertia.

The moment of inertia is a scalar physical quantity, a measure of the inertia of a body in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion. It is characterized by the distribution of masses in the body: the moment of inertia is equal to the sum of the products of elementary masses by the square of their distances to the base set (point, line or plane).

SI unit: kg · m².

Designation: I or J.

I = (sign of sums) mh ^ 2 or I = (integral) ph ^ 2dV,

where mi are the masses of points of the body, hi are their distances from the z axis, r is the mass density, V is the volume of the body. The quantity Iz is a measure of the inertness of the body when it rotates around the axis /

There are several moments of inertia - depending on the manifold from which the distance of the points is measured.

MOMENT OF FORCES ?? WHO HAS THE OLD LECTURES?