A vector connecting the beginning and end of a path. A displacement is a vector connecting the starting and ending points of a trajectory

Weight is a property of a body that characterizes its inertia. Under the same influence from surrounding bodies, one body can quickly change its speed, while another, under the same conditions, can change much more slowly. It is customary to say that the second of these two bodies has greater inertia, or, in other words, the second body has greater mass.

If two bodies interact with each other, then as a result the speed of both bodies changes, i.e., in the process of interaction, both bodies acquire acceleration. The ratio of the accelerations of these two bodies turns out to be constant under any influence. In physics, it is accepted that the masses of interacting bodies are inversely proportional to the accelerations acquired by the bodies as a result of their interaction.

Force is a quantitative measure of the interaction of bodies. Force causes a change in the speed of a body. In Newtonian mechanics, forces can have a different physical nature: friction force, gravity force, elastic force, etc. Force is vector quantity. The vector sum of all forces acting on a body is called resultant force.

To measure forces it is necessary to set standard of strength And comparison method other forces with this standard.

As a standard of force, we can take a spring stretched to a certain specified length. Force module F 0 with which this spring, at a fixed tension, acts on a body attached to its end is called standard of strength. The way to compare other forces with a standard is as follows: if the body, under the influence of the measured force and the reference force, remains at rest (or moves uniformly and rectilinearly), then the forces are equal in magnitude F = F 0 (Fig. 1.7.3).

If the measured force F greater (in absolute value) than the reference force, then two reference springs can be connected in parallel (Fig. 1.7.4). In this case the measured force is 2 F 0 . Forces 3 can be measured similarly F 0 , 4F 0, etc.

Measuring forces less than 2 F 0, can be performed according to the scheme shown in Fig. 1.7.5.

The reference force in the International System of Units is called newton(N).

A force of 1 N imparts an acceleration of 1 m/s to a body weighing 1 kg 2

In practice, there is no need to compare all measured forces with a standard. To measure forces, springs calibrated as described above are used. Such calibrated springs are called dynamometers . The force is measured by the stretch of the dynamometer (Fig. 1.7.6).

Newton's laws of mechanics - three laws underlying the so-called. classical mechanics. Formulated by I. Newton (1687). First Law: “Every body continues to be maintained in its state of rest or uniform and rectilinear motion until and unless it is compelled by applied forces to change that state.” Second law: “The change in momentum is proportional to the applied driving force and occurs in the direction of the straight line along which this force acts.” Third law: “An action always has an equal and opposite reaction, otherwise, the interactions of two bodies on each other are equal and directed in opposite directions.” 1.1. Law of inertia (Newton's First Law) : a free body, which is not acted upon by forces from other bodies, is in a state of rest or uniform linear motion (the concept of speed here is applied to the center of mass of the body in the case of non-translational motion). In other words, bodies are characterized by inertia (from the Latin inertia - “inactivity”, “inertia”), that is, the phenomenon of maintaining speed if external influences on them are compensated. Reference systems in which the law of inertia is satisfied are called inertial reference systems (IRS). The law of inertia was first formulated by Galileo Galilei, who, after many experiments, concluded that for a free body to move at a constant speed, no external cause is needed. Before this, a different point of view (going back to Aristotle) ​​was generally accepted: a free body is at rest, and to move at a constant speed it is necessary to apply a constant force. Newton subsequently formulated the law of inertia as the first of his three famous laws. Galileo's principle of relativity: in all inertial frames of reference, all physical processes proceed in the same way. In a reference system brought to a state of rest or uniform rectilinear motion relative to an inertial reference system (conventionally, “at rest”), all processes proceed in exactly the same way as in a system at rest. It should be noted that the concept of an inertial reference system is an abstract model (a certain ideal object considered instead of a real object. Examples of an abstract model are an absolutely rigid body or a weightless thread), real reference systems are always associated with some object and the correspondence of the actually observed motion of bodies in such systems with the calculation results will be incomplete. 1.2 Law of motion - a mathematical formulation of how a body moves or how a more general type of motion occurs. In classical mechanics of a material point, the law of motion represents three dependences of three spatial coordinates on time, or a dependence of one vector quantity (radius vector) on time, type. The law of motion can be found, depending on the problem, either from the differential laws of mechanics or from the integral ones. Law of energy conservation - the basic law of nature, which is that the energy of a closed system is conserved over time. In other words, energy cannot arise from nothing and cannot disappear into anything; it can only move from one form to another. The law of conservation of energy is found in various branches of physics and manifests itself in the conservation of various types of energy. For example, in classical mechanics the law is manifested in the conservation of mechanical energy (the sum of potential and kinetic energies). In thermodynamics, the law of conservation of energy is called the first law of thermodynamics and speaks of the conservation of energy in addition to thermal energy. Since the law of conservation of energy does not apply to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it is more correct to call it not a law, but the principle of conservation of energy. A special case is the Law of Conservation of Mechanical Energy - the mechanical energy of a conservative mechanical system is conserved over time. Simply put, in the absence of forces such as friction (dissipative forces), mechanical energy does not arise from nothing and cannot disappear anywhere. Ek1+Ep1=Ek2+Ep2 The law of conservation of energy is an integral law. This means that it consists of the action of differential laws and is a property of their combined action. For example, it is sometimes said that the impossibility of creating a perpetual motion machine is due to the law of conservation of energy. But that's not true. In fact, in every perpetual motion machine project, one of the differential laws is triggered and it is this that makes the engine inoperative. The law of conservation of energy simply generalizes this fact. According to Noether's theorem, the law of conservation of mechanical energy is a consequence of the homogeneity of time. 1.3. Law of conservation of momentum (Law of conservation of momentum, Newton's 2nd law) states that the sum of the momenta of all bodies (or particles) of a closed system is a constant value. From Newton's laws it can be shown that when moving in empty space, momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces. In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. However, this conservation law is also true in cases where Newtonian mechanics is not applicable (relativistic physics, quantum mechanics). Like any of the conservation laws, the law of conservation of momentum describes one of the fundamental symmetries - the homogeneity of space Newton's third law explains what happens to two interacting bodies. Let us take for example a closed system consisting of two bodies. The first body can act on the second with a certain force F12, and the second can act on the first with a force F21. How do the forces compare? Newton's third law states: the action force is equal in magnitude and opposite in direction to the reaction force. Let us emphasize that these forces are applied to different bodies, and therefore are not compensated at all. The law itself: Bodies act on each other with forces directed along the same straight line, equal in magnitude and opposite in direction: . 1.4. Inertia forces Newton's laws, strictly speaking, are valid only in inertial frames of reference. If we honestly write down the equation of motion of a body in a non-inertial frame of reference, then it will differ in appearance from Newton’s second law. However, often, to simplify the consideration, a certain fictitious “force of inertia” is introduced, and then these equations of motion are rewritten in a form very similar to Newton’s second law. Mathematically, everything here is correct (correct), but from the point of view of physics, the new fictitious force cannot be considered as something real, as a result of some real interaction. Let us emphasize once again: “force of inertia” is only a convenient parameterization of how the laws of motion differ in inertial and non-inertial reference systems. 1.5. Law of viscosity Newton's law of viscosity (internal friction) is a mathematical expression relating the internal friction stress τ (viscosity) and the change in the velocity of the medium v ​​in space (strain rate) for fluid bodies (liquids and gases): where the value η is called the coefficient of internal friction or dynamic coefficient of viscosity (GHS unit - poise). The kinematic viscosity coefficient is the value μ = η / ρ (CGS unit is Stokes, ρ is the density of the medium). Newton's law can be obtained analytically using methods of physical kinetics, where viscosity is usually considered simultaneously with thermal conductivity and the corresponding Fourier law for thermal conductivity. In the kinetic theory of gases, the coefficient of internal friction is calculated by the formula Where< u >is the average speed of thermal motion of molecules, λ is the average free path.

Trajectory(from Late Latin trajectories - related to movement) is the line along which a body (material point) moves. The trajectory of movement can be straight (the body moves in one direction) and curved, that is, mechanical movement can be rectilinear and curvilinear.

Straight-line trajectory in this coordinate system it is a straight line. For example, we can assume that the trajectory of a car on a flat road without turns is straight.

Curvilinear movement is the movement of bodies in a circle, ellipse, parabola or hyperbola. An example of curvilinear motion is the movement of a point on the wheel of a moving car or the movement of a car in a turn.

The movement can be difficult. For example, the trajectory of a body at the beginning of its journey can be rectilinear, then curved. For example, at the beginning of the journey a car moves along a straight road, and then the road begins to “wind” and the car begins to move in a curved direction.

Path

Path is the length of the trajectory. Path is a scalar quantity and is measured in meters (m) in the SI system. Path calculation is performed in many physics problems. Some examples will be discussed later in this tutorial.

Move vector

Move vector(or simply moving) is a directed straight line segment connecting the initial position of the body with its subsequent position (Fig. 1.1). Displacement is a vector quantity. The displacement vector is directed from the starting point of movement to the ending point.

Motion vector module(that is, the length of the segment that connects the starting and ending points of the movement) can be equal to the distance traveled or less than the distance traveled. But the magnitude of the displacement vector can never be greater than the distance traveled.

The magnitude of the displacement vector is equal to the distance traveled when the path coincides with the trajectory (see sections Trajectory and Path), for example, if a car moves from point A to point B along a straight road. The magnitude of the displacement vector is less than the distance traveled when a material point moves along a curved path (Fig. 1.1).

Rice. 1.1. Displacement vector and distance traveled.

In Fig. 1.1:

Another example. If the car drives in a circle once, it turns out that the point at which the movement begins will coincide with the point at which the movement ends, and then the displacement vector will be equal to zero, and the distance traveled will be equal to the length of the circle. Thus, path and movement are two different concepts.

Vector addition rule

The displacement vectors are added geometrically according to the vector addition rule (triangle rule or parallelogram rule, see Fig. 1.2).

Rice. 1.2. Addition of displacement vectors.

Figure 1.2 shows the rules for adding vectors S1 and S2:

a) Addition according to the triangle rule
b) Addition according to the parallelogram rule

Motion vector projections

When solving problems in physics, projections of the displacement vector onto coordinate axes are often used. Projections of the displacement vector onto the coordinate axes can be expressed through the differences in the coordinates of its end and beginning. For example, if a material point moves from point A to point B, then the displacement vector (Fig. 1.3).

Let us choose the OX axis so that the vector lies in the same plane with this axis. Let's lower the perpendiculars from points A and B (from the starting and ending points of the displacement vector) until they intersect with the OX axis. Thus, we obtain the projections of points A and B onto the X axis. Let us denote the projections of points A and B, respectively, as A x and B x. The length of the segment A x B x on the OX axis is displacement vector projection on the OX axis, that is

S x = A x B x

IMPORTANT!
I remind you for those who do not know mathematics very well: do not confuse a vector with the projection of a vector onto any axis (for example, S x). A vector is always indicated by a letter or several letters, above which there is an arrow. In some electronic documents, the arrow is not placed, as this may cause difficulties when creating an electronic document. In such cases, be guided by the content of the article, where the word “vector” may be written next to the letter or in some other way they indicate to you that this is a vector, and not just a segment.

Rice. 1.3. Projection of the displacement vector.

The projection of the displacement vector onto the OX axis is equal to the difference between the coordinates of the end and beginning of the vector, that is

S x = x – x 0 Similarly, the projections of the displacement vector on the OY and OZ axes are determined and written: S y = y – y 0 S z = z – z 0

Here x 0 , y 0 , z 0 are the initial coordinates, or the coordinates of the initial position of the body (material point); x, y, z - final coordinates, or coordinates of the subsequent position of the body (material point).

The projection of the displacement vector is considered positive if the direction of the vector and the direction of the coordinate axis coincide (as in Fig. 1.3). If the direction of the vector and the direction of the coordinate axis do not coincide (opposite), then the projection of the vector is negative (Fig. 1.4).

If the displacement vector is parallel to the axis, then the modulus of its projection is equal to the modulus of the Vector itself. If the displacement vector is perpendicular to the axis, then the modulus of its projection is equal to zero (Fig. 1.4).

Rice. 1.4. Motion vector projection modules.

The difference between the subsequent and initial values ​​of some quantity is called the change in this quantity. That is, the projection of the displacement vector onto the coordinate axis is equal to the change in the corresponding coordinate. For example, for the case when the body moves perpendicular to the X axis (Fig. 1.4), it turns out that the body DOES NOT MOVE relative to the X axis. That is, the movement of the body along the X axis is zero.

Let's consider an example of body motion on a plane. The initial position of the body is point A with coordinates x 0 and y 0, that is, A(x 0, y 0). The final position of the body is point B with coordinates x and y, that is, B(x, y). Let's find the modulus of body displacement.

From points A and B we lower perpendiculars to the coordinate axes OX and OY (Fig. 1.5).

Rice. 1.5. Movement of a body on a plane.

Let us determine the projections of the displacement vector on the OX and OY axes:

S x = x – x 0 S y = y – y 0

In Fig. 1.5 it is clear that triangle ABC is a right triangle. It follows from this that when solving the problem one can use Pythagorean theorem, with which you can find the module of the displacement vector, since

AC = s x CB = s y

According to the Pythagorean theorem

S 2 = S x 2 + S y 2

Where can you find the module of the displacement vector, that is, the length of the body’s path from point A to point B:

And finally, I suggest you consolidate your knowledge and calculate a few examples at your discretion. To do this, enter some numbers in the coordinate fields and click the CALCULATE button. Your browser must support the execution of JavaScript scripts and script execution must be enabled in your browser settings, otherwise the calculation will not be performed. In real numbers, the integer and fractional parts must be separated by a dot, for example, 10.5.

Mechanical movement. Relativity of motion. Elements of kinematics. material point. Galileo's transformations. The classical law of addition of velocities

Mechanics is a branch of physics that studies the laws of motion and interaction of bodies. Kinematics is a branch of mechanics that does not study the causes of the movement of bodies.

Mechanical movement is a change in the position of a body in space relative to other bodies over time.

A material point is a body whose dimensions can be neglected under given conditions.

Translational is a movement in which all points of the body move equally. Translational is a movement in which any straight line drawn through the body remains parallel to itself.

Kinematic characteristics of movement

Trajectory – line of movement. S - path – path length.

S – displacement – ​​a vector connecting the initial and final position of the body.

Relativity of motion. Reference system - a combination of a reference body, a coordinate system and a device for measuring time (hours)

Rectilinear uniform motion is a motion in which a body makes equal movements in any equal intervals of time. Speed ​​is a physical quantity equal to the ratio of the displacement vector to the period of time during which this displacement occurred. The speed of uniform rectilinear motion is numerically equal to displacement per unit time.

The displacement of a body is a directed segment of a straight line connecting the initial position of the body with its subsequent position. Displacement is a vector quantity.

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Basic concepts of kinematics

Kinematics is a branch of mechanics in which the movement of bodies is considered without identifying the causes of this movement.

Mechanical movement bodies are called changes in position in space relative to other bodies over time.

Mechanical movement relatively. The motion of the same body relative to different bodies turns out to be different. To describe the movement of a body, it is necessary to indicate in relation to which body the movement is being considered. This body is called reference body.

The coordinate system associated with the reference body and the clock for counting time form reference system , allowing you to determine the position of a moving body at any time.

In the International System of Units (SI), the unit of length is meter, and per unit of time – second.

Every body has certain dimensions. Different parts of the body are in different places in space. However, in many mechanics problems there is no need to indicate the positions of individual parts of the body. If the dimensions of a body are small compared to the distances to other bodies, then this body can be considered ᴇᴦο material point. This can be done, for example, when studying the movement of planets around the Sun.

If all parts of the body move equally, then such movement is called progressive . For example, cabins in the “Giant Wheel” attraction, a car on a straight section of track, etc. move translationally. With translational motion of a body, ᴇᴦο can also be considered as a material point.

A body whose dimensions can be neglected under given conditions is called material point .

The concept of a material point plays an important role in mechanics.

Moving over time from one point to another, a body (material point) describes a certain line, which is called body movement trajectory .

The position of a material point in space at any time ( law of motion ) can be determined either using the dependence of coordinates on time x = x(t), y = y(t), z = z(t) (coordinate method), or using the time dependence of the radius vector (vector method) drawn from the origin to a given point (Fig. 1.1.1).

The movement of a body is a directed segment of a straight line connecting the initial position of the body with its subsequent position. Displacement is a vector quantity.

The displacement of a body is a directed segment of a straight line connecting the initial position of the body with its subsequent position. Displacement is a vector quantity. - concept and types. Classification and features of the category "The displacement of a body is a directed segment of a straight line connecting the initial position of the body with its subsequent position. Displacement is a vector quantity." 2015, 2017-2018.

Definition 1

Body trajectory is a line that was described by a material point when moving from one point to another over time.

There are several types of movements and trajectories of a rigid body:

• progressive;
• rotational, that is, movement in a circle;
• flat, that is, movement along a plane;
• spherical, characterizing movement on the surface of a sphere;
• free, in other words, arbitrary.

Picture 1 . Defining a point using coordinates x = x (t), y = y (t) , z = z (t) and the radius vector r → (t) , r 0 → is the radius vector of the point at the initial time

The position of a material point in space at any time can be specified using the law of motion, determined by the coordinate method, through the dependence of the coordinates on time x = x (t) , y = y (t) , z = z (t) or from the time of the radius vector r → = r → (t) drawn from the origin to a given point. This is shown in Figure 1.

S → = ∆ r 12 → = r 2 → - r 1 → – a directed straight line segment connecting the start and end points of the body’s trajectory. The value of the distance traveled l is equal to the length of the trajectory traveled by the body over a certain period of time t.

Figure 2. Distance traveled l and the displacement vector s → for curvilinear movement of the body, a and b are the starting and ending points of the path, accepted in physics

Definition 3

Figure 2 shows that when a body moves along a curved path, the magnitude of the displacement vector is always less than the distance traveled.

Path is a scalar quantity. Counts as a number.

The sum of two successive movements from point 1 to point 2 and from point 2 to point 3 is the movement from point 1 to point 3, as shown in Figure 3.

Drawing 3 . The sum of two consecutive movements ∆ r → 13 = ∆ r → 12 + ∆ r → 23 = r → 2 - r → 1 + r → 3 - r → 2 = r → 3 - r → 1

When the radius vector of a material point at a certain moment of time t is r → (t), at the moment t + ∆ t is r → (t + ∆ t), then its displacement ∆ r → during the time ∆ t is equal to ∆ r → = r → (t + ∆ t) - r → (t) .

The displacement ∆ r → is considered a function of time t: ∆ r → = ∆ r → (t) .

Example 1

According to the condition, a moving airplane is given, shown in Figure 4. Determine the type of trajectory of point M.

Drawing 4

Solution

It is necessary to consider the reference system I, called “Airplane” with the trajectory of the point M in the form of a circle.

The reference system II “Earth” will be specified with the trajectory of the existing point M in a spiral.

Example 2

Given a material point that moves from A to B. The value of the radius of the circle is R = 1 m. Find S, ∆ r →.

Solution

While moving from A to B, a point travels a path that is equal to half a circle, written by the formula:

We substitute the numerical values ​​and get:

S = 3.14 · 1 m = 3.14 m.

The displacement ∆ r → in physics is considered to be a vector connecting the initial position of a material point with the final one, that is, A with B.

Substituting numerical values, we calculate:

∆ r → = 2 R = 2 · 1 = 2 m.

Answer: S = 3.14 m; ∆ r → = 2 m.

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