How does movement differ from a path? Define the concepts: movement, path, trajectory

Trajectory- this is the line that the body describes when moving.

Bee trajectory

Path is the length of the trajectory. That is, the length of that possibly curved line along which the body moved. Path is a scalar quantity ! Moving- vector quantity ! This is a vector drawn from the initial point of departure of the body to the final point. Has a numerical value equal to the length of the vector. Path and displacement are significantly different physical quantities.

You may come across different path and movement designations:

Amount of movements

Let the body make a movement s 1 during the period of time t 1, and move s 2 during the next period of time t 2. Then for the entire time of movement the displacement s 3 is the vector sum

Uniform movement

Movement with constant speed in magnitude and direction. What does it mean? Consider the motion of a car. If she drives in a straight line, the speedometer shows the same speed value (velocity module), then this movement is uniform. As soon as the car changes direction (turn), it will mean that the velocity vector has changed its direction. The speed vector is directed in the same direction as the car is going. Such movement cannot be considered uniform, despite the fact that the speedometer shows the same number.

The direction of the velocity vector always coincides with the direction of movement of the body

Can the movement on a carousel be considered uniform (if there is no acceleration or braking)? It’s impossible, the direction of movement is constantly changing, and therefore the velocity vector. From the reasoning we can conclude that uniform motion is it is always moving in a straight line! This means that with uniform motion, the path and displacement are the same (explain why).

It is not difficult to imagine that with uniform motion, over any equal periods of time, the body will move the same distance.

Displacement, shift, movement, migration, movement, rearrangement, regrouping, transfer, transportation, transition, relocation, transfer, travel; shifting, moving, telekinesis, epeirophoresis, relocation, rolling, waddle,... ... Dictionary of synonyms

MOVEMENT, movement, cf. (book). 1. Action under Ch. move move. Moving within the service. 2. Action and condition according to Ch. move move. Movement of layers of the earth's crust. Ushakov's explanatory dictionary. D.N. Ushakov. 1935 1940 ... Ushakov's Explanatory Dictionary

In mechanics, a vector connecting the positions of a moving point at the beginning and end of a certain period of time; The P vector is directed along the chord of the point's trajectory. Physical encyclopedic dictionary. M.: Soviet Encyclopedia. Editor-in-Chief A.M.... ... Physical encyclopedia

MOVE, eat, eat; still (yon, ena); owls, who what. Place, transfer to another place. P. scenery. P. brigade to another site. Displaced persons (persons forcibly displaced from their country). Ozhegov's explanatory dictionary. S.I.... ... Ozhegov's Explanatory Dictionary

- (relocation) Relocation of an office, enterprise, etc. to another place. Often it is caused by a merger or acquisition. Sometimes employees receive a relocation allowance, which is intended to encourage them to stay in their current location... ... Dictionary of business terms

moving- - Telecommunications topics, basic concepts EN redeployment... Technical Translator's Guide

Moving,- Displacement, mm, the amount of change in the position of any point of an element of a window block (usually a frame impost or vertical bars of sashes) in the direction normal to the plane of the product under the influence of wind load. Source: GOST... ...

moving- Migration of material in the form of a solution or suspension from one soil horizon to another... Dictionary of Geography

moving- 3.14 transfer (in relation to storage location): Changing the storage location of a document. Source: GOST R ISO 15489 1 2007: System of information standards... Dictionary-reference book of terms of normative and technical documentation

moving- ▲ change of position, motionless movement in space change of position in space; transformation of a figure that preserves the distances between points of the figure; moving to another place. movement. forward motion... ... Ideographic Dictionary of the Russian Language

Books

• GESNm 81-03-40-2001. Part 40. Additional movement of equipment and material resources. State estimate standards. State elemental estimate standards for installation of equipment (hereinafter referred to as GESNm) are intended to determine the need for resources (labor costs of workers,...
• Movement of people and cargo in near-Earth space through technical ferrographitization, R. A. Sizov. This publication is the second applied edition to the books by R. A. Sizov “Matter, Antimatter and Energy Environment - Physical Triad of the Real World”, in which, based on the discovered…

Mechanics.

weight(kg)

Electric charge(C)

Trajectory

Distance traveled or just the path( l) -

Moving- this is a vectorS

Define and indicate the unit of measurement for speed.

Speed- vector physical quantity characterizing the speed of movement of a point and the direction of this movement. [V]=m s

Define and indicate the unit of measurement for acceleration.

Acceleration- vector physical quantity characterizing the speed of change in the magnitude and direction of velocity and equal to the increment of the velocity vector per unit time:

Define and indicate the unit of measurement for radius of curvature.

Radius of curvature- a scalar physical quantity inverse to the curvature C at a given point of the curve and equal to the radius of the circle tangent to the trajectory at this point. The center of such a circle is called the center of curvature for a given point on the curve. The radius of curvature is determined: R = C -1 = , [R]=1m/rad.

Define and indicate the unit of measurement of curvature

Trajectories.

Path curvature– physical quantity equal to , where is the angle between the tangents drawn at 2 points of the trajectory; - the length of the trajectory between these points. How< , тем кривизна меньше. В окружности 2 пи радиант = .

Define and indicate the unit of measurement for angular velocity.

Angular velocity- vector physical quantity characterizing the speed of change in angular position and equal to the angle of rotation per unit. time: . [w]= 1 rad/s=1s -1

Define and indicate the unit of measure for the period.

Period(T) is a scalar physical quantity equal to the time of one full revolution of a body around its axis or the time of a full revolution of a point along a circle. where N is the number of revolutions in a time equal to t. [T]=1c.

Define and indicate the unit of frequency.

Frequency- scalar physical quantity equal to the number of revolutions per unit time: . =1/s.

Define and indicate the unit of measurement of body impulse (amount of motion).

Pulse– vector physical quantity equal to the product of mass and velocity vector. . [p]=kg m/s.

Define and indicate the unit of measurement for force impulse.

Impulse force– vector physical quantity equal to the product of force and the time of its action. [N]=N·s.

Define and indicate the unit of measurement for work.

Work of force- a scalar physical quantity characterizing the action of a force and equal to the scalar product of the force vector and the displacement vector: where is the projection of the force onto the direction of displacement, is the angle between the directions of force and displacement (velocity). [A]= =1N m.

Define and indicate the unit of measurement for power.

Power- a scalar physical quantity characterizing the speed of work and equal to the work done per unit of time: . [N]=1 W=1J/1s.

Define potential forces.

Potential or conservative forces - forces whose work when moving a body is independent of the trajectory of the body and is determined only by the initial and final positions of the body.

Define dissipative (non-potential) forces.

Non-potential forces are forces, when acting on a mechanical system, its total mechanical energy decreases, turning into other non-mechanical forms of energy.

Define leverage.

Shoulder of strength called distance between the axis and the straight line along which the force acts(distance x measured along the O axis x perpendicular to the given axis and force).

Define the moment of force about a point.

Moment of force about a certain point O- a vector physical quantity equal to the vector product of the radius vector drawn from a given point O to the point of application of the force and the force vector. M= r * F= . [M] SI = 1 N m = 1 kg m 2 / s 2

Define an absolutely rigid body.

Absolutely solid body- a body whose deformations can be neglected.

Conservation of momentum.

Law of conservation of momentum:the momentum of a closed system of bodies is a constant quantity.

Mechanics.

1. Indicate the unit of measurement for the concepts: force (1 N = 1 kg m/s 2)

weight(kg)

Electric charge(C)

Define the concepts: movement, path, trajectory.

Trajectory- an imaginary line along which the body moves

Distance traveled or just the path( l) -length of the path along which the body moved

Moving- this is a vectorS, directed from the starting point to the ending point

Path is a physical quantity equal to the length

trajectories between the initial position of the body and

its final position. Designated l.

Path units are units of length (m, cm, km,...)

but the basic unit of length is the SI meter. It is written like this

The distance between points A and C is not equal to the length of the path. This is another physical quantity. It's called displacement. Movement has not only a numerical value, but also a certain direction, which depends on the location of the starting and ending points of body movement. Quantities that have not only a modulus (numerical value), but also a direction are called vector quantities or just vectors.

Moving – this is a vector physical quantity that characterizes the change in the position of a body in space, equal to the length of the segment connecting the point of the initial position of the body with the point of its final position. The movement is directed from the initial position to the final one.

Denoted by . Unit.

Quantities that have no direction, such as path, mass, temperature, are called scalar quantities or scalars.

Can path and movement be equal?

If a body or a material point (MP) moves along a straight line, and always in the same direction, then the path and displacement coincide, i.e. numerically they are equal. So if a stone falls vertically into a gorge 100 m deep, then its movement will be directed downward and s = 100 m. Path l =100 m.

If a body makes several movements, then they are added, but not in the same way as numerical values ​​are added, but according to other rules, according to the rules for adding vectors. You will soon go through them in your math course. For now, let's look at an example.

To get to the bus stop, Pyotr Sergeevich walks first through the courtyard 300 m to the west, and then along the avenue 400 m to the north. Find the displacement of Pyotr Sergeevich and compare it with the distance traveled.

Given: s 1 = 300 m; s 2 = 400 m.

______________________

Solution:

l = s 1 + s 2 = 300 m +400 m = 700 m.

To find the displacement, you need to find out the length of the segment connecting the initial position of the body and the final position. This is the length of the vector s.

Before us is a right triangle with known legs (300 and

400 m). Let's use the Pythagorean theorem to find the length of the hypotenuse s:

Thus, the path traveled by a person is greater than the displacement by 200 m.

If, suppose, Pyotr Sergeevich, having reached the stop, suddenly decided to turn back and moved in the opposite direction, then the length of his path would be 1400 m, and the displacement would be 0 m.

Reference system.

To solve the basic problem of mechanics means to indicate where the body will be at any given moment in time. In other words, calculate the coordinates of the body. But here’s the catch: where will we count the coordinates from?

You can, of course, take geographic coordinates - longitude and latitude, but! Firstly, the body (MT) can move outside the planet Earth. Secondly, the geographic coordinate system does not take into account the three-dimensionality of our space.

First you need to choose reference body. This is so important that otherwise we will find ourselves in a situation similar to that presented in R. Stevenson's novel “Treasure Island”. Having buried the main part of the treasure, Captain Flint left a map and description of the place.

Tall tree of Spy Mountain. The direction is from the tree along the shade at noon. Walk a hundred feet. Turn towards west. Walk ten fathoms. Dig to a depth of ten inches.

The disadvantage of describing the place where the treasure lies is that the tree, which in this problem is the reference body, cannot be found using the specified characteristics.

This example shows the importance of choice bodies of reference – any body from which the coordinates of the position of a moving material point are measured.

Look at the drawing. As a moving object, take: 1) a yacht; 2) seagull. Take as a body of reference: a) a rock on the shore; b) captain of the yacht; c) a flying seagull. How does the nature of the movement of a moving object and its coordinates depend on the choice of the reference body?

When describing the features of the movement of a particular body, it is important to indicate in relation to which body of reference the characteristics are given.

Let's try to enter the coordinates of the body or MT. Let's use a rectangular Cartesian XYZ coordinate system with the origin at point O. We place the origin of the reference system where the reference body is located. From this point we draw three mutually perpendicular coordinate axes OX, OY, OZ. Now the coordinates of the material point (x;y;z) can be indicated relative to the reference body.

To study body movement (BMT), you also need a watch or a device for measuring time. We will associate the start of the countdown with a specific event. Most often this is the beginning of body movement (MT).

The combination of a reference body, a coordinate system associated with the reference body and a device for measuring time intervals is called reference system (CO) .

If a stationary body is chosen as the reference body, then the reference system will also be stationary (NSO). Most often, the surface of the Earth is chosen as a stationary body of reference. You can choose a moving body as the reference body and get moving frame of reference(PSO).

Look at Figure 1. A three-dimensional coordinate system allows you to specify the position in space of any point. For example, the coordinates of point F located on the column are equal to (6; 3; 1).

Think! Which coordinate system will you choose when solving problems related to movement:

1) a cyclist participates in competitions on a cycling track;

2) a fly crawls on the glass;

3) a fly flies around the kitchen;

4) the truck is moving along a straight section of the highway;

5) a person goes up in an elevator;

6) the projectile takes off and flies from the muzzle of the gun.

Exercise 1.

1. Select in Fig. 3 the cases in which mechanical movement occurs.

3.There are two operators at the flight control center. One controls the orbital parameters of the Mir station, and the other docks the Progress spacecraft with this station. Which operator can consider the Mir station to be a material point?

4. To study the movement of a fighter plane and a hot air balloon (Fig. 4), the rectangular coordinate system XOYZ was chosen. Describe the frame of reference that is used here. Could simpler coordinate systems be used?

5. The athlete ran a 400-meter distance (Fig. 5). Find the movement of the athlete and the path traveled by him.

6. Figure 6 shows a leaf of a plant on which a snail is crawling. Using a scale grid, calculate the path traveled by the snail from point A to point B and from point B to point C.

7. The car, having driven along a straight section of the highway from a gas station to the nearest populated area, returned back. Calculate the modulus of displacement of the machine and the distance traveled by it. What can be said about the relationship between the displacement module and the distance traveled if the car only traveled from a gas station to a populated area?

The position of a material point is determined in relation to some other, arbitrarily chosen body, called reference body. Contacts him frame of reference– a set of coordinate systems and clocks associated with a reference body.

In the Cartesian coordinate system, the position of point A at a given time relative to this system is characterized by three coordinates x, y and z or a radius vector r – a vector drawn from the origin of the coordinate system to a given point. When a material point moves, its coordinates change over time. r=r(t) or x=x(t), y=y(t), z=z(t) – kinematic equations of a material point.

The main task of mechanics– knowing the state of the system at some initial moment of time t 0 , as well as the laws governing the movement, determine the state of the system at all subsequent moments of time t.

Trajectory movement of a material point - a line described by this point in space. Depending on the shape of the trajectory, there are rectilinear And curvilinear point movement. If the trajectory of a point is a flat curve, i.e. lies entirely in one plane, then the motion of the point is called flat.

The length of the section of the trajectory AB traversed by the material point since the start of time is called path lengthΔs is a scalar function of time: Δs=Δs(t). Unit of measurement – meter(m) – the length of the path traveled by light in a vacuum in 1/299792458 s.

IV. Vector method of specifying movement

Radius vector r – a vector drawn from the origin of the coordinate system to a given point. Vector Δ r=r-r 0 , drawn from the initial position of a moving point to its position at a given time is called moving(increment of the radius vector of a point over the considered period of time).

Average speed vector< v> called the increment ratio Δ r radius vector of a point to the time interval Δt: (1). The direction of the average speed coincides with the direction Δ r.With an unlimited decrease in Δt, the average speed tends to a limiting value, which is called instantaneous speedv. Instantaneous speed is the speed of a body at a given moment of time and at a given point of the trajectory: (2). Instantaneous speed v is a vector quantity equal to the first derivative of the radius vector of a moving point with respect to time.

To characterize the speed of change of speed v points in mechanics, a vector physical quantity called acceleration.

Medium acceleration uneven motion in the interval from t to t+Δt is called a vector quantity equal to the ratio of the change in speed Δ v to the time interval Δt:

Instantaneous acceleration a material point at time t will be the limit of average acceleration: (4). Acceleration A is a vector quantity equal to the first derivative of speed with respect to time.

V. Coordinate method of specifying movement

The position of point M can be characterized by the radius vector r or three coordinates x, y and z: M(x,y,z). The radius vector can be represented as the sum of three vectors directed along the coordinate axes: (5).

From the definition of speed (6). Comparing (5) and (6) we have: (7). Taking into account (7), formula (6) can be written (8). The speed module can be found:(9).

Similarly for the acceleration vector:

(10),

(11),

A natural way to define movement (describing movement using trajectory parameters)

The movement is described by the formula s=s(t). Each point of the trajectory is characterized by its value s. The radius vector is a function of s and the trajectory can be given by the equation r=r(s). Then r=r(t) can be represented as a complex function r. Let's differentiate (14). Value Δs – distance between two points along the trajectory, |Δ r| - the distance between them in a straight line. As the points get closer, the difference decreases. , Where τ – unit vector tangent to the trajectory. , then (13) has the form v=τ v(15). Therefore, the speed is directed tangentially to the trajectory.

Acceleration can be directed at any angle to the tangent to the trajectory of motion. From the definition of acceleration (16). If τ is tangent to the trajectory, then is a vector perpendicular to this tangent, i.e. directed normally. Unit vector, in the normal direction is denoted n. The value of the vector is 1/R, where R is the radius of curvature of the trajectory.

A point located at a distance from the path and R in the direction of the normal n, is called the center of curvature of the trajectory. Then (17). Taking into account the above, formula (16) can be written: (18).

The total acceleration consists of two mutually perpendicular vectors: directed along the trajectory of motion and called tangential, and acceleration directed perpendicular to the trajectory along the normal, i.e. to the center of curvature of the trajectory and called normal.

We find the absolute value of the total acceleration: (19).

Lecture 2 Movement of a material point in a circle. Angular displacement, angular velocity, angular acceleration. Relationship between linear and angular kinematic quantities. Vectors of angular velocity and acceleration.

Lecture outline

Kinematics of rotational motion

In rotational motion, the measure of displacement of the entire body over a short period of time dt is the vector dφ elementary body rotation. Elementary turns (denoted by or) can be considered as pseudovectors (as it were).

Angular movement - a vector quantity whose magnitude is equal to the angle of rotation, and the direction coincides with the direction of translational motion right screw (directed along the axis of rotation so that when viewed from its end, the rotation of the body appears to be occurring counterclockwise). The unit of angular displacement is rad.

The rate of change in angular displacement over time is characterized by angular velocity ω . The angular velocity of a rigid body is a vector physical quantity that characterizes the rate of change in the angular displacement of a body over time and is equal to the angular displacement performed by the body per unit time:

Directed vector ω along the axis of rotation in the same direction as dφ (according to the right screw rule). Unit of angular velocity - rad/s

The rate of change in angular velocity over time is characterized by angular acceleration ε

(2).

The vector ε is directed along the axis of rotation in the same direction as dω, i.e. with accelerated rotation, with slow rotation.

The unit of angular acceleration is rad/s 2 .

During the time dt an arbitrary point of a rigid body A move to dr, having walked the path ds. From the figure it is clear that dr equal to the vector product of the angular displacement dφ to radius – point vector r : dr =[ dφ · r ] (3).

Linear speed of a point is related to the angular velocity and radius of the trajectory by the relation:

In vector form, the formula for linear speed can be written as vector product: (4)

By definition of the vector product its module is equal to , where is the angle between the vectors and, and the direction coincides with the direction of translational motion of the right propeller as it rotates from to.

Let's differentiate (4) with respect to time:

Considering that - linear acceleration, - angular acceleration, and - linear velocity, we obtain:

The first vector on the right side is directed tangent to the trajectory of the point. It characterizes the change in linear velocity modulus. Therefore, this vector is the tangential acceleration of the point: a τ =[ ε · r ] (7). The tangential acceleration module is equal to a τ = ε · r. The second vector in (6) is directed towards the center of the circle and characterizes the change in the direction of linear velocity. This vector is the normal acceleration of the point: a n =[ ω · v ] (8). Its modulus is equal to a n =ω·v or taking into account that v = ω· r, a n = ω 2 · r = v 2 / r (9).

Special cases of rotational motion

With uniform rotation: , hence .

Uniform rotation can be characterized rotation period T- the time it takes for a point to complete one full revolution,

Rotational speed - the number of full revolutions made by a body during its uniform motion in a circle, per unit of time: (11)

Speed ​​unit - hertz (Hz).

With uniformly accelerated rotational motion :

Lecture 3 Newton's first law. Strength. The principle of independence of acting forces. Resultant force. Weight. Newton's second law. Pulse. Law of conservation of momentum. Newton's third law. Moment of impulse of a material point, moment of force, moment of inertia.

Lecture outline

Newton's first law

Newton's second law

Newton's third law

Moment of impulse of a material point, moment of force, moment of inertia

Newton's first law. Weight. Strength

Newton's first law: There are reference systems relative to which bodies move rectilinearly and uniformly or are at rest if no forces act on them or the action of the forces is compensated.

Newton's first law is satisfied only in the inertial frame of reference and asserts the existence of the inertial frame of reference.

Inertia- this is the property of bodies to strive to keep their speed constant.

Inertia call the property of bodies to prevent a change in speed under the influence of an applied force.

Body weight– this is a physical quantity that is a quantitative measure of inertia, it is a scalar additive quantity. Additivity of mass is that the mass of a system of bodies is always equal to the sum of the masses of each body separately. Weight– the basic unit of the SI system.

One form of interaction is mechanical interaction. Mechanical interaction causes deformation of bodies, as well as a change in their speed.

Strength– this is a vector quantity that is a measure of the mechanical impact on the body from other bodies, or fields, as a result of which the body acquires acceleration or changes its shape and size (deforms). Force is characterized by its module, direction of action, and point of application to the body.