When the path and movement coincide. Differences of moving and path

If we take into account the physical processes in the household sphere, then many of them seem very well. Therefore, the concepts of the path and movement are perceived as the same thing, the difference is only that the first is a description of the action, and the second is the result of action. But if you ask for a refinement to information sources, it will be possible to immediately find a significant difference between these operations.

What is the path?

The path is a movement as a result, which is the change in the location of the subject or person. This value refers to scalar, therefore it does not have directions, but it can be used to determine the distance traveled.

The path can be performed by the following images:

In a straight line.
Curvilinear.
Round.
Other methods (for example, a zigzag trajectory) are possible.

The way can never be negative and decrease during the time. Measuring the path is carried out in meters. Most often, in physics, the letter is used to designate the path S.In rare cases, the letter L. Using the path it is impossible to foresee where the subject we need will be at a certain point in time.

Features of movement

Movement is the difference between the initial and final point of the arrangement of a person or an item in space after some way was overcome.

The movement value is always positive, and also has a clear direction.

The coincidence between the movement and the way is possible only if the path was carried out in a straight line, and the direction did not change.

With the help of displacement, you can calculate where a person or item was located at a certain stage of time.

To refer to the movement, the letter S is used, but since the movement is a vector value that the arrow is installed above this letter, which indicates that moving is a vector. Unfortunately, it increases the confusion between the way and the movement is still the fact that both concepts can also be denoted by the letter L.

What is common between concepts path and movement?

Despite the fact that the path and movement are absolutely different concepts there are certain elements that contribute to the fact that the concepts are confused:

The path and movement can always be only positive values.
To indicate the path and movement, one and the same letter L.

Even considering the fact that these concepts have only two common elements, their meaning is so great that makes the confusion of many people. Especially arising from schoolchildren during the study of physics.

The main differences between the concepts of the path and movement?

These concepts have a number of differences that will always help determine which value to be in front of you, path or movement:

The path is the primary concept, and the movement is secondary. For example, moving determines the difference between the initial and final point of the person in space after overcoming some way. Accordingly, it is impossible to get the amount of movement without using the initial path.
For the path plays a huge role. The beginning of the movement plays, and to determine the movement, the beginning of the movement is absolutely no need.
The main difference between the values \u200b\u200bis that the path does not have directions, and it has it. For example, the path is carried out only directly - forward, and moving allows and movement back.
In addition, concepts are different. The path refers to a scalar value, and moving to the vector.
Executive method. For example, the path is calculated using a common segment passed, and moving, in turn, is calculated by changing the location of the object in space.
The way can never be zero, and a value is allowed to zero.

Having studied these differences, you can immediately understand what the difference between the concepts of the path and movement is, and no longer be confused.

The difference between the way and moving on the examples

In order to quickly understand the difference between the way and moving, you can use certain examples:

Machine was made by 2 meters forward and 2 meters ago. The path is the sum of all distance traveled, respectively, it is 4 meters. A movement is the initial and end point, so in this case it is zero.
In addition, the difference between the way and moving can be considered on its own experience. It is necessary to stand up on the start of the 400-meter treadmill and run two circles (the second round will end at the starting point). As a result, it turns out that the path was 800 meters (400 + 400), and moving is 0, since the initial and endpoints coincide.
Abandoned up the ball has reached a height of 15 meters, and then fell to the ground. In this case, the path will be equal to 30 meters, because 15 meters up and 15 meters down plot. And the movement will be equal to 0, due to the fact that the ball returned to its original position.

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

The trajectory of the bee

Way - This is the length of the trajectory. That is, the length of that, perhaps the curve of the line, according to which the body was moving. Path scalar value! Move - Vector value! This is a vector that is spent from the initial point of body departure to the end point. It has a numerical value equal to the length of the vector. The path and movement are essentially different physical quantities.

Way and displacement designations You can meet Miscellaneous:

Amount of movements

Suppose for a period of time T 1, the body was moving S 1, and during the next period of time T 2 - movement S 2. Then for all the time movement movement S 3 - this is a vector sum

Uniform traffic

Movement with a constant module and speed. What does it mean? Consider the movement of the machine. If it goes in a straight line, on the speedometer the same speed value (speed module), then this is a uniform movement. It is worth a machine to change the direction (turn), it will mean that the speed vector has changed its direction. The speed vector is directed there by the same car. Such a 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 the body movement

Is it possible to read the carousel to be uniform (if there is no acceleration or braking)? It is impossible, the direction of movement is constantly changing, which means the speed vector. From reasoning, we can conclude that uniform movement - it is always movement in a straight line! So, with uniform movement, the path and movement are the same (explain why).

It is easy to imagine that with a uniform movement for any equal intervals, the body will move to the same distance.

Class: 9

Objectives lesson:

Educational:
- Enter the concepts of "movement", "path", "trajectory".
Developing:
- develop logical thinking, proper physical speech, use the appropriate terminology.
Educational:
- achieve high class activity, attention, focus of students.

Equipment:

plastic bottle with a capacity of 0.33 liters with water and with a scale;
medical bottle with a capacity of 10 ml (or small test tube) with a scale.

Demonstrations: Determining the movement and the path traveled.

During the classes

1. Actualization of knowledge.

- Hello guys! Sit down! Today we will continue to study the topic "The laws of interaction and movement of the bodies" and at the lesson we will get acquainted with three new concepts (terms) concerning this topic. In the meantime, check your homework fulfillment with this lesson.

2. Checking homework.

Before the lesson, one student discharges the following homework on the board:

Two students are distributed cards with individual tasks that are performed during the verification of the UPR. 1 page 9 textbook.

1. What coordinate system (one-dimensional, two-dimensional, three-dimensional) should be selected to determine the position of the phone:

a) the tractor in the field;
b) helicopter in the sky;
c) train
d) chess piece on the board.

2. An expression is given: S \u003d υ 0 · T + (A · T 2) / 2, Express: A, υ 0

1. What coordinate system (one-dimensional, two-dimensional, three-dimensional) should be selected to determine the position of such tel:

a) chandelier in the room;
b) elevator;
c) submarine;
D) Airplane on the runway.

2. An expression is given: S \u003d (υ 2 - υ 0 2) / 2 · A, express: υ 2, υ 0 2.

3. Studying a new theoretical material.

Changes in the coordinates of the body is associated with the value entered to describe the movement - Move.

The movement of the body (material point) is a vector connecting the initial position of the body with its subsequent position.

Move is accepted to signify the letter. In SI, movement is measured in meters (m).

- [M] - meter.

Movement - Value vector those. In addition to the numerical value, the direction also has. Vector image is depicted in the form Cutwhich begins at some point and ends with the edge indicating the direction. Such a cut-arrow is called vector.

- vector spent from point m in m 1

Know the travel vector - it means to know its direction and module. The vector module is a scalar, i.e. Numerical value. Knowing the initial position and vector movement of the body, you can determine where the body is located.

In the process of movement, the material point occupies various positions in space relative to the selected reference system. In this case, the moving point "describes" in the space some line. Sometimes this line is visible, - for example, a highly flying plane can leave a trace in the sky. More familiar example - a trail of a piece of chalk on the board.

Imaginary line in space for which the body is moving is called Trajectory Body movement.

The trajectory of the body movement is the continuous line that the moving body describes (considered as a material point) relative to the selected reference system.

Movement in which all Points body moving in Equestrian trajectories, called additional.

Very often the trajectory is an invisible line. Trajectory moving point maybe straightor crookedline. Accordingly, the form of the trajectory traffic it happens straight and kriviolinene.

Trajectory length is WAY. The path is a scalar value and is denoted by the letter L. The path increases if the body is moving. And remains unchanged if the body rests. In this way, the path cannot be reduced over time.

The movement module and the path may match the value, only if the body moves along the straight line in one direction.

What is the path from moving? These two concepts are often mixed, although in fact they differ very much from each other. Consider these differences: ( Appendix 3.) (distributed in the form of cards to each student)

The path is a scalar value and is characterized only by a numerical value.
Moving is a vector value and is characterized by both a numerical value (module) and direction.
When the body moves, the path can only increase, and the movement module can both increase and decrease.
If the body returned to the starting point, its movement is zero, and the path is not equal to the zero.

	Way	Move
Definition	The length of the trajectory described by the body for a certain time	The vector connecting the initial position of the body with its subsequent position
Designation	l [m]	S [m]
Character of physical quantities	Scalar, i.e. Determined only by numeric value	Vector, i.e. determined by the numerical value (module) and direction
The need for administration	Knowing the initial position of the body and the path L, passed during the period of T, it is impossible to determine the position of the body at a given moment T	Knowing the initial position of the body and S over time t, the position of the body is uniquely determined at a given moment of time T
	l \u003d S in the case of a straight line without returns

4. Demonstration of experience (students perform independently in their places outside the parties, the teacher, together with students, performs the demonstration of this experience)

Fill with water to the neck plastic bottle with a scale.
The bottle with a scale fill with water to 1/5 of its volume.
Tilt the bottle so that the water goes to the neck, but did not flow out of the bottle.
Quickly lower the bottler with water in a bottle (without closing it with a plug) so that the neck of the bottle is in the water. The bottle floats on the surface of the water in the bottle. Part of the water at the same time out of the bottle
Screw the bottle cap.
Squeezing the side walls of the bottle, lower the float on the bottom of the bottle.

Relaxing the pressure on the walls of the bottle, achieve the float. Determine the path and moving the float: ________________________________________________________
Lower the float to the bottom of the bottle. Determine the path and moving the float: ______________________________________________________________________________
Cut the float to float and drown. What is the path and moving the float in this case? _______________________________________________________________________________________

5. Exercises and questions for repetition.

Way or movement we pay when traveling to a taxi? (Way)
The ball fell from a height of 3 m, bounced off the floor and was caught at an altitude of 1 m. Find the path and moving the ball. (Path - 4 m, movement - 2 m)

6. The result of the lesson.

Repetition of the concepts of the lesson:

- movement;
- trajectory;
- way.

7. Homework.

§ 2 textbooks, questions after paragraph, exercise 2 (p.12) textbook, repeat the performance of the lesson's experience.

Bibliography

1. Pryrickin A.V., Godnik EM. Physics. 9 CL: student. For the general formation. Measories - 9th ed., Stereotype. - M.: Drop, 2005.

Section 1 Mechanics

Chapter 1: O S N O V I N E M A T and K and

Mechanical movement. Trajectory. Path and movement. Addition of speeds

Mechanical movement of bodyit is called the change in its position in space relative to other bodies over time.

Mechanical movement bodies studies mechanics. The section of mechanics describing the geometric properties of the movement without taking into account the masses of bodies and the current forces is called kinematics .

Mechanical movement relative. To determine the position of the body in space, you need to know its coordinates. To determine the coordinates of the material point, it follows, first of all, choose the body of reference and associate the coordinate system with it.

Body referencethe body is called relative to which the position of other bodies is determined. The point of reference is chosen arbitrarily. This can be anything: land, building, car, motor ship, etc.

The coordinate system, the reference body with which it is connected, and the pointing of the time is formed system reference , regarding which the body movement is considered (Fig.1.1).

Body, dimensions, shape and structure of which can be neglected when studying this mechanical movement, is called material point . The material point can be considered the body, the dimensions of which are much less than the distances characteristic of the movement under consideration.

Trajectory This line is moving the body.

Depending on the type of the trajectory of the movement, they are divided into straight and curvilinear

Way- this is the length of the trajectory ℓ (m) (fig.1.2)

The vector conducted from the initial position of the particle into its final position is called movement this particle size is given time.

Unlike the path, moving is not scalar, and the vector value, as it shows not only for what distance, but in which direction the body has shifted for this time.

Module of travel vector (That is, the length of the segment that connects the initial and ending points of the movement) can be equal to the path traveled or be less than the path traveled. But never the move module can not be more traveled path. For example, if the car moves from the point A to the point b, then the module of the movement vector is less than the path passed. The path and the movement module turn out to be equal only in one single case when the body moves in a straight line.

Speed - This is a vector quantitative characteristic of body movement

average speed - this is a physical value equal to the ratio of the movement of the point to the time interval

The direction of the midway vector coincides with the direction of the movement vector.

Instant speed That is, the speed at the moment time is the vector physical value equal to the limit to which the average speed is striving for an infinite decrease in the period of time Δt.

The instantaneous velocity vector is aimed at tangent to the trajectory of the movement (Fig. 1.3).

In the system, the speed is measured in meters per second (m / s), that is, the unit of speed is considered to be the speed of such a uniform straight movement, at which in one second the body goes into one meter. Often speed is measured in kilometers per hour.

or 1.

Addition of speeds

Any mechanical phenomena are considered in any reference system: the movement makes sense only relative to other bodies. When analyzing the movement of the same body in different reference systems, all the kinematic characteristics of the movement (path, trajectory, moving, speed, acceleration) turn out to be different.

For example, a passenger train moves along the railway at a speed of 60km / h. On the car of this train there is a person with a speed of 5km / h. If you consider the railway stationary and take it for the reference system, then the speed of man is relative to the railway, will be equal to the addition of the speeds of the train and man, that is

60km / h + 5 km / h \u003d 65 km / h, if a person goes in the same direction as the train and

60km / h - 5 km / h \u003d 55 km / h, if a person comes against the direction of the train movement.

However, this is true only in this case, if a person and train move on one line. If a person will move at an angle, then it is necessary to take into account this angle, and the fact that speed is a vector magnitude.

Consider the example described above in more detail - with the details and pictures.

So, in our case, the railway is a fixed reference system. The train that moves along this road is a movable reference system. The car in which a person goes is part of the train. The human speed relative to the car (relative to the movable reference system) is 5km / h. Denote her letter. The speed of the train, (and hence the car) relative to the fixed reference system (that is, relative to the railway) is 60 km / h. Denote her letter. In other words, the speed of the train is the speed of the movable reference system relative to the fixed reference system.

The speed of man relative to the railway (relatively fixed reference system) is still unknown. Denote her letter.

We connect with a fixed reference system (Fig. 1.4) Hoy coordinate system, and with a movable reference system - x n. We will now determine the speed of a person relative to the fixed reference system, that is, relative to the railway.

Over the short period of time, the following events occur:

· Man moves relative to the car for distance

· Car moves relative to the railway for distance

Then over this time, the movement of man relative to the railway:

it the law of addition of movements . In our example, the movement of man relative to the railway is equal to the sum of human movements relative to the car and the car on the railway.

Dividing both parts of equality at a small period of time DT, for which the move occurred:

We get:

Figure 1.3.

This is the law speed addition: with the bodies of the body relative to the fixed reference system is equal to the amount of body velocities in the movable reference system and the speed of the mobile reference system relatively fixed. This term has other values, see Movement (values).

Move (in kinematics) - change in the position of the physical body in space over time relative to the selected reference system.

In relation to the motion of the material point movement Call a vector characterizing this change. It has the property of additivity. It is usually denoted by the S → (\\ DisplayStyle (\\ VEC (S)) symbol) - from Ital. s.postAmento.

The module of the S → (\\ DisplayStyle (\\ VEC (S))) is a movement module, in the international system of units (C) is measured in meters; In the SGS system - in centimeters.

You can determine the movement as a change in the radius-vector point: Δ R → (\\ DisplayStyle \\ Delta (\\ VEC (R))).

The movement module coincides with the passage passed in that and only if the speed direction does not change when driving. At the same time, the trajectory will be a straight line. In any other case, for example, in curvilinear movement, it follows from a triangle inequality that the path is strictly greater.

Instant point speed is defined as the limit of the relationship of the movement to a small period of time for which it is performed. More strictly:

V → \u003d LIM Δ T → 0 Δ R → Δ T \u003d DR → DT (\\ displayStyle (\\ VEC (V)) \u003d \\ LIM \\ Limits _ (\\ Delta T \\ To 0) (\\ FRAC (\\ Delta (\\ VEC (r))) (\\ Delta T)) \u003d (\\ FRAC (D (\\ VEC (R))) (DT))).

III. Trajectory, path and movement

The position of the material point is determined in relation to any other, an arbitrarily selected body called body reference. Associated with him reference system - A combination of the coordinate system and clock-related coupling.

In the Cartesian coordinate system, the position of the point and at the moment the time in relation to this system is characterized by three x, y and z coordinates or a radius-vector r.– The vector spent from the start of the coordinate system at this point. When the material point is moved, its coordinate changes over time. r.=r.(t) or x \u003d x (t), y \u003d y (t), z \u003d z (t) - kinematic equations of material point.

The main task of mechanics - Knowing the state of the system in some initial time t 0, as well as the laws controlling the movement, determine the state states at all subsequent points in time t.

Trajectory The motion of the material point is the line described by this point in space. Depending on the form of the trajectory distinguish straightforward and curvilinear Movement point. If the path trajectory is a flat curve, i.e. entirely lies in the same plane, then the movement of the point is called flat.

The length of the trajectory of the AB trajectory passed by the material point since the start of time is called long path ΔS and is a scalar time function: ΔS \u003d ΔS (T). Unit of measurement - meter (m) - the length of the path passing by light in vacuo for 1/299792458 p.

IV. Vector Movement Movement Movement

Radius vector r.– The vector spent from the start of the coordinate system at this point. Vector Δ. r.=r.-r. 0 conducted from the initial position of the moving point to its position at the moment is called movement (The increment of the radius-vector point during the time period in question).

The vector of the average velocity V\u003e is the ratio of the increment of ΔR of the radius-vector point to the time interval Δt: (1). The direction of the average speed coincides with the direction ΔR. In an unlimited decrease in ΔT, the average rate strive to the limit value, which is called instantaneous speed V. Instant speed is the body rate at a given time and at this point of the trajectory: (2). Instant speed There is a vector magnitude equal to the first derivative of the radius-vector moving point in time.

To characterize speed change v.points in mechanics introduced vector physical quantity called acceleration.

Average acceleration Uneven motion in the interval from T to T + ΔT is called a vector value equal to the ratio of the speed change Δ v. By time interval Δt:

Instant acceleration A. The material point at time t will be the limit of the average acceleration: (4). Acceleration but There is a vector value equal to the first time derivative of time.

V. Coordinate Movement Movement Method

Position point m can be characterized by a radius - vector r. or three coordinates x, y and z: m (x, y, z). 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). Considering (7), formula (6) can be recorded (8). Speed \u200b\u200bmodule can be found: (9).

Similar to the acceleration vector:

(10),

(11),

Natural way to set the movement (motion descriptions using the path parameters)

The movement is described by the formula S \u003d S (T). Each point of the trajectory is characterized by its value s. Radius - vector is a function from S and the trajectory can be set by the equation r.=r.(s). Then r.=r.(t) can be represented as a complex function r.. Differentiation (14). The value ΔS is the distance between two points along the trajectory, | δ r.| - Distance between them in a straight line. As the points are rapprocheted, the difference is reduced. where τ - single vector tangent to the trajectory. , then (13) has the view v.=τ v (15). Consequently, the speed is aimed at tangent to the trajectory.

Acceleration can be directed at any angle to the tangent to the trajectory of the movement. From the definition of acceleration (sixteen). If a τ - tangent to the trajectory, then - vector perpendicular to this tangential, i.e. Directed by normal. Unit vector, in the direction of the Normal is indicated n.. The vector value is 1 / R, where R is the radius of the curvature of the trajectory.

Point, distinguished from the trajectory at a distance and r in the direction of normal n.is called the center of the curvature of the trajectory. Then (17). Considering the above formula (16) you can write: (18).

Complete acceleration consists of two mutually perpendicular vectors: directed along the trajectory of movement and called tangential, and acceleration directed perpendicular to the path of normal, i.e. To the center of curvature of the trajectory and called normal.

The absolute value of complete acceleration will find: (19).

Lecture 2 Movement of the material point around the circumference. Angular movement, angular speed, angular acceleration. Communication between linear and angular kinematic values. Angular velocity and acceleration vectors.

Plan lectures

Kinematics of the rotational motion

With rotational motion, the measure of the movement of the entire body over a short period of time DT is vector dφ. Elementary body rotation. Elementary turns (designated or) can be considered as pseudoors (as it were).

Corner moving - vector quantity, the module of which is equal to the angle of rotation, and the direction coincides with the direction of the translational movement right screw (directed along the axis of rotation so that if you look from its end, then the rotation of the body seems to be against the clockwise). The unit of angular movement is happy.

The speed of changes in the angular movement over time characterizes angular velocity ω . The angular velocity of the solid is a vector physical value that characterizes the speed of changes in the angular movement of the body over time and equal to the angular movement committed by the body per unit of time:

Directed vector ω along the axis of rotation to the same side as dφ. (according to the rule of the right screw). Angle speed is rare / s

The speed of changes in the angular velocity is characterized by angular acceleration ε.

(2).

The vector ε is directed along the axis of rotation to the same side as DΩ, i.e. With accelerated rotation, during slow.

Unit of angular acceleration - Rad / C2.

During dt. arbitrary point of solid body A moving to dr., having passed the way ds.. From the figure it is clear that dr. equal to the vector product of angular movement dφ. On the radius - vector point r. : dr. =[ dφ. · r. ] (3).

Linear speed point associated with angular velocity and radius of the trajectory by the ratio:

In vector formula for linear speed can be written as vector art: (4)

By definition of vector work Its module is equal, where - the angle between vectors and, and the direction coincides with the direction of the forward movement of the right screw during its rotation from K.

Differentiation (4) in time:

Considering that - linear acceleration, - angular acceleration, and - linear speed, we get:

The first vector in the right part is aimed at a tangent to the trajectory of the point. It characterizes the change in linear speed module. Consequently, this vector is a tangential point acceleration: a. τ =[ ε · r. ] (7). Tangential acceleration module is equal 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 speed. This vector is normal point acceleration: a. n. =[ ω · v. ] (eight). The module is equal to a n \u003d ω · v or considering that v.= ω· r., a. n. = ω 2 · r.= v.2 / r. (9).

Private cases of rotational motion

With uniform rotation: , hence .

Uniform rotation can be characterized rotation period T. - time for which the point makes one full revolution,

Rotation frequency - the number of full revolutions performed by the body with its uniform movement around the circumference, per unit of time: (11)

Unit of rotation frequency - Hertz (Hz).

With an equilibrium rotational motion :

(13), (14) (15).

Lecture 3 First Law Newton. Force. The principle of independence of the current forces. Revulting force. Weight. The second law of Newton. Pulse. The law of preserving the impulse. The third law of Newton. The moment of the momentum of the material point, the moment of strength, the moment of inertia.

Plan lectures

First Law Newton

Second Newton Law

Third Law Newton

Moment of impulse material point, moment of strength, moment of inertia

The first law of Newton. Weight. Force

The first law of Newton: There are reference systems relative to which the bodies move straight and evenly or rest if the forces or the action of the forces are compensated for them.

The first Newton law is performed only in the inertial reference system and approves the existence of an inertial reference system.

Inertia - This property of bodies strive to maintain the speed unchanged.

Inertia Call the property of the bodies to prevent the change in the speed under the action of the applied force.

Body mass - This is a physical value that is a quantitative measure of inertia, this is a scalar additive value. The additivity of the mass It is that the mass of body system is always equal to the sum of the masses of each body separately. Weight - The main unit of the SI system.

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

Force - This is a vector value that is a measure of mechanical impact on the body from other bodies, or fields, as a result of which the body acquires acceleration or changes its shape and dimensions (deformed). The force is characterized by a module, direction of action, an application point to the body.

General methods for determining movements

 1 \u003d x 1  11 + x 2  12 + x 3  13 + ...

 2 \u003d x 1  21 + x 2  22 + x 3  23 + ...

 3 \u003d x 1  31 + x 2  32 + x 3  33 + ...

Abote for permanent forces: A \u003d R R, R - generalized power - any load (focused force, concentrated moment, distributed load),  R - generalized moving (deflection, rotation angle). The designation  Mn means moving in the direction of the generalized force "M", which is caused by the action of the force of the generalized "n". Full move caused by several power factors:  p \u003d  p +  p Q +  P M. Displacement caused by a single force or a single moment:  - specific move . If the unit force p \u003d 1 caused the movement  P, then the total movement caused by the force p, will be:  P \u003d Р R. If the power factors acting on the system, indicate x 1, x 2, x 3, etc. , then move towards each of them:

where x 1  11 \u003d +  11; X 2  12 \u003d +  12; X i  m i \u003d +  m i. The dimension of specific movements:

, J-Joji dimension of 1J \u003d 1NM.

The work of external forces, valid on the elastic system:

.

- Active work with a static action of generalized force on the elastic system is equal to half the product of the final value of force on the final value of the corresponding movement. The work of the internal forces (forces of elasticity) in the case of flat bending:

,

k is a coefficient that takes into account the uneven distribution of tangent stresses along the cross-sectional area depends on the form of section.

Based on the law of conservation of energy: the potential energy U \u003d a.

Theorem on the reciprocity of work (Bethley Theorem) . Two states of elastic CITEME:

 1

1 - moving sent. Forces P 1 from the action of force p 1;

 12 - moving sent. force p 1 from the action of force p 2;

 21 - Moving sent. Forces p 2 from the action of force p 1;

 22 - moving sent. Forces p 2 from force p 2.

A 12 \u003d P 1  12 is the operation of the force p 1 of the first state on the movement in its direction caused by the force of p 2 of the second state. Similarly: a 21 \u003d p 2  21 - the operation of the force p 2 of the second state on the movement in its direction caused by the force of p 1 of the first state. A 12 \u003d A 21. The same result is obtained with any number of forces and moments. Theorem on the reciprocity of work: P 1  12 \u003d p 2  21.

The work of the forces of the first state of movements in their directions caused by the forces of the second state is equal to the work of the forces of the second state on movements in their directions caused by the forces of the first state.

Theorem on the reciprocity of movements (Theorem Maxwell) If p 1 \u003d 1 and p 2 \u003d 1, then p 1  12 \u003d p 2  21, i.e.  12 \u003d  21, in the general case  Mn \u003d  Nm.

For two unit states of the elastic system, the movement in the direction of the first unit force caused by the second unit force equals to move towards the second unit force caused by the first force.

Niversal method for determining movements (linear and rotation angles) - mOLE METHOD. The system is applied by a single generalized force at a point for which a generalized movement is searched. If deflection is determined, the unit force is an immensurate focused force if the angle of rotation is determined, then a dimensionless single moment. In the case of a spatial system, there are six internal components. The generalized movement is determined by the formula (formula or integral of Mora):

The trait above m, q and n indicates that these internal efforts are caused by the action of single force. To calculate the integrals included in the formula, multiply the plots of the relevant effort. The procedure for determining the movement: 1) for a given (actual or cargo) system find expressions M n, n n and q n; 2) in the direction of the desired movement, apply the corresponding unit force (force or moment) corresponding to it; 3) determine efforts

from the action of a single force; 4) Found expressions are substituted into the Mora integral and integrate on specified areas. If the resulting Мn\u003e 0, then the movement coincides with the selected direction of the unit force if

For flat design:

Typically, in determining displacements, they neglect the effect of longitudinal deformations and a shift, which are caused by the longitudinal N and the transverse q forces, only the movements caused by the bend are taken into account. For a flat system will be:

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IN

Mora's integral pathmethod Vereshchagin . Integral

For the case when the plot of a given load has an arbitrary outline, and from a single - straightforward is convenient to determine the graph-analytical method proposed by Vereshchagin.

, where - the area of \u200b\u200bEpur M p from the external load, Y C - the ordinate of the epira from the unit load under the center of gravity of the EPUR M R. The result of multiplication of the EPUR is equal to the product of the area of \u200b\u200bone of the EPUR on the ordinate of another epuras, taken under the center of gravity of the area of \u200b\u200bthe first epura. The ordinate must be taken from a straight line. If both epures are straightforward, then the ordinate can be taken from any.

P

Remote:

. The calculation according to this formula is made in areas, each of which the straight line must be without fractures. The complex Eppura M R is divided into simple geometric shapes, for which it is easier to determine the coordinates of the centers of gravity. With multiplying two EPUs, having a type of trapezium, it is convenient to use the formula:

. The same formula is also suitable for triangular EPUR, if we substitute the appropriate ordinate \u003d 0.

P

Efforts of a uniformly distributed load on the articulated focus of the EPURA beam is built in the form of a convex quadratic parabola whose area

(For fig.

.

, x C \u003d L / 2).

D.

la "deaf" sealing with evenly distributed load we have a concave quadratic parabola for which

;

,

, x C \u003d 3L / 4. Also, you can also get if the Eppure is to present the difference in the triangle area and the area of \u200b\u200bconvex quadratic parabola:

. The "missing" area is considered negative.

Kastigaliano theorem .

- Move the point of application of the generalized force in the direction of its action is equal to a private derivative of potential energy on this force. Neglecting the influence on the movement of axial and transverse forces, we have potential energy:

From!

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What is moving in physics definition?

Sad Roger.

In physics, the movement is the absolute vector of the vector spent from the initial point of the body path to the final. At the same time, the shape of the path on which movement passed (that is, the trajectory is saccinously), as well as the magnitude of this path, does not matter. Let's say, moving Magellan's ships - well, at least that in the end returned (one of three), - equally zero, although the path has passed.

Trifon Lie

Moving can be viewed in two hypostatas. 1. Changing the position of the body in space. Moreover, regardless of the coordinates. 2. Process of movement, i.e. Change position over time. According to claim 1, you can argue, but for this you need to recognize the existence of an absolute (initial) coordinate coordinates.

Movement - Changing the location of a certain physical body in space relative to the reference system used.

This definition is set in kinematics - a subsection of mechanics, studying the movement of the bodies and a mathematical description of the movement.

Movement is the absolute value of the vector (that is, straight) connecting two points of the path (from point A to point b). Moving differs from the way that this is a vector value. This means that if the object came to the same point from which he began, then the movement is zero. And there is no way. The path is the distance that the object overcomed as a result of its movement. To better understand Look at the picture:

What is the path and moving, from the point of view of physics? And in what the difference between them ....

very necessary) I ask you to answer)

User deleted

Alexander Kalapats

The path is a scalar physical value that determines the length of the trajectory site passed by the body for a specified time. The path is a non-negative and non-breaking function of time.
Movement - directional segment (vector) connecting the position of the body at the initial moment of time with its position at the end point in time.
I explain. If you leave home, go to visit to a friend, and come back home, then your path will be equal to the distance between your home and a friend's home, multiplied by two (there and back), and your movement will be zero, since. At the end moment, you will find yourself there, where in the initial, i.e. at home. The path is the distance, length, i.e. the value is scalar, having no directions. Movement - directional, vector quantity, and the direction is given by the sign, i.e. the movement may be negative (if we assume that you reach your friend to a friend you made a move, then when you come from each other, you make moving where minus denotes that you walked in the opposite direction to the one in which it was from home to a friend).

Forserr33 V.