Path and movement. Addition speeds. Mechanical movement. Trajectory. Way. Move

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).

Vector mid speed< v.> called the relation of increment Δ r. Radius-vector points to the time interval Δt: (1). The direction of medium speed coincides with the direction Δ r.. And an unlimited decrease in Δt average speed to strive for the limit value called instant speedv.. Instant speed is the body rate at a given time and at this point of the trajectory: (2). Instant speed v. 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). Unit angular speed - Run / 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.

Angle acceleration unit - Run / C 2.

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 pointassociated 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, - an angular acceleration, a- 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 :

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 massit 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.

Displacement, shift, movement, migration, movement, permutation, rearrangement, transfer, transportation, transition, moving, transmission, travel; Shift, adjustment, telecision, epeyrophoresis, relocation, rolling, pulling, ... ... Synonym dictionary

Moving, movement, cf. (Book.). 1. Action on ch. Move move. Moving in service. 2. Action and condition according to ch. Move moving. Moving the layers of the earth's crust. Explanatory dictionary of Ushakov. D.N. Ushakov. 1935 1940 ... Explanatory Dictionary Ushakov

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

Move, still estay; Street (Yong, Ena); owls. "Who is. Place, translate to another place. P. Decoration. P. Brigada to another site. Displaced persons (persons forcibly resettled from their country). Explanatory dictionary of Ozhegov. S.I. ... ... Explanatory dictionary of Ozhegov

- (Relocation) Moving office, enterprises, etc. To another place. Often, his cause is a merger, absorption. Sometimes employees receive a transfer allowance (Relocation Allowance), which should stimulate them to stay in the service in this ... ... Business Terms Dictionary

move - - Television Themes, Basic Concepts en redupployment ... Technical translator directory

Moving - Move, mm, the value of changes in the position of any point of the element of the window block (as a rule, the imposite of the box or vertical bars of the sash) in the direction of normal to the plane of the product under the influence of wind load. Source: GOST ... ...

move - Material migration in the form of a solution or suspension from one soil horizon in another ... Dictionary on geography

move - 3.14 Movement (TRANSFER) (in relation to the storage location): Change the storage location of the document. Source: GOST R ISO 15489 1 2007: System of standards for information ... Dictionary directory terms of regulatory and technical documentation

move - ▲ Change position, in space Fixed movement Change position in space; transformation of the figure, which retains the distances between the figures; Movement to another place. movement. Progressive movement ... ... The ideographic dictionary of the Russian language

Books

Moving people and cargo in the near-earth space through technical ferrographation, R. A. Sizov. This publication is the second applied edition to the books R. A. Sizova "Matter, Antimatterium and Energy Protection - the Physical Triad of the Real World", in which on the basis of the detected ...
Moving people and cargo in the near-earth space through technical ferrogravitation, Sizov R.A. This publication is the second applied edition to the books R. A. Sizova `Matter, antimatterium and energy promenade - the physical triad of the real world`, in which based on the detected ...

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.

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.

Basic units of measuring values \u200b\u200bin the SI system Such:

a unit of length - meter (1 m),
time - second (1 s),
masses - kilogram (1 kg),
the amount of substance is mole (1 mol),
temperatures - Kelvin (1 K),
electric current forces - amp (1 A),
Reference: Light forces - candela (1 cd, is actually not used in solving school tasks).

When calculating the calculations in the system, the angles are measured in radians.

If the task in physics is not specified, in which units need to be answered, it needs to be given in units of SI system or in derivatives from them the values \u200b\u200bcorresponding to the physical size of which is asked in the task. For example, if the task needs to find speed, and does not say what it needs to be expressed, the answer must be given in m / s.

For convenience, the tasks in physics often have to use dollars (reduction) and multiple (increasing) consoles. They can be applied to any physical size. For example, MM is a millimeter, CT - kilotonne, ns - nanosecond, mg - megagrams, mmol - millimol, MCA - microamper. Remember that there are no double consoles in physics. For example, the ICG is a microgram, and not millikilograms. Note that when adding and subtracting values, you can only operate only the values \u200b\u200bof the same dimension. For example, kilograms can only be folded with kilograms, you can only deduct millimeters from millimeters, and so on. When transferring values, use the following table.

Path and moving

Kinematics They call the section of mechanics in which the movement of bodies is considered without clarifying the reasons for this movement.

Mechanical movement Bodies call the change in its position in space relative to other bodies over time.

Every body has defined dimensions. However, in many tasks mechanics there is no need to indicate the positions of individual parts of the body. If the sizes of the body are small compared to distances to other bodies, then this body can be considered material point. So when moving a car over long distances, you can neglect it long, since the length of the car is small compared to the distances it passes.

It is intuitive that the characteristics of the movement (speed, trajectory, etc.) depend on where we look at it. Therefore, the concept of the reference system is introduced to describe the movement. Reference system (CO) - A combination of the reference body (it is considered absolutely solid), tied to it by the coordinate system, ruler (device measuring), hours and time synchronizer.

Moving over time from one point to another, the body (material point) describes in a given line called the trajectory of the movement of the body.

Displacement of body They call the directional cut line connecting the initial position of the body with its end position. Move there is a vector magnitude. Movement can increase the movement in the process, decrease and becomes zero.

Passed way equal to the length of the trajectory passed by the body for a while. The path is a scalar value. The path cannot be reduced. The path only increases either remains constant (if the body does not move). When the body moves along the curvilinear trajectory, the module (length) of the movement vector is always less than the path traveled.

For uniform (at constant speed) movement path L. Can be found by the formula:

where: v. - body speed, t. - Time for which it moved. When solving kinematics tasks, the movement is usually made of geometric considerations. Often, geometric considerations for the location of the move require the knowledge of the Pythagores theorem.

average speed

Speed - vector quantity characterizing the speed of moving the body in space. Speed \u200b\u200bis medium and instantaneous. Instant speed describes the movement at this particular point in this particular point of space, and the average speed characterizes all the movement as a whole, in general, without describing the details of the movement at each specific site.

Average speed path - This is the ratio of all the way to the whole time of movement:

where: L. full - all the way that the body passed, t. Full - all the time of movement.

The average speed of movement - This is the ratio of all movement to the whole time of movement:

This value is directed as well as the full movement of the body (that is, from the starting point of movement to the end point). At the same time, do not forget that complete movement is not always equal to the algebraic amount of movements at certain stages of movement. The full movement vector is equal to the vector sum of movements at separate stages of the movement.

When solving problems on kinematics, do not make a very common error. The average speed is usually not equal to the average arithmetic body velocity at each stage of movement. The arithmetic average is obtained only in some particular cases.
And even more, the average speed is not equal to one of the velocities that the body moves during the movement, even if this speed had about an intermediate value relative to the other velocities with which the body was moving.

Equal asked straight movement

Acceleration - Vector physical quantity that determines the speed of changing the body of the body. The acceleration of the body is called the ratio of speed change by the period during which the speed change occurred:

where: v. 0 - initial body speed, v. - final body speed (i.e. after a period of time t.).

Further, unless otherwise indicated in the condition of the problem, we believe that if the body moves with acceleration, this acceleration remains constant. Such a body movement is called equalious (or equal). With an equilibrium movement, the body speed varies to the same value for any equal periods of time.

Equal asked movement is actually accelerated when the body increases the speed, and slow down when the speed decreases. For simplicity, tasks are convenient for slow motion to take acceleration with the sign "-".

From the previous formula, another more common formula should be described changing speed with time With an equilibrium movement:

Move (but not the way) At an equalized movement, the formulas are calculated by the formulas:

In the last formula, one feature of an equivalent movement was used. With an equilibrium movement, the average speed can be calculated as the arithmetic average and final velocity (this property is very convenient to use when solving some tasks):

With the calculation of the path everything is more complicated. If the body did not change the direction of movement, then with an equilibriated straight movement, the path is numerically equal to movement. And if it changed - it is necessary to read the path separately before stopping (turning the moment) and the path after stopping (the moment of reversal). And just a time resistance in the formula for moving in this case will lead to a typical error.

Coordinate At an equilibrium movement, changes under the law:

Projection speed With an equilibrium movement, it changes according to such a law:

Similar formulas are obtained for the remaining coordinate axes.

Free drop by vertical

On all the bodies in the field of land, the power of gravity acts. In the absence of support or suspension, this power causes the bodies to fall to the surface of the Earth. If you neglect the air resistance, then the movement of the body only under the action of gravity is called a free drop. The force of gravity reports to any bodies, regardless of their shape, mass and sizes, the same acceleration, called the acceleration of free fall. Near the surface of the Earth acceleration of gravity Amount:

This means that the free drop in all bodies near the surface of the Earth is equivalent (but not necessarily straightforward) movement. First, consider the simplest case of free fall when the body moves strictly vertically. Such a movement is an equilibrium straightforward movement, so all the previously studied patterns and focuses of such a movement are suitable for free fall. Only acceleration is always equal to the acceleration of free fall.

Traditionally, with a free fall, the directed vertically axis of Oy is used. There is nothing terrible here. Just need in all formulas instead of the index " h.»Write" w." The meaning of this index and the rule of definition of signs is preserved. Where to direct the Oy axis - your choice depending on the convenience of solving the problem. Options 2: up or down.

Let us give a few formulas that are solving some specific kinematics tasks for free drop by vertical. For example, the speed with which the body falling down from the height falls h. Without initial speed:

Body fall time from height h. Without initial speed:

The maximum height of which the body will rise, abandoned vertically up at the initial speed v. 0, lifting time of this body for maximum height, and the full flight time (before returning to the starting point):

Horizontal throw

With a horizontal throw at the initial speed v. 0 The movement of the body is conveniently considered as two movements: uniform along the axis oh (along the axis oh there are no strength of impeding or helping the movement) and the equilibrium movement along the Oy axis.

The speed at any time is directed by the trajectory. It can be decomposed into two components: horizontal and vertical. The horizontal component always remains unchanged and equal v. x \u003d. v. 0. And the vertical increases according to the laws of accelerated movement v. y \u003d. gT.. Wherein full body speed Can be found by formulas:

It is important to understand that the time of the body's fall to the ground does not depend on what horizontal speed it was thrown, but is determined only by the height with which the body was thrown. The time of falling the body to the ground is by the formula:

While the body drops, it is simultaneously moving along the horizontal axis. Hence, body flight range or the distance that the body can fly along the OH axis, will be equal to:

The corner of each horizon and body speed is easy to find from the ratio:

Sometimes sometimes in tasks can ask about the time at which the full body speed will be tilted at a certain angle to vertical. Then this angle will be from the ratio:

It is important to understand which angle appears in the task (with a vertical or horizontal). This will help you choose the right formula. If you solve this problem with the coordinate method, then the general formula for the law of changes in the coordinate with an equilibrium movement:

It is converted to the next law of movement along the Oy axis for the body of an abandoned horizontal:

With her help, we can find the height on which the body will be located at any time. At the same time, at the time of the body fall to the earth coordinate of the body along the Oy axis will be zero. Obviously, along the axis of Oh, the body is moving evenly, therefore, within the coordinate method, the horizontal coordinate will change by law:

Throw at an angle to the horizon (from the ground to earth)

Maximum lifting height when throwing at an angle to the horizon (relative to the initial level):

Lifting time to maximum height when throwing at an angle to the horizon:

The flight range and the full time of the body of the body abandoned at an angle to the horizon (provided that the flight ends at the same height with which the body was thrown, for example, from land to earth):

The minimum body rate of the body abandoned at an angle to the horizon is in the highest point of the lifting, and is equal to:

The maximum body rate abandoned at an angle to the horizon is at the moments of throwing and falling to the ground, and is equal to the initial one. This statement is true only for throwing from Earth to Earth. If the body continues to fly below the level from which it was thrown, it will be there to acquire an increasing and more speed.

Addition of speeds

Tel movement can be described in various reference systems. From the point of view of kinematics, all reference systems are equal. However, the kinematic characteristics of motion, such as the trajectory, moving, speed, are different in different systems. The values \u200b\u200bdepending on the choice of the reference system in which their measurement is made are called relative. Thus, peace and movement of the body are relative.

Thus, the absolute body rate is equal to the vector sum of its speed relative to the movable coordinate system and the speed of the mobile reference system itself. Or, in other words, the body speed in the fixed reference system is equal to the vector amount of body velocity in the movable reference system and the speed of the movable reference system relatively fixed.

Uniform Movement around the circle

The body movement around the circumference is a special case of curvilinear movement. This type of movement is also considered in kinematics. With curved motion, the body velocity vector is always aimed at tangent to the trajectory. The same thing happens when driving around the circle (see Figure). The uniform movement of the body around the circle is characterized by a number of values.

Period - Time for which the body, moving around the circle, makes one full turn. Unit of measurement - 1 s. The period is calculated by the formula:

Frequency - The number of revolutions that made the body by moving around the circumference, per unit of time. Unit of measurement - 1 rev / s or 1 Hz. Frequency is calculated by the formula:

In both formulas: N. - the number of revolutions during the time t.. As can be seen from the above formulas, the period and frequency of the magnitude of the interpretation:

For uniform rotation speed The bodies will be defined as follows:

where: l. - Length of the circle or path passed by the body during an equal period T.. When the body is moving around the circle, it is convenient to consider the angular movement φ (or angle of rotation), measured in radians. Angular speed ω Bodies at this point are called the ratio of a small angular movement Δ φ to a small period of time Δ t.. Obviously, during an equal period T. The body will pass an angle equal to 2 π Therefore, with a uniform movement around the circle, formulas are performed:

The angular speed is measured in rad / s. Do not forget to transfer corners from degrees to radians. Dougie Length l. associated with an angle of rotation by the ratio:

Communication between linear speed module v. and angular speed ω :

When the body is moving around the circle with a constant modulo, only the direction of the velocity vector changes, so the movement of the body around the circumference with a constant velocity by the speed is the movement with acceleration (but not equal to), since the direction of speed changes. In this case, the acceleration is directed along the radius to the center of the circle. It is called normal, or centripetal accelerationSince the acceleration vector at any point of the circle is directed to its center (see Figure).

Module of centripetal acceleration Linear is associated v. and corner ω Speed \u200b\u200bratios:

Please note that if the bodies (points) are on a rotating disk, a ball, rod, and so on, in one word on the same rotating object, then all bodies have the same period of rotation, angular speed and frequency.