To use the natural method of specifying the movement of a point, its trajectory must be known. Trajectory can be set different ways :
Equations (possibly with inequalities), for example,
Verbally, for example, the radius of a circle is 3m;
In the form of a graph to scale.
For the task law of motion of a point along a known trajectory it is necessary:
- 
select on the trajectory the beginning of the distance count – point O and indicate the direction of the positive count (the “+” sign);
Select start time t =0, usually the beginning of time is taken to be either the beginning of movement or the moment in time when the moving point M passes through a point ABOUT.
Law of point motion M along the trajectory looks like:
where is a continuous twice differentiable function, and this expression determines the position of a point on the trajectory, but not the path it has traveled.
.
If at , then
.
If the law of motion of a point in Cartesian coordinates is known, then
,
where the sign “+” or “–” is determined by the choice of positive or negative direction for counting distances along the trajectory. This expression sets connection between the natural way of specifying the movement of a point and the coordinate one.
Point speed is equal to:
,
But 
the unit vector is directed tangent to the trajectory in the direction of the point’s movement M, therefore, the speed of the point M is directed tangent to the trajectory in the direction of movement and is equal to
Let us align the origin of the moving coordinate system with the point M moving along the trajectory – axes of natural trihedron
Mtnb. Axis Mt- tangent
Let's direct it tangentially to the trajectory in the direction of the point's movement. Axis Mn – main normal
let's direct it perpendicularly Mt towards the concavity of the trajectory so that these axes form touching plane
. Axis Mb- binormal
let's direct it perpendicular to the contacting plane in the direction from which the rotation from the axis Mt to the axis Mn visible counterclockwise. Two more coordinate planes were formed: Mnb - normal
And Mtb – straightening
Let the point M moved to position M 1. Its velocity vectors at these points form angle of adjacency φ .
,
k – curvature of the curve at the point M,
ρ – radius of curvature curve at a point M.
Point acceleration M equals:
,
but therefore
.
The acceleration vector of point M is decomposed into two mutually perpendicular components lying in an adjacent plane.
Thursday, October 09, 2014 13:07 + to quote book
Many of you who work on flash drives know how to do classic motion animation. This places an object at a specific point in the first key frame and then moves it to another key point. On the Timeline, a certain number of simple intermediate frames are created between these key points.
The animation object moves from one key point to another strictly in a straight line.
How to make an animation object move along a given path. To do this, this trajectory, firstly, must, of course, be specified. Secondly, bind our object to this trajectory. Such a trajectory in flash is called a Guide.
And so, let’s look in more detail at how to create the movement of an object along a given trajectory. We will animate autumn leaves.
For this we will create new flash Action Script 3 document
File - Create
Next, on the Timeline of the Main working field (Editing frame 1), create 2 layers
1. Background
2. Leaves
The Main Workspace timeline (Editing Frame 1) will look like this.
Save the created project under some name, for example “Falling Leaf”
We import any picture with an autumn background and the AI file “Leaf” into the Program Library, which can be downloaded from the attachment below 
The attachment:
File - Import - Import to Library
After downloading, the Library will contain the following files
Using the "Arrow" tool, drag the background image onto the "Background" layer on the main Working field Editing Frame 1 from the Library and either change the size of the Working field to fit the size of the picture, or transform the size of the picture to the size of the Working field.
After you click “OK” in the dialog box for creating a new symbol, you will be taken to the “Sheet” symbol editing window. Rename Layer 1 on the Timeline to "Animation Sheet"
From the Library, using the "Arrow" tool, add to the Working field of the "Sheet" symbol graphic symbol"Sheet".
On the Timeline on the "Animation Sheet" layer, click on frame 140 and call context menu, select "Insert Key Frame".
Intermediate frames appeared between frames 1 and 140. Now click on any frame between the first and one hundred and fortieth and, in the context menu that appears, select the “Create classic motion animation” item. After this, such an animation will be created automatically.
We do not touch the leaf added to the Workspace for now, but continue to work with the Timeline.
On the Timeline, click the 140th (last and key) frame of our animation and, calling up the context menu, also select the “Create classic motion tween” item. This way we include the last 140 key frame in the classic tween we created.
Now it’s time to create the trajectory along which our “Falling Leaf” object will fly.
To do this, click on the “Animation Sheet” layer and, calling up the context menu, select “Add a classic animation guide.”
After this, we will see that a new layer has appeared on the Timeline - “Guide”, and the “Animation Sheet” layer is “subordinate” to this layer.
Now, the trajectory for the movement of the object, created on the "Guide" layer, will be a guide to the action (movement) for the "Animation Sheet" layer, that is, all the classic motion animation created on the "Animation Sheet" layer will occur along the trajectory depicted on the "Animation Sheet" layer. Guide".
The “Guide” layer is a working layer and all graphics placed on it will not be displayed when publishing a flash video.
So let’s create (draw) on the “Guide” layer some kind of trajectory for the autumn leaf to fall from top to bottom.
To draw a trajectory, use the “Pencil” tool in the “Pencil mode with anti-aliasing” mode and, selecting the “Guide” layer and its first frame, draw a curved line for the required trajectory. 
After the guide line is created, go to the “Leaf Animation” layer, select the first key frame and begin creating a classic motion animation for a falling leaf. To do this, use the Arrow Tool to place the leaf at the beginning of the trajectory we drew. In this case, the registration point of our leaf (in our case, the registration point is in the center) must be!!! is located on the guide path line.
At the same time, click again on the first frame of the “Animation Sheet” layer and make sure that it is selected. Now open the "Properties" tab and find the "Animation" section there. Check the checkboxes as shown in the picture.
On the "Leaf Animation" layer, click the 140th (last) key frame and use the Arrow Tool to place our autumn leaf at the end of the drawn path. In this case, the registration point of the leaf graphic object must also be located on the line of the movement trajectory.
Or, if snapping to the “Guide” works for you without any problems, then when you click the last frame of the animation, the leaf will automatically move to the end of the guide.
After that, click on frame 140 again to make sure that it is selected and by opening the “Properties” tab in the “Animation” section, also check the checkboxes as indicated in the picture.
Click again on any frame on the “Leaf Animation” layer, for example frame 40, and make sure that our leaf moves strictly along the drawn path.
If everything is fine, then the animation of the “Falling Leaf” along the given trajectory has been created and you can return to the Main Working Scene - Editing Frame 1.
While on Editing Frame 1, select the “leaves” layer and using the Arrow Tool, drag the “Leaf” video from the Library onto it, placing it at the very top of the background image.
For the "Leaf" video, apply the "Shadow" filter with the following parameters. Let me remind you that the "Filters" section can be found on the "Properties" tab.
By holding down the Shift+Ctrl keys, you can multiply the “Leaf” video and get several falling leaves. You can use the Free Transform Tool to resize and rotate the Leaf video so that the leaves don't fall exactly the same way.
By holding down the Ctrl+Enter keys, we view the resulting flash video. If everything suits you, then save the flash drive as a project in FLA format
File - Save
Exporting the flash video for further publication
File - Export - Export video
Basic concepts of kinematics and kinematic characteristics
Human movement is mechanical, that is, it is a change in the body or its parts relative to other bodies. Relative movement is described by kinematics.
Kinematics – a branch of mechanics in which mechanical motion is studied, but the causes of this motion are not considered. The description of the movement of both the human body (its parts) in various sports and various sports equipment is an integral part of sports biomechanics and in particular kinematics.
Whatever material object or phenomenon we consider, it turns out that nothing exists outside of space and outside of time. Any object has spatial dimensions and shape, and is located in some place in space in relation to another object. Any process in which material objects participate has a beginning and an end in time, how long it lasts in time, and can occur earlier or later than another process. This is precisely why there is a need to measure spatial and temporal extent.
Basic units of measurement of kinematic characteristics in the international system of measurements SI.
Space. One forty-millionth of the length of the earth's meridian passing through Paris was called a meter. Therefore, length is measured in meters (m) and its multiple units: kilometers (km), centimeters (cm), etc.
Time– one of the fundamental concepts. We can say that this is what separates two successive events. One way to measure time is to use any regularly repeated process. One eighty-six thousandth of an earthly day was chosen as a unit of time and was called the second (s) and its multiple units (minutes, hours, etc.).
In sports, special time characteristics are used:
Moment of time(t)- this is a temporary measure of the position of a material point, links of a body or system of bodies. Moments of time indicate the beginning and end of a movement or any part or phase of it.
Movement duration(∆t) – this is its temporary measure, which is measured by the difference between the moments of the end and the beginning of movement∆t = tcon. – tbeg.
Movement speed(N) – it is a temporal measure of the repetition of movements repeated per unit of time. N = 1/∆t; (1/s) or (cycle/s).
Rhythm of movements – this is a temporary measure of the relationship between parts (phases) of movements. It is determined by the ratio of the duration of the parts of the movement.
The position of a body in space is determined relative to a certain reference system, which includes a reference body (that is, relative to which the movement is considered) and a coordinate system necessary to describe at a qualitative level the position of the body in one or another part of space.
The beginning and direction of measurement are associated with the reference body. For example, in a number of competitions, the origin of coordinates can be chosen as the starting position. Various competitive distances in all cyclic sports are already calculated from it. Thus, in the selected “start-finish” coordinate system, the distance in space that the athlete will move when moving is determined. Any intermediate position of the athlete’s body during movement is characterized by the current coordinate within the selected distance interval.
To accurately determine a sports result, the competition rules stipulate at what point (reference point) the count is taken: along the toe of a skater’s skate, at the protruding point of a sprinter’s chest, or along the back edge of the landing long jumper’s track.
In some cases, to accurately describe the movement of the laws of biomechanics, the concept of a material point is introduced.
Material point – this is a body whose dimensions and internal structure can be neglected under given conditions.
The movement of bodies can be different in nature and intensity. To characterize these differences, a number of terms are introduced in kinematics, presented below.
Trajectory – a line described in space by a moving point of a body. When biomechanical analysis of movements, first of all, the trajectories of movements of characteristic points of a person are considered. As a rule, such points are the joints of the body. Based on the type of movement trajectories, they are divided into rectilinear (straight line) and curvilinear (any line other than a straight line).
Moving – is the vector difference of finite and initial position body. Therefore, displacement characterizes the final result of the movement.
Path – this is the length of the trajectory section traversed by a body or a point of the body during a selected period of time.
KINEMATICS OF A POINT
Introduction to Kinematics
Kinematics is a branch of theoretical mechanics that studies the motion of material bodies from a geometric point of view, regardless of the applied forces.
The position of a moving body in space is always determined in relation to any other unchanging body, called reference body. A coordinate system invariably associated with a reference body is called reference system. In Newtonian mechanics, time is considered absolute and not related to moving matter. In accordance with this, it proceeds identically in all reference systems, regardless of their motion. The basic unit of time is the second (s).
If the position of the body relative to the chosen frame of reference does not change over time, then it is said that body relative to a given frame of reference is at rest. If a body changes its position relative to the chosen reference system, then it is said to move relative to this system. A body can be at rest in relation to one reference system, but move (and in completely different ways) in relation to other reference systems. For example, a passenger sitting motionless on the bench of a moving train is at rest relative to the frame of reference associated with the car, but is moving with respect to the frame of reference associated with the Earth. A point lying on the rolling surface of the wheel moves in relation to the reference system associated with the car in a circle, and in relation to the reference system associated with the Earth, in a cycloid; the same point is at rest with respect to the coordinate system associated with the wheel pair.
Thus, the movement or rest of a body can be considered only in relation to any chosen frame of reference. Set the motion of a body relative to some reference system -means to give functional dependencies with the help of which one can determine the position of the body at any time relative to this system. Different points of the same body move differently in relation to the chosen reference system. For example, in relation to the system associated with the Earth, the tread surface point of the wheel moves along a cycloid, and the center of the wheel moves in a straight line. Therefore, the study of kinematics begins with the kinematics of a point.
§ 2. Methods for specifying the movement of a point
The movement of a point can be specified in three ways:natural, vector and coordinate.
With the natural way The movement assignment is given by a trajectory, i.e., a line along which the point moves (Fig. 2.1). On this trajectory, a certain point is selected, taken as the origin. The positive and negative directions of reference of the arc coordinate, which determines the position of the point on the trajectory, are selected. As the point moves, the distance will change. Therefore, to determine the position of a point at any time, it is enough to specify the arc coordinate as a function of time:
This equality is called equation of motion of a point along a given trajectory .
So, the movement of a point in the case under consideration is determined by a combination of the following data: the trajectory of the point, the position of the origin of the arc coordinate, the positive and negative directions of the reference and the function .
With the vector method of specifying the movement of a point, the position of the point is determined by the magnitude and direction of the radius vector drawn from the fixed center to a given point (Fig. 2.2). When a point moves, its radius vector changes in magnitude and direction. Therefore, to determine the position of a point at any time, it is enough to specify its radius vector as a function of time:
This equality is called vector equation of motion of a point .
With the coordinate method
specifying the motion, the position of the point in relation to the selected reference system is determined using a rectangular Cartesian coordinate system (Fig. 2.3). When a point moves, its coordinates change over time. Therefore, to determine the position of a point at any time, it is enough to specify the coordinates , ,
as a function of time:
These equalities are called equations of motion of a point in rectangular Cartesian coordinates . The motion of a point in a plane is determined by two equations of system (2.3), rectilinear motion by one.
There is a mutual connection between the three described methods of specifying movement, which allows you to move from one method of specifying movement to another. This is easy to verify, for example, when considering the transition from the coordinate method of specifying movement to vector.
Let us assume that the motion of a point is given in the form of equations (2.3). Bearing in mind that
can be written down
And this is an equation of the form (2.2).
Task 2.1.
Find the equation of motion and the trajectory of the middle point of the connecting rod, as well as the equation of motion of the slider of the crank-slider mechanism (Fig. 2.4), if 
;
.
Solution. The position of a point is determined by two coordinates and . From Fig. 2.4 it is clear that
, .
Then from and:
; ; 
.
Substituting values , 
and , we obtain the equations of motion of the point:
; 
.
To find the equation for the trajectory of a point in explicit form, it is necessary to exclude time from the equations of motion. For this purpose, we will carry out the necessary transformations in the equations of motion obtained above:
; .
By squaring and adding the left and right sides of these equations, we obtain the trajectory equation in the form
.
Therefore, the trajectory of the point is an ellipse.
The slider moves in a straight line. The coordinate , which determines the position of the point, can be written in the form
.
Speed and acceleration
Point speed
In the previous article, the movement of a body or point is defined as a change in position in space over time. In order to more fully characterize the qualitative and quantitative aspects of movement, the concepts of speed and acceleration were introduced.
Velocity is a kinematic measure of the movement of a point, characterizing the speed of change of its position in space.
Velocity is a vector quantity, that is, it is characterized not only by its magnitude (scalar component), but also by its direction in space.
As is known from physics, with uniform motion, speed can be determined by the length of the path traveled per unit time: v = s/t = const
(it is assumed that the origin of the path and time are the same).
During rectilinear motion, the speed is constant both in magnitude and direction, and its vector coincides with the trajectory.
Unit of speed in system SI is determined by the length/time ratio, i.e. m/s .
Obviously, with curvilinear movement, the speed of the point will change in direction.
In order to establish the direction of the velocity vector at each moment of time during curvilinear motion, we divide the trajectory into infinitesimal sections of the path, which can be considered (due to their smallness) rectilinear. Then at each section the conditional speed v p
such a rectilinear motion will be directed along the chord, and the chord, in turn, with an infinite decrease in the length of the arc ( Δs
tends to zero) will coincide with the tangent to this arc.
It follows from this that during curvilinear motion the velocity vector at each moment of time coincides with the tangent to the trajectory (Fig. 1a). Rectilinear motion can be represented as a special case of curvilinear motion along an arc whose radius tends to infinity (trajectory coincides with tangent).
When a point moves unevenly, the magnitude of its velocity changes over time.
Let us imagine a point whose movement is given in a natural way equation s = f(t)
.
If in a short period of time Δt the point has passed the way Δs , then its average speed is:
vav = Δs/Δt.
Average speed does not give an idea of the true speed at each this moment time (true speed is otherwise called instantaneous). Obviously, the shorter the time period for which the average speed is determined, the closer its value will be to the instantaneous speed.
True (instantaneous) speed is the limit to which the average speed tends as Δt tends to zero:
v = lim v av at t→0 or v = lim (Δs/Δt) = ds/dt.
Thus, the numerical value of the true speed is v = ds/dt
.
The true (instantaneous) speed for any movement of a point is equal to the first derivative of the coordinate (i.e., the distance from the origin of the movement) with respect to time.
At Δt tending to zero, Δs also tends to zero, and, as we have already found out, the velocity vector will be directed tangentially (i.e., it coincides with the true velocity vector v ). It follows from this that the limit of the conditional speed vector v p , equal to the limit of the ratio of the point's displacement vector to an infinitesimal period of time, is equal to the vector of the point's true speed.
Fig.1
Let's look at an example. If a disk, without rotating, can slide along an axis fixed in a given reference system (Fig. 1, A), then in a given reference frame it obviously has only one degree of freedom - the position of the disk is uniquely determined, say, by the x coordinate of its center, measured along the axis. But if the disk, in addition, can also rotate (Fig. 1, b), then it acquires one more degree of freedom - to the coordinate x the rotation angle φ of the disk around the axis is added. If the axis with the disk is clamped in a frame that can rotate around a vertical axis (Fig. 1, V), then the number of degrees of freedom becomes equal to three - to x and φ the frame rotation angle is added ϕ .
A free material point in space has three degrees of freedom: for example, Cartesian coordinates x, y And z. The coordinates of a point can also be determined in cylindrical ( r, 𝜑, z) and spherical ( r, 𝜑, 𝜙) reference systems, but the number of parameters that uniquely determine the position of a point in space is always three.
A material point on a plane has two degrees of freedom. If we select a coordinate system in the plane xOy, then the coordinates x And y determine the position of a point on the plane, coordinate z is identically equal to zero.
A free material point on a surface of any kind has two degrees of freedom. For example: the position of a point on the Earth's surface is determined by two parameters: latitude and longitude.
A material point on a curve of any kind has one degree of freedom. The parameter that determines the position of a point on a curve can be, for example, the distance along the curve from the origin.
Consider two material points in space connected by a rigid rod of length l(Fig. 2). The position of each point is determined by three parameters, but a connection is imposed on them.
Fig.2
The equation l 2 =(x 2 -x 1) 2 +(y 2 -y 1) 2 +(z 2 -z 1) 2 is the coupling equation. From this equation, any one coordinate can be expressed in terms of the other five coordinates (five independent parameters). Therefore, these two points have (2∙3-1=5) five degrees of freedom.
Let us consider three material points in space that do not lie on the same straight line, connected by three rigid rods. The number of degrees of freedom of these points is (3∙3-3=6) six.
Free solid body in general case has 6 degrees of freedom. Indeed, the position of a body in space relative to any reference system is determined by specifying three of its points that do not lie on the same straight line, and the distances between points in a rigid body remain unchanged during any of its movements. According to the above, the number of degrees of freedom should be six.
Forward movement
In kinematics, as in statistics, we will consider all rigid bodies as absolutely rigid.
Absolutely solid body is a material body whose geometric shape and dimensions do not change under any mechanical influences from other bodies, and the distance between any two of its points remains constant.
Kinematics of a rigid body, as well as the dynamics of a rigid body, is one of the most difficult sections of the course in theoretical mechanics.
Rigid body kinematics problems fall into two parts:
1) setting the movement and determining the kinematic characteristics of the movement of the body as a whole;
2) determination of the kinematic characteristics of the movement of individual points of the body.
There are five types of rigid body motion:
1) forward movement;
2) rotation around a fixed axis;
3) flat movement;
4) rotation around a fixed point;
5) free movement.
The first two are called the simplest motions of a rigid body.
Let's start by considering the translational motion of a rigid body.
Progressive is the movement of a rigid body in which any straight line drawn in this body moves while remaining parallel to its initial direction.
Translational motion should not be confused with rectilinear motion. When a body moves forward, the trajectories of its points can be any curved lines. Let's give examples.
1. The car body on a straight horizontal section of the road moves forward. In this case, the trajectories of its points will be straight lines.
2. Sparnik AB(Fig. 3) when the cranks O 1 A and O 2 B rotate, they also move translationally (any straight line drawn in it remains parallel to its initial direction). The points of the partner move in circles.
Fig.3
The pedals of a bicycle move progressively relative to its frame during movement, the pistons in the cylinders of an internal combustion engine move relative to the cylinders, and the cabins of Ferris wheels in parks (Fig. 4) relative to the Earth.
Fig.4
The properties of translational motion are determined by the following theorem: during translational motion, all points of the body describe identical (overlapping, coinciding) trajectories and at each moment of time have the same magnitude and direction of velocity and acceleration.
To prove this, consider a rigid body undergoing translational motion relative to the reference frame Oxyz. Let's take two arbitrary points in the body A And IN, whose positions at the moment of time t are determined by radius vectors and (Fig. 5).
Fig.5
Let's draw a vector connecting these points.
In this case, the length AB constant, like the distance between points of a rigid body, and the direction AB remains unchanged as the body moves forward. So the vector AB remains constant throughout the body's movement ( AB=const). As a result, the trajectory of point B is obtained from the trajectory of point A by parallel displacement of all its points by a constant vector. Therefore, the trajectories of the points A And IN will really be the same (when superimposed, coinciding) curves.
To find the velocities of points A And IN Let us differentiate both sides of the equality with respect to time. We get
But the derivative of a constant vector AB equal to zero. Derivatives of vectors and with respect to time give the velocities of points A And IN. As a result, we find that
those. what are the speeds of the points A And IN bodies at any moment of time are identical both in magnitude and direction. Taking derivatives with respect to time from both sides of the resulting equality:
Therefore, the accelerations of the points A And IN bodies at any moment of time are also identical in magnitude and direction.
Since the points A And IN were chosen arbitrarily, then from the results found it follows that for all points of the body their trajectories, as well as velocities and accelerations at any time, will be the same. Thus, the theorem is proven.
It follows from the theorem that the translational motion of a rigid body is determined by the movement of any one of its points. Consequently, the study of the translational motion of a body comes down to the problem of the kinematics of a point, which we have already considered.
During translational motion, the speed common to all points of the body is called the speed of translational motion of the body, and acceleration is called the acceleration of translational motion of the body. Vectors and can be depicted as applied at any point of the body.
Note that the concept of speed and acceleration of a body makes sense only in translational motion. In all other cases, the points of the body, as we will see, move with at different speeds and accelerations, and terms<<скорость тела>> or<<ускорение тела>> these movements lose their meaning.
Fig.6
During the time ∆t, the body, moving from point A to point B, makes a displacement equal to the chord AB and covers a path equal to the length of the arc l.
The radius vector rotates through an angle ∆φ. The angle is expressed in radians.
The speed of movement of a body along a trajectory (circle) is directed tangent to the trajectory. It is called linear speed. The modulus of linear velocity is equal to the ratio of the length of the circular arc l to the time interval ∆t during which this arc is passed:
A scalar physical quantity, numerically equal to the ratio of the angle of rotation of the radius vector to the period of time during which this rotation occurred, is called angular velocity:
The SI unit of angular velocity is radian per second.
With uniform motion in a circle, the angular velocity and the linear velocity module are constant values: ω=const; v=const.
The position of the body can be determined if the modulus of the radius vector and the angle φ it makes with the Ox axis (angular coordinate) are known. If at the initial moment of time t 0 =0 the angular coordinate is equal to φ 0, and at the moment of time t it is equal to φ, then the angle of rotation ∆φ of the radius vector during the time ∆t=t-t 0 is equal to ∆φ=φ-φ 0. Then from the last formula we can obtain the kinematic equation of motion of a material point in a circle:
It allows you to determine the position of the body at any time t.
Considering that , we get:
Formula for the relationship between linear and angular velocity.
The time period T during which the body makes one full revolution is called the period of rotation:
Where N is the number of revolutions made by the body during time Δt.
During the time ∆t=T the body travels the path l=2πR. Hence,
At ∆t→0, the angle is ∆φ→0 and, therefore, β→90°. The perpendicular to the tangent to the circle is the radius. Therefore, it is directed radially towards the center and is therefore called centripetal acceleration:
Module , direction changes continuously (Fig. 8). Therefore, this movement is not uniformly accelerated.
Fig.8
Fig.9
Then the position of the body at any moment of time is uniquely determined by the angle φ between these half-planes taken with the appropriate sign, which we will call the angle of rotation of the body. We will consider the angle φ to be positive if it is plotted from the fixed plane in a counterclockwise direction (for an observer looking from the positive end of the Az axis), and negative if it is clockwise. We will always measure the angle φ in radians. To know the position of the body at any moment in time, you need to know the dependence of the angle φ on time t, i.e.
The equation expresses the law of rotational motion of a rigid body around a fixed axis.
During rotational motion of an absolutely rigid body around a fixed axis the angles of rotation of the radius vector of different points of the body are the same.
The main kinematic characteristics of the rotational motion of a rigid body are its angular velocity ω and angular acceleration ε.
If during a period of time ∆t=t 1 -t the body rotates through an angle ∆φ=φ 1 -φ, then the numerically average angular velocity of the body during this period of time will be . In the limit at ∆t→0 we find that
Thus, the numerical value of the angular velocity of a body at a given time is equal to the first derivative of the angle of rotation with respect to time. The sign of ω determines the direction of rotation of the body. It is easy to see that when rotation occurs counterclockwise, ω>0, and when clockwise, then ω<0.
The dimension of angular velocity is 1/T (i.e. 1/time); the unit of measurement is usually rad/s or, which is the same, 1/s (s -1), since the radian is a dimensionless quantity.
The angular velocity of a body can be represented as a vector whose modulus is equal to | | and which is directed along the axis of rotation of the body in the direction from which the rotation can be seen occurring counterclockwise (Fig. 10). Such a vector immediately determines the magnitude of the angular velocity, the axis of rotation, and the direction of rotation around this axis.
Fig.10
The angle of rotation and angular velocity characterize the motion of the entire absolutely rigid body as a whole. The linear speed of any point on an absolutely rigid body is proportional to the distance of the point from the axis of rotation:
With uniform rotation of an absolutely rigid body, the angles of rotation of the body for any equal periods of time are the same, there are no tangential accelerations at various points of the body, and the normal acceleration of a point of the body depends on its distance to the axis of rotation:
The vector is directed along the radius of the point's trajectory towards the axis of rotation.
Angular acceleration characterizes the change in the angular velocity of a body over time. If over a period of time ∆t=t 1 -t the angular velocity of a body changes by the amount ∆ω=ω 1 -ω, then the numerical value of the average angular acceleration of the body over this period of time will be . In the limit at ∆t→0 we find,
Thus, the numerical value of the angular acceleration of a body at a given time is equal to the first derivative of the angular velocity or the second derivative of the angle of rotation of the body with respect to time.
The dimension of angular acceleration is 1/T 2 (1/time 2); the unit of measurement is usually rad/s 2 or, what is the same, 1/s 2 (s-2).
If the module of angular velocity increases with time, the rotation of the body is called accelerated, and if it decreases, it is called slow. It is easy to see that the rotation will be accelerated when the quantities ω and ε have the same signs, and slowed down when they are different.
The angular acceleration of a body (by analogy with angular velocity) can also be represented as a vector ε directed along the axis of rotation. Wherein
The direction of ε coincides with the direction of ω when the body rotates at an accelerated rate (Fig. 10, a), and is opposite to ω when the body rotates at a slow speed (Fig. 10, b).
Fig.11 Fig. 12
2. Acceleration of body points. To find the acceleration of a point M let's use the formulas
In our case ρ=h. Substituting the value v into the expressions a τ and a n, we get:
or finally:
The tangential component of acceleration a τ is directed tangentially to the trajectory (in the direction of motion during accelerated rotation of the body and in the opposite direction during slow rotation); the normal component a n is always directed along the radius MS to the axis of rotation (Fig. 12). Total point acceleration M will
The deviation of the total acceleration vector from the radius of the circle described by the point is determined by the angle μ, which is calculated by the formula
Substituting the values of a τ and a n here, we get
Since ω and ε have the same value for all points of the body at a given moment in time, the accelerations of all points of a rotating rigid body are proportional to their distances from the axis of rotation and form at a given moment in time the same angle μ with the radii of the circles they describe . The acceleration field of points of a rotating rigid body has the form shown in Fig. 14.
Fig.13 Fig.14
3. Vectors of velocity and acceleration of body points. To find expressions directly for vectors v and a, let’s draw from an arbitrary point ABOUT axes AB radius vector of a point M(Fig. 13). Then h=r∙sinα and by the formula
So I can
Special guide layers allow you to define motion paths for animated instances, groups, or text blocks. You can link multiple layers of objects to a single guide layer so that all objects follow the same path. The normal layer associated with the guide layer becomes the slave layer.
Rice. 4.12. Snap an object to a path
Let's consider the sequence of actions when creating animation with the movement of an object along a given path:
- Let's create motion animation using one of the methods discussed earlier.
- When checking the box Orient to Path(Path Orientation) The baseline of a group of animated objects will move parallel to the specified path. To fix the registration point of an object on the trajectory, set the checkbox Snap(Binding).
- Execute the command Insert › Motion Guide(Insert › Motion trajectory). Flash creates a new layer above the selected layer with a guide icon to the left of its name.
- We use any drawing tool to depict the desired trajectory. In the first frame, we fix the object at the starting point of the line, and in the last frame, at the end of the line, moving the object with the mouse beyond its registration point.
- To make the path invisible, click at the intersection of the row of the guide layer and the column marked with the eye icon.
Rice. 4.13. Movement along a given trajectory
Rice. 4.14. Layer Properties Window
To link a layer to an existing guide layer, you can do one of the following:
- Move the layer with objects under the layer with the guide. All animated objects on it are automatically snapped to the path, as indicated by shifting the layer name to the right.
- Create a new layer under the guide layer. Objects placed on this layer to which animation will be applied using the frame calculation method (tweened), are automatically attached to the trajectory.
- Select the layer below the guide layer and execute the command Modify › Layer Guided(Managed) for layer type in the dialog box Layer Properties(Layer properties).
- Click on a layer while holding down the key ALT.
To unlink a layer from a guide layer, do one of the following:
- Select the layer whose link you want to break and drag it above the guide layer.
- Run the command Modify › Layer(Edit › Layer) with a choice of value Normal(Normal) for layer type in window Layer Properties(Layer properties).
- Click on the layer while holding down the key ALT.
Trajectory(from Late Latin trajectories - related to movement) is the line along which a body (material point) moves. The trajectory of movement can be straight (the body moves in one direction) and curved, that is, mechanical movement can be rectilinear and curvilinear.
Straight-line trajectory in this coordinate system it is a straight line. For example, we can assume that the trajectory of a car on a flat road without turns is straight.
Curvilinear movement is the movement of bodies in a circle, ellipse, parabola or hyperbola. An example of curvilinear motion is the movement of a point on the wheel of a moving car or the movement of a car in a turn.
The movement can be difficult. For example, the trajectory of a body at the beginning of its journey can be rectilinear, then curved. For example, at the beginning of the journey a car moves along a straight road, and then the road begins to “wind” and the car begins to move in a curved direction.
Path
Path is the length of the trajectory. Path is a scalar quantity and is measured in meters (m) in the SI system. Path calculation is performed in many physics problems. Some examples will be discussed later in this tutorial.
Move vector
Move vector(or simply moving) is a directed straight line segment connecting the initial position of the body with its subsequent position (Fig. 1.1). Displacement is a vector quantity. The displacement vector is directed from the starting point of movement to the ending point.
Motion vector module(that is, the length of the segment that connects the starting and ending points of the movement) can be equal to the distance traveled or less than the distance traveled. But the magnitude of the displacement vector can never be greater than the distance traveled.
The magnitude of the displacement vector is equal to the distance traveled when the path coincides with the trajectory (see sections and ), for example, if a car moves from point A to point B along a straight road. The magnitude of the displacement vector is less than the distance traveled when a material point moves along a curved path (Fig. 1.1).
Rice. 1.1. Displacement vector and distance traveled.
In Fig. 1.1:
Another example. If the car drives in a circle once, it turns out that the point at which the movement begins will coincide with the point at which the movement ends, and then the displacement vector will be equal to zero, and the distance traveled will be equal to the length of the circle. Thus, path and movement are two different concepts.
Vector addition rule
The displacement vectors are added geometrically according to the vector addition rule (triangle rule or parallelogram rule, see Fig. 1.2).
Rice. 1.2. Addition of displacement vectors.
Figure 1.2 shows the rules for adding vectors S1 and S2:
a) Addition according to the triangle rule
b) Addition according to the parallelogram rule
Motion vector projections
When solving problems in physics, projections of the displacement vector onto coordinate axes are often used. Projections of the displacement vector onto the coordinate axes can be expressed through the differences in the coordinates of its end and beginning. For example, if a material point moves from point A to point B, then the displacement vector (see Fig. 1.3).
Let us choose the OX axis so that the vector lies in the same plane with this axis. Let's lower the perpendiculars from points A and B (from the starting and ending points of the displacement vector) until they intersect with the OX axis. Thus, we obtain the projections of points A and B onto the X axis. Let us denote the projections of points A and B, respectively, as A x and B x. The length of the segment A x B x on the OX axis is displacement vector projection on the OX axis, that is
S x = A x B x
IMPORTANT!
I remind you for those who do not know mathematics very well: do not confuse a vector with the projection of a vector onto any axis (for example, S x). A vector is always indicated by a letter or several letters, above which there is an arrow. In some electronic documents, the arrow is not placed, as this may cause difficulties when creating an electronic document. In such cases, be guided by the content of the article, where the word “vector” may be written next to the letter or in some other way they indicate to you that this is a vector, and not just a segment.
Rice. 1.3. Projection of the displacement vector.
The projection of the displacement vector onto the OX axis is equal to the difference between the coordinates of the end and beginning of the vector, that is
S x = x – x 0
The projections of the displacement vector on the OY and OZ axes are determined and written similarly:
S y = y – y 0 S z = z – z 0
Here x 0 , y 0 , z 0 are the initial coordinates, or the coordinates of the initial position of the body (material point); x, y, z - final coordinates, or coordinates of the subsequent position of the body (material point).
The projection of the displacement vector is considered positive if the direction of the vector and the direction of the coordinate axis coincide (as in Fig. 1.3). If the direction of the vector and the direction of the coordinate axis do not coincide (opposite), then the projection of the vector is negative (Fig. 1.4).
If the displacement vector is parallel to the axis, then the modulus of its projection is equal to the modulus of the Vector itself. If the displacement vector is perpendicular to the axis, then the modulus of its projection is equal to zero (Fig. 1.4).
Rice. 1.4. Motion vector projection modules.
The difference between the subsequent and initial values of some quantity is called the change in this quantity. That is, the projection of the displacement vector onto the coordinate axis is equal to the change in the corresponding coordinate. For example, for the case when the body moves perpendicular to the X axis (Fig. 1.4), it turns out that the body DOES NOT MOVE relative to the X axis. That is, the movement of the body along the X axis is zero.
Let's consider an example of body motion on a plane. The initial position of the body is point A with coordinates x 0 and y 0, that is, A(x 0, y 0). The final position of the body is point B with coordinates x and y, that is, B(x, y). Let's find the modulus of body displacement.
From points A and B we lower perpendiculars to the coordinate axes OX and OY (Fig. 1.5).
Rice. 1.5. Movement of a body on a plane.
Let us determine the projections of the displacement vector on the OX and OY axes:
S x = x – x 0 S y = y – y 0
In Fig. 1.5 it is clear that triangle ABC is a right triangle. It follows from this that when solving the problem one can use Pythagorean theorem, with which you can find the module of the displacement vector, since
AC = s x CB = s y
According to the Pythagorean theorem
S 2 = S x 2 + S y 2
Where can you find the module of the displacement vector, that is, the length of the body’s path from point A to point B:
And finally, I suggest you consolidate your knowledge and calculate a few examples at your discretion. To do this, enter some numbers in the coordinate fields and click the CALCULATE button. Your browser must support the execution of JavaScript scripts and script execution must be enabled in your browser settings, otherwise the calculation will not be performed. In real numbers, the integer and fractional parts must be separated by a dot, for example, 10.5.