Kinematics of a point, kinematics of a rigid body, translational motion, rotational motion, plane-parallel motion, velocity projection theorem, instantaneous center of velocities, determination of velocity and accelerations of points of a flat body, complex motion of a point

Rigid Body Kinematics

To uniquely determine the position of a rigid body, you need to specify three coordinates (x A , y A , z A ) one of the points A of the body and three angles of rotation. Thus, the position of a rigid body is determined by six coordinates. That is solid has six degrees of freedom.

In the general case, the dependence of the coordinates of the points of a rigid body relative to a fixed coordinate system is determined by rather cumbersome formulas. However, the velocities and accelerations of points are determined quite simply. To do this, you need to know the dependence of the coordinates on time of one, arbitrarily chosen, point A and the angular velocity vector . Differentiating with respect to time, we find the speed and acceleration of point A and the angular acceleration of the body:
; ; .
Then the speed and acceleration of a point of the body with a radius vector is determined by the formulas:
(1) ;
(2) .
Here and below, the products of vectors in square brackets mean vector artwork.

Note that the angular velocity vector is the same for all points of the body. It does not depend on the coordinates of the points of the body. Also the angular acceleration vector is the same for all points of the body.

See derivation of formulas (1) and (2) on the page: Velocity and acceleration of points of a rigid body > > >

Translational motion of a rigid body

At forward movement, the angular velocity is zero. The velocities of all points of the body are equal. Any straight line drawn in the body moves while remaining parallel to its initial direction. Thus, to study the motion of a rigid body during translational motion, it is sufficient to study the motion of any one point of this body. See section.

Uniformly accelerated motion

Consider the case of uniformly accelerated motion. Let the projection of the acceleration of the point of the body on the x-axis be constant and equal to a x . Then the projection of the speed v x and x - the coordinate of this point depend on the time t according to the law:
v x = v x 0 + a x t;
,
where v x 0 and x 0 - speed and coordinate of the point at the initial time t = 0 .

Rotational motion of a rigid body

Consider a body that rotates around a fixed axis. We choose a fixed coordinate system Oxyz centered at the point O . Let's direct the z-axis along the rotation axis. We consider that z - coordinates of all points of the body remain constant. Then the movement occurs in the xy plane. Angular velocity ω and angular acceleration ε are directed along the z axis:
; .
Let φ be the angle of rotation of the body, which depends on the time t. Differentiating with respect to time, we find projections of angular velocity and angular acceleration on the z-axis:
;
.

Consider the motion of a point M , which is located at a distance r from the axis of rotation. The trajectory of movement is a circle (or an arc of a circle) of radius r.
Point speed:
v = ω r .
The velocity vector is directed tangentially to the trajectory.
Tangential acceleration:
a τ = ε r .
The tangential acceleration is also directed tangentially to the trajectory.
Normal acceleration:
.
It is directed towards the axis of rotation O.
Full acceleration:
.
Since the vectors and are perpendicular to each other, then acceleration module:
.

Uniformly accelerated motion

In the case of uniformly accelerated motion, in which the angular acceleration is constant and equal to ε, the angular velocity ω and the angle of rotation φ change with time t according to the law:
ω = ω 0 + εt;
,
where ω 0 and φ 0 - angular velocity and angle of rotation at the initial time t = 0 .

Plane-parallel motion of a rigid body

Plane-parallel or flat called such a motion of a rigid body in which all its points move parallel to some fixed plane. Let's choose a rectangular coordinate system Oxyz . The x and y axes will be located in the plane in which the points of the body move. Then all z - coordinates of body points remain constant, z - components of velocities and accelerations are equal to zero. The vectors of angular velocity and angular acceleration, on the contrary, are directed along the z axis. Their x and y components are zero.

The projections of the velocities of two points of a rigid body on the axis passing through these points are equal to each other.
v A cos α = v B cos β.

Instantaneous center of speed

Instant center of speeds is a point on a plane figure whose velocity in this moment equals zero.

To determine the position of the instantaneous center of velocities P of a plane figure, you only need to know the directions of the velocities and its two points A and B. To do this, we draw a straight line through point A perpendicular to the direction of speed. Through point B we draw a line perpendicular to the direction of velocity. The point of intersection of these lines is the instantaneous center of velocities P . Angular velocity of body rotation:
.

If the speeds of two points are parallel to each other, then ω = 0 . The speeds of all points of the body are equal to each other (at a given time).

If the speed of any point A of a flat body and its angular speed ω are known, then the speed of an arbitrary point M is determined by the formula (1) , which can be represented as the sum of translational and rotational motion:
,
where is the speed of the rotational movement of the point M relative to the point A. That is, the speed that point M would have when rotating along a circle of radius |AM| with angular velocity ω if point A were fixed.
Relative velocity module:
v MA = ω |AM| .
The vector is directed tangentially to the circle of radius |AM| centered at point A.

Determination of accelerations of points of a flat body is performed using the formula (2) . The acceleration of any point M is equal to the vector sum of the acceleration of some point A and the acceleration of point M during rotation around the point A, considering the point A fixed:
.
can be decomposed into tangent and normal accelerations:
.
The tangential acceleration is directed tangentially to the trajectory. Normal acceleration is directed from point M to point A. Here ω and ε are the angular velocity and angular acceleration of the body.

Complex point movement

Let O 1 x 1 y 1 z 1- fixed rectangular coordinate system. The speed and acceleration of the point M in this coordinate system will be called absolute speed and absolute acceleration.

Let Oxyz be a moving rectangular coordinate system, say, rigidly connected to some rigid body moving relative to the frame O 1 x 1 y 1 z 1. The speed and acceleration of the point M in the coordinate system Oxyz will be called relative speed and relative acceleration . Let be the angular velocity of rotation of the system Oxyz with respect to O 1 x 1 y 1 z 1.

Let us consider a point that coincides, at a given moment of time, with point M and is fixed relative to the Oxyz system (a point rigidly connected to a rigid body). The speed and acceleration of such a point in the coordinate system O 1 x 1 y 1 z 1 we will call the portable speed and portable acceleration .

Velocity addition theorem

The absolute speed of a point is equal to the vector sum of the relative and translational speeds:
.

Acceleration addition theorem (Coriolis theorem)

The absolute acceleration of a point is equal to the vector sum of the relative, translational and Coriolis accelerations:
,
where
- Coriolis acceleration.

References:
S. M. Targ, Short course theoretical mechanics, graduate School", 2010.

Speed ​​is one of the main characteristics. It expresses the very essence of movement, i.e. determines the difference that exists between a stationary body and a moving body.

The SI unit for speed is m/s.

It is important to remember that speed is a vector quantity. The direction of the velocity vector is determined by the movement. The velocity vector is always directed tangentially to the trajectory at the point through which the moving body passes (Fig. 1).

For example, consider the wheel of a moving car. The wheel rotates and all points of the wheel move in circles. Spray flying from the wheel will fly along tangents to these circles, indicating the direction of the velocity vectors of the individual points of the wheel.

Thus, the speed characterizes the direction of motion of the body (the direction of the velocity vector) and the speed of its movement (modulus of the velocity vector).

Negative speed

Can the speed of a body be negative? Yes maybe. If the speed of the body is negative, this means that the body is moving in the direction opposite to the direction of the coordinate axis in the selected frame of reference. Figure 2 shows the movement of the bus and the car. The speed of the car is negative and the speed of the bus is positive. It should be remembered that speaking of the sign of the velocity, we mean the projection of the velocity vector onto the coordinate axis.

Uniform and uneven movement

In general, the speed depends on time. According to the nature of the dependence of speed on time, the movement is uniform and uneven.

DEFINITION

Uniform movement is a movement with a constant modulo speed.

In the case of uneven movement, they talk about:

Examples of solving problems on the topic "Speed"

EXAMPLE 1

EXAMPLE 2

Position material point in space at a given time is determined in relation to some other body, which is called reference body.

Contacts him frame of reference- a set of coordinate systems and clocks associated with the body, in relation to which the movement of some other material points is being studied. The choice of reference system depends on the objectives of the study. In kinematic studies, all frames of reference are equal (Cartesian, polar). In problems of dynamics, a predominant role is played by inertial systems reference, in relation to which differential equations movements are simpler.

In the Cartesian coordinate system, the position of the point BUT at a given time with respect to this system is determined by three coordinates X, at and z, or the radius vector (Fig. 1.1). When a material point moves, its coordinates change over time. In the general case, its motion is determined by the equations

or vector equation

=(t). (1.2)

These equations are called kinematic equations of motion material point.

Excluding time t in the system of equations (1.1), we obtain the equation movement trajectories material point. For example, if the kinematic equations of motion of a point are given in the form:

then, excluding t, we get:

those. point moves in a plane z= 0 along an elliptical trajectory with semi-axes equal to a and b.

Trajectory of movement material point is the line described by this point in space. Depending on the shape of the trajectory, the movement can be straightforward and curvilinear.

Consider the motion of a material point along an arbitrary trajectory AB(Fig. 1.2). Let's start counting the time from the moment when the point was in the position BUT (t= 0). Trajectory section length AB passed by the material point from the moment t= 0 is called path length and is a scalar function of time. The vector drawn from the initial position of the moving point to its current position is called displacement vector. With rectilinear motion, the displacement vector coincides with the corresponding section of the trajectory and its module is equal to the distance traveled.

Speed is a vector physical quantity, introduced to determine the speed of movement and its direction at a given time.

Let the material point move along a curvilinear trajectory and at the moment of time t it corresponds to the radius vector . (Fig. 1.3). For a short period of time, the point will pass the way and get an infinitesimal displacement . Distinguish between average and instantaneous speed.

Average speed vector is the ratio of the increment of the radius-vector of a point to the time interval:

The vector is directed in the same way as . With an unlimited decrease in , the average speed tends to a limit value, which is called instant speed or simply speed:

Thus, speed is a vector quantity equal to the first derivative of the radius-vector of a moving point with respect to time. Since the secant coincides with the tangent in the limit, the velocity vector is directed tangentially to the trajectory in the direction of motion.

As the length of the arc decreases, it approaches the length of the chord that subtends it more and more, i.e. the numerical value of the speed of a material point is equal to the first derivative of the length of its path with respect to time:

In this way,

From expression (1.5) we obtain Integrating over time from to , we find the length of the path traveled by a material point in time :

If the direction of the instantaneous velocity vector does not change during the motion of a material point, this means that the point moves along a trajectory, the tangents to which at all points have the same direction. Only rectilinear trajectories have this property. So the movement in question is straightforward.

If the direction of the velocity vector of a material point changes over time, the point will describe curvilinear trajectory.

If the numerical value of the instantaneous velocity of a point remains constant during the movement, then such a movement is called uniform. In this case

This means that for arbitrary equal intervals of time, a material point passes paths of equal length.

If for arbitrary equal intervals of time a point passes paths of different lengths, then the numerical value of its speed changes over time. Such a movement is called uneven. In this case, a scalar value is used, called average speed of uneven movement on this part of the trajectory. It is equal to the numerical value of the speed of such a uniform movement, at which the same time is spent on the passage of the path, as with a given uneven movement:

If a material point simultaneously participates in several movements, then according to the law of independence of motion its resulting displacement is equal to the vector sum of the displacements performed by it in the same time in each of the movements separately. Therefore, the speed of the resulting motion is found as vector sum speeds of all those movements in which the material point participates.

In nature, movements are most often observed in which the speed changes both in magnitude (modulus) and in direction, i.e. dealing with uneven movements. To characterize the change in the speed of such movements, the concept is introduced acceleration.

Let the moving point move from the position BUT into position AT(Fig. 1.4). Vector specifies the speed of a point at a position BUT. Pregnant AT the point acquired a speed different from both in magnitude and in direction and became equal to . Move the vector to a point BUT and find .

Average acceleration non-uniform movement in the time interval from to is called a vector quantity equal to the ratio of the change in speed to the time interval:

Obviously, the vector coincides in direction with the velocity change vector.

Instant acceleration or acceleration material point at time will be the limit of the average acceleration:

Thus, acceleration is a vector quantity equal to the first derivative of velocity with respect to time.

Let's decompose the vector into two components. For this, from the point BUT in the direction of velocity, we set aside the vector equal in absolute value to . Then the vector equal to determines the change in speed modulo(value) for time , i.e. . The second component of the vector characterizes the change in speed over time towards - .

The component of acceleration, which determines the change in speed in magnitude, is called tangential component. Numerically, it is equal to the first time derivative of the velocity modulus:

Let us find the second component of acceleration, called normal component. Let's say the point AT close enough to the point BUT, so the path can be considered an arc of a circle of some radius r, little different from a chord AB. From the similarity of triangles AOB and EAD follows that

whence In the limit at so the second component of the acceleration is equal to:

It is in the direction and is directed to the center of curvature of the trajectory along the normal. She is also called centripetal acceleration.

Full acceleration body is the geometric sum of the tangential and normal components:

From fig. 1.5 it follows that the total acceleration module is equal to:

The direction of full acceleration is determined by the angle between the vectors and . It's obvious that

Depending on the values ​​of the tangential and normal components of the acceleration, the movement of the body is classified differently. If (the magnitude of the velocity does not change in magnitude), the movement is uniform. If > 0, the movement is called accelerated, if< 0 - slow. If = const0, then the movement is called equally variable. Finally, in any rectilinear motion (no change in direction of speed).

Thus, the movement of a material point can be of the following types:

1) - rectilinear uniform movement ();

2) - rectilinear uniform motion. With this type of movement

If the initial moment of time , and the initial speed , then, denoting and , we get:

where . (1.16)

Integrating this expression from zero to an arbitrary point in time, we obtain a formula for finding the length of the path traveled by a point during uniformly variable motion:

3) - rectilinear movement with variable acceleration;

4) - the modulo speed does not change, which shows that the radius of curvature must be constant. Therefore, this circular motion is uniform;

5) - uniform curvilinear motion;

6) - curvilinear uniform motion;

7) - curvilinear movement with variable acceleration.

Kinematics of rotational motion of a rigid body

As already noted, the rotational motion of an absolutely rigid body around a fixed axis is such a movement in which all points of the body move in planes perpendicular to a fixed straight line, called the axis of rotation, and describe circles whose centers lie on this axis.

Consider a rigid body that rotates around a fixed axis (Fig. 1.6). Then individual points of this body will describe circles of different radii, the centers of which lie on the axis of rotation. Let some point A move along a circle of radius R. Its position after a period of time is set by the angle .

angular velocity of rotation is a vector numerically equal to the first derivative of the angle of rotation of the body with respect to time and directed along the axis of rotation according to the rule of the right screw:

The unit of measure for angular velocity is radians per second (rad/s).

Thus, the vector determines the direction and speed of rotation. If , then the rotation is called uniform.

The angular velocity can be associated with the linear velocity of an arbitrary point A. Let the point pass along the arc of a circle in time, the length of the path. Then the linear speed of the point will be equal to:

With uniform rotation, it can be characterized rotation period T- the time for which the point of the body makes one complete revolution, i.e. rotates through an angle of 2π:

Number full revolutions performed by the body during uniform motion in a circle, per unit time is called speed:

To characterize the non-uniform rotation of a body, the concept is introduced angular acceleration. Angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect to time:

When a body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the angular velocity vector (Fig. 1.7); during accelerated motion, the vector is directed in the same direction as , and in the opposite direction during slow rotation.

Let us express the tangential and normal components of the point acceleration BUT rotating body in terms of angular velocity and angular acceleration:

In the case of equally variable movement of a point along a circle ():

where is the initial angular velocity.

The translational and rotational motions of a rigid body are only the simplest types of its motion. In general, the motion of a rigid body can be quite complex. However, in theoretical mechanics it is proved that any complex motion of a rigid body can be represented as a set of translational and rotational movements.

The kinematic equations of translational and rotational motions are summarized in Table. 1.1.

Table 1.1

Brief conclusions:

The part of physics that studies the laws of mechanical motion and the causes that cause or change this motion is called mechanics. Classical mechanics (Newton-Galilean mechanics) studies the laws of motion of macroscopic bodies whose speeds are small compared to the speed of light in vacuum.

- Kinematic- a branch of mechanics, the subject of which is the movement of bodies without considering the causes by which this movement is due.

In mechanics, to describe the motion of bodies, depending on the conditions of specific problems, various physical models : material point, absolutely rigid body, absolutely elastic body, absolutely inelastic body.

The movement of bodies occurs in space and time. Therefore, in order to describe the motion of a material point, it is necessary to know in what places in space this point was and at what moments in time it passed one or another position. The set of the reference body, the coordinate system associated with it and the clocks synchronized with each other is called reference system.

The vector drawn from the initial position of the moving point to its position at a given time is called displacement vector. The line described by a moving material point (body) relative to the chosen reference system is called trajectory. Depending on the shape of the trajectory, there are rectilinear and curvilinear traffic. The length of the section of the trajectory traversed by a material point in a given period of time is called path length.

- Speed is a vector physical quantity that characterizes the speed of movement and its direction at a given time. Instant Speed is determined by the first derivative of the radius-vector of the moving point with respect to time:

The instantaneous velocity vector is directed tangentially to the trajectory in the direction of motion. The module of the instantaneous velocity of a material point is equal to the first derivative of the length of its path with respect to time:

- Acceleration- vector physical quantity for the characteristic uneven movement. It determines the rate of change of speed in magnitude and direction. Instant Boost- vector quantity equal to the first derivative of the speed with respect to time:

Tangential component of acceleration characterizes the rate of change of speed in size(directed tangentially to the motion path):

Normal component of acceleration characterizes the rate of change of speed towards(directed towards the center of curvature of the trajectory):

Full acceleration with curvilinear motion - the geometric sum of the tangential and normal components:

3. What is the frame of reference? What is a displacement vector?

4. What movement is called translational? Rotational?

5. What characterizes speed and acceleration? Give definitions of average speed and average acceleration, instantaneous speed and instantaneous acceleration.

6. Write an equation for the trajectory of a body thrown horizontally at a speed v 0 from a certain height. Air resistance is ignored.

7. What characterize the tangential and normal components of acceleration? What are their modules?

8. How can motion be classified depending on the tangential and normal components of acceleration?

9. What is called angular velocity and angular acceleration? How are directions determined?

10. What formulas are related to linear and angular characteristics of movement?

Examples of problem solving

Task 1. Neglecting air resistance, determine the angle at which the body is thrown to the horizon if the maximum height of the body is equal to 1/4 of its flight range (Fig. 1.8).

And why is it needed. We already know what a frame of reference, relativity of motion and a material point are. Well, it's time to move on! Here we will review the basic concepts of kinematics, bring together the most useful formulas on the basics of kinematics, and give a practical example of solving the problem.

Let's solve the following problem: A point moves in a circle with a radius of 4 meters. The law of its motion is expressed by the equation S=A+Bt^2. A=8m, B=-2m/s^2. At what point in time is the normal acceleration of a point equal to 9 m/s^2? Find the speed, tangential and total acceleration of the point for this moment in time.

Solution: we know that in order to find the speed, we need to take the first time derivative of the law of motion, and the normal acceleration is equal to the private square of the speed and the radius of the circle along which the point moves. Armed with this knowledge, we find the desired values.

Need help solving problems? A professional student service is ready to provide it.

Based on the definition of speed, we can say that speed is a vector. It is directly expressed in terms of a displacement vector referred to a time interval, and must have all the properties of a displacement vector.

The direction of the velocity vector, as well as the direction of the physically small displacement vector, is determined from the trajectory drawing. This can be seen clearly in simple examples.

If you touch a rotating grindstone with an iron plate, then the sawdust removed by it will acquire the speed of those points of the stone that the plate touched, and then fly away in the direction of the vector of this speed. All points of the stone move in circles. During the experiment, it is clearly seen that the incandescent particles-sawdust that come off go along the tangents to these circles, indicating the directions of the velocity vectors of individual points of the rotating grindstone.

Pay attention to how the outlet pipes are located at the casing of the centrifugal water pump or at the milk separator. In these machines, fluid particles are forced to move in circles and then allowed to exit into a hole located in the direction of the vector of the velocity that they have at the moment of exit. The direction of the velocity vector at this moment coincides with the direction of the tangent to the trajectory of the fluid particles. And the outlet pipe is also directed along this tangent.

In the same way, they provide the exit of particles in modern accelerators of electrons and protons in nuclear research.

So, we have seen that the direction of the velocity vector is determined by the trajectory of the body. The velocity vector is always directed along the tangent to the trajectory at the point through which the moving body passes.

In order to determine in which direction the velocity vector is directed along the tangent and what its module is, one must refer to the law of motion. Let us assume that the law of motion is given by the graph shown in Fig. 1.54. Let's take the path length increment corresponding to the small vector by which the velocity vector is determined. Let us remember that the sign indicates

the direction of movement along the trajectory, and therefore determines the orientation of the velocity vector along the tangent. Obviously, the speed modulus will be determined through the modulus of this path length increment.

Thus, the modulus of the velocity vector and the orientation of the velocity vector along the tangent to the trajectory can be determined from the relation

Here is an algebraic quantity, the sign of which indicates in which direction the velocity vector is directed tangentially to the trajectory.

So, we have seen that the modulus of the velocity vector can be found from the graph of the law of motion. The ratio determines the slope of the a tangent in this graph. The slope of the tangent on the graph of the law of motion will be the greater, the greater, i.e., the greater the speed of movement at the selected moment.

Let us once again pay attention to the fact that for a complete determination of the speed, simultaneous knowledge of the trajectory and the law of motion is required. The drawing of the trajectory allows you to determine the direction of the speed, and the graph of the law of motion - its module and sign.

If we now turn again to the definition of mechanical motion, we will be convinced that after the introduction of the concept of speed, nothing more is required for a complete description of any motion. Using the concepts of radius vector, displacement vector, velocity vector, path length, trajectory and law of motion, you can get answers to all questions related to determining the features of any movement. All these concepts are interrelated with each other, and knowledge of the trajectory and the law of motion allows you to find any of these quantities.