Rotational movement solid. Rotational motion is the motion of a rigid body in which all its points lying on a certain straight line, called the axis of rotation, remain motionless.

During rotational motion, all other points of the body move in planes perpendicular to the axis of rotation and describe circles whose centers lie on this axis.

To determine the position of a rotating body, we draw two half-planes through the z-axis: half-plane I - stationary and half-plane II - connected to the rigid body and rotating with it (Fig. 2.4). Then the position of the body at any moment of time will be uniquely determined by the angle j between these half-planes, taken with the corresponding sign, which is called the angle of rotation of the body.

When a body rotates, the angle of rotation j changes depending on time, i.e., it is a function of time t:

This equation is called equation rotational motion of a rigid body.

The main kinematic characteristics of the rotational motion of a rigid body are its angular velocity w and angular acceleration e.

If during time D t= t1 + t the body makes a turn by Dj = j1 –j, then the average angular velocity of the body during this period of time will be equal to

(1.16)

To determine the value of the angular velocity of a body at a given time t let's find the limit of the ratio of the increment of the rotation angle Dj to the time interval D t as the latter tends to zero:

(2.17)

Thus, the angular velocity of the body at a given time is numerically equal to the first derivative of the angle of rotation with respect to time. The sign of the angular velocity w coincides with the sign of the angle of rotation of the body j: w > 0 at j > 0, and vice versa, if j < 0. then w < 0. The dimension of angular velocity is usually 1/s, so radians are dimensionless.

Angular velocity can be represented as a vector w , the numerical value of which is equal to dj/dt which is directed along the axis of rotation of the body in the direction from which the rotation can be seen occurring counterclockwise.

The change in the angular velocity of a body over time is characterized by angular acceleration e. By analogy with finding the average value of angular velocity, we will find an expression for determining the value of the average acceleration:

(2.18)

Then the acceleration of the rigid body at a given time is determined from the expression

(2.19)

i.e., the angular acceleration of the 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/s 2.

The angular acceleration of a rigid body, like the angular velocity, can be represented as a vector. The angular acceleration vector coincides in direction with the angular velocity vector during the accelerated movement of a solid top and is directed towards the opposite side in slow motion.

Having established the characteristics of the motion of a rigid body as a whole, let us move on to studying the motion of its individual points. Let's consider some point M solid body located at a distance h from the axis of rotation r (Fig. 2.3).

When the body rotates, the point M will describe a circular point of radius h centered on the axis of rotation and lying in a plane perpendicular to this axis. If during the time dt an elementary whipping of the body occurs at an angle dj , then point M at the same time makes an elementary movement along its trajectory dS = h*dj ,. Then the speed of point M was determined from the expression

(2.20)

The speed is called the linear or circumferential speed of point M.

Thus, the linear velocity of a point on a rotating rigid body is numerically equal to the product of the angular velocity of the body and the distance from this point to the axis of rotation. Since for all points of the body the angular velocity w; has the same value, then from the formula for linear speed it follows that the linear speeds of the points of a rotating body are proportional to their distances from the axis of rotation. The linear velocity of a point of a rigid body is a vector n directed tangentially to the circle described by the point M.

Beli the distance from the axis of rotation of a solid pel to a certain point M considered as the radius vector h of point M, then the linear velocity vector of point v can be represented as the vector product of the angular velocity vector w radius vector h:

V = w * h (2/21)

Indeed, the result vector product(2.21) is a vector equal in magnitude to the product w*h and directed (Fig. 2.5) perpendicular to the plane in which the two factors lie, in the direction from which the closest combination of the first factor with the second is observed to occur counterclockwise, i.e. tangent to the trajectory of point M.

Thus, the vector resulting from the vector product (2.21) corresponds in magnitude and direction to the linear velocity vector of point M.

Rice. 2.5

To find an expression for acceleration A point M, we differentiate with respect to time the expression (2.21) for the speed of the point

(2.22)

Taking into account that dj/dt=e, and dh/dt = v, we write expression (2.22) in the form

where аг and аn are, respectively, the tangent and normal components of the total acceleration of a point of a body during rotational motion, determined from the expressions

The tangential component of the total acceleration of a body point (tangential acceleration) at characterizes the change in the velocity vector in magnitude and is directed tangentially to the trajectory of the body point in the direction of the velocity vector during accelerated motion or in the opposite direction during slow motion. The magnitude of the tangential acceleration vector of a point of a body during rotational motion of a rigid body is determined by the expression

(2,25)

Normal component of total acceleration (normal acceleration) A" arises due to a change in the direction of the velocity vector of a point when painting a solid body. As follows from expression (2.24) for normal acceleration, this acceleration is directed along the radius h to the center of the circle along which the point moves. The modulus of the normal acceleration vector of a point during rotational motion of a rigid body is determined taking into account (2.20) by the expression

Rotational motion of a rigid body around a fixed axis is such a motion in which any two points belonging to the body (or invariably associated with it) remain motionless throughout the movement(Fig. 2.2) .

Figure 2.2

Passing through fixed points A And IN the straight line is called axis of rotation. Since the distance between the points of a rigid body must remain unchanged, it is obvious that during rotational motion all points belonging to the axis will be motionless, and all others will describe circles, the planes of which are perpendicular to the axis of rotation, and the centers lie on this axis. To determine the position of a rotating body, we draw through the axis of rotation along which the axis is directed Az, half-plane І – fixed and half-plane ІІ embedded in the body itself and rotating with it. Then the position of the body at any moment of time is uniquely determined by the angle taken with the corresponding sign φ between these planes, which we call body rotation angle. We will consider the angle φ positive if it is delayed from a fixed plane in a counterclockwise direction (for an observer looking from the positive end of the axis Az), and negative if clockwise. Measure angle φ We'll be in radians. To know the position of a body at any moment in time, you need to know the dependence of the angle φ from time to time t, i.e.

.

This equation expresses the law of rotational motion of a rigid body around a fixed axis.

The main kinematic characteristics of the rotational motion of a rigid body are its angular velocity ω and angular acceleration ε.

9.2.1. Angular velocity and angular acceleration of a body

The quantity characterizing the rate of change in the angle of rotation φ over time is called angular velocity.

If during a period of time
the body rotates through an angle
, then the numerically average angular velocity of the body during this period of time will be
. In the limit at
we get

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.

Sign Rule: When rotation occurs counterclockwise, ω> 0, and when clockwise, then ω< 0.

or, since radian is a dimensionless quantity,
.

In theoretical calculations it is more convenient to use the angular velocity 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 is visible counterclockwise. This vector immediately determines the magnitude of the angular velocity, the axis of rotation, and the direction of rotation around this axis.

The quantity that characterizes the rate of change in angular velocity over time is called the angular acceleration of the body.

If during a period of time
the increment in angular velocity is equal to
, then the relation
, i.e. determines the value of the average acceleration of a rotating body over time
.

When striving
we obtain the magnitude of the angular acceleration at the moment t:

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 in time.

The unit of measurement is usually used or, which is also,
.

If the modulus of angular velocity increases with time, the rotation of the body is called accelerated, and if it decreases, - slow When the values ω And ε have the same signs, then the rotation will be accelerated, when they are different, it will be slowed down. By analogy with angular velocity, angular acceleration can also be represented as a vector , directed along the axis of rotation. At the same time

.

If a body rotates in an accelerated direction coincides with , and opposite with slow rotation.

If the angular velocity of a body remains constant during movement ( ω= const), then the rotation of the body is called uniform.

From
we have
. Hence, considering that at the initial moment of time
corner
, and taking the integrals to the left of to , and on the right from 0 to t, we will finally get

.

With uniform rotation, when =0,
And
.

The speed of uniform rotation is often determined by the number of revolutions per minute, denoting this value by n rpm Let's find the relationship between n rpm and ω 1/s. With one revolution the body will rotate by 2π, and with n rpm at 2π n; this turn is done in 1 minute, i.e. t= 1min=60s. It follows from this that

.

If the angular acceleration of a body remains constant throughout its motion (ε = const), then the rotation is called equally variable.

At the initial moment of time t=0 angle
, and the angular velocity
(- initial angular velocity).
;
=ε
. Integrating the left side of to , and the right one from 0 to t, we'll find

Angular velocityω of this rotation
. If ω and ε have the same signs, the rotation will be uniformly accelerated, and if different - equally slow.

And Savelyeva.

During the forward motion of a body (§ 60 in the textbook by E. M. Nikitin), all its points move along identical trajectories and at each given moment they have equal speeds and equal accelerations.

Therefore, the translational motion of a body is determined by the movement of any one point, usually the movement of the center of gravity.

When considering the movement of a car (problem 147) or a diesel locomotive (problem 141) in any problem, we actually consider the movement of their centers of gravity.

The rotational motion of a body (E.M. Nikitin, § 61) cannot be identified with the movement of any one of its points. The axis of any rotating body (diesel flywheel, electric motor rotor, machine spindle, fan blades, etc.) during movement occupies the same place in space relative to the surrounding stationary bodies.

Movement of a material point or forward movement bodies are characterized depending on time linear quantities s (path, distance), v (speed) and a (acceleration) with its components a t and a n.

Rotational movement bodies depending on time t characterize angular values: φ (angle of rotation in radians), ω (angular velocity in rad/sec) and ε (angular acceleration in rad/sec 2).

The law of rotational motion of a body is expressed by the equation
φ = f(t).

Angular velocity- a quantity characterizing the speed of rotation of a body is defined in the general case as the derivative of the angle of rotation with respect to time
ω = dφ/dt = f" (t).

Angular acceleration- a quantity characterizing the rate of change of angular velocity is defined as the derivative of angular velocity
ε = dω/dt = f"" (t).

When starting to solve problems on the rotational motion of a body, it is necessary to keep in mind that in technical calculations and problems, as a rule, angular displacement is expressed not in radians φ, but in revolutions φ about.

Therefore, it is necessary to be able to move from the number of revolutions to the radian measurement of angular displacement and vice versa.

Since one full turn corresponds to 2π rad, then
φ = 2πφ about and φ about = φ/(2π).

Angular speed in technical calculations is very often measured in revolutions produced per minute (rpm), so it is necessary to clearly understand that ω rad/sec and n rpm express the same concept - the speed of rotation of a body (angular speed) , but in different units - in rad/sec or in rpm.

The transition from one unit of angular velocity to another is made according to the formulas
ω = πn/30 and n = 30ω/π.

When a body rotates, all its points move in circles, the centers of which are located on one fixed straight line (the axis of the rotating body). When solving the problems given in this chapter, it is very important to clearly understand the relationship between the angular quantities φ, ω and ε, which characterize the rotational motion of the body, and the linear quantities s, v, a t and an, characterizing the movement of various points of this body (Fig. 205).

If R is the distance from the geometric axis of a rotating body to any point A (in Fig. 205 R=OA), then the relationship between φ - the angle of rotation of the body and s - the distance traveled by a point of the body in the same time is expressed as follows:
s = φR.

The relationship between the angular velocity of a body and the velocity of a point at each given moment is expressed by the equality
v = ωR.

The tangential acceleration of a point depends on the angular acceleration and is determined by the formula
a t = εR.

The normal acceleration of a point depends on the angular velocity of the body and is determined by the relationship
a n = ω 2 R.

When solving the problem given in this chapter, it is necessary to clearly understand that rotation is the movement of a rigid body, not a point. Taken separately material point does not rotate, but moves in a circle - makes a curvilinear movement.

§ 33. Uniform rotational motion

If the angular velocity is ω=const, then the rotational motion is called uniform.

The uniform rotation equation has the form
φ = φ 0 + ωt.

In the particular case when the initial angle of rotation φ 0 =0,
φ = ωt.

Angular velocity of a uniformly rotating body
ω = φ/t
can be expressed like this:
ω = 2π/T,
where T is the period of rotation of the body; φ=2π - angle of rotation for one period.

§ 34. Uniformly alternating rotational motion

Rotational motion with variable angular velocity is called uneven (see below § 35). If the angular acceleration ε=const, then the rotational motion is called equally variable. Thus, uniform rotation of the body is special case uneven rotational movement.

Equation of uniform rotation
(1) φ = φ 0 + ω 0 t + εt 2 /2
and the equation expressing the angular velocity of a body at any time,
(2) ω = ω 0 + εt
represent a set of basic formulas for the rotational uniform motion of a body.

These formulas include only six quantities: three constants for a given problem φ 0, ω 0 and ε and three variables φ, ω and t. Consequently, the condition of each problem for uniform rotation must contain at least four specified quantities.

For the convenience of solving some problems, two more auxiliary formulas can be obtained from equations (1) and (2).

Let us exclude angular acceleration ε from (1) and (2):
(3) φ = φ 0 + (ω + ω 0)t/2.

Let us exclude time t from (1) and (2):
(4) φ = φ 0 + (ω 2 - ω 0 2)/(2ε).

In the particular case of uniformly accelerated rotation starting from a state of rest, φ 0 =0 and ω 0 =0. Therefore, the above basic and auxiliary formulas take the following form:
(5) φ = εt 2 /2;
(6) ω = εt;
(7) φ = ωt/2;
(8) φ = ω 2 /(2ε).

§ 35. Uneven rotational motion

Let's consider an example of solving a problem in which non-uniform rotational motion of a body is specified.

Progressive is the movement of a rigid body in which any straight line invariably associated with this body remains parallel to its initial position.

Theorem. During the translational motion of a rigid body, all its points describe identical trajectories and at each given moment have equal velocity and acceleration in magnitude and direction.

Proof. Let's draw through two points and , a linearly moving body segment
and consider the movement of this segment in position
. At the same time, the point describes the trajectory
, and point – trajectory
(Fig. 56).

Considering that the segment
moves parallel to itself, and its length does not change, it can be established that the trajectories of points And will be the same. This means that the first part of the theorem is proven. We will determine the position of the points And vector method relative to a fixed origin . Moreover, these radii - vectors are dependent
. Because. neither the length nor the direction of the segment
does not change when the body moves, then the vector

. Let's move on to determining the velocities using dependence (24):

, we get
.

Let's move on to determining accelerations using dependence (26):

, we get
.

From the proven theorem it follows that the translational motion of a body will be completely determined if the motion of only one point is known. Therefore, the study of the translational motion of a rigid body comes down to the study of the movement of one of its points, i.e. to the point kinematics problem.

Topic 11. Rotational motion of a rigid body

Rotational This is the movement of a rigid body in which two of its points remain motionless throughout the entire movement. In this case, the straight line passing through these two fixed points is called axis of rotation.

During this movement, each point of the body that does not lie on the axis of rotation describes a circle, the plane of which is perpendicular to the axis of rotation, and its center lies on this axis.

We draw through the axis of rotation a fixed plane I and a movable plane II, invariably connected to the body and rotating with it (Fig. 57). The position of plane II, and accordingly the entire body, in relation to plane I in space, is completely determined by the angle . When a body rotates around an axis this angle is a continuous and unambiguous function of time. Therefore, knowing the law of change of this angle over time, we can determine the position of the body in space:

- law of rotational motion of a body. (43)

In this case, we will assume that the angle measured from a fixed plane in the direction opposite to the clockwise movement, when viewed from the positive end of the axis . Since the position of a body rotating around a fixed axis is determined by one parameter, such a body is said to have one degree of freedom.

Angular velocity

The change in the angle of rotation of a body over time is called angular body speed and is designated
(omega):

.(44)

Angular velocity, just like linear velocity, is a vector quantity, and this vector built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the rotation of the body counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (44). Application point on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action. If we denote the orth-vector of the rotation axis by , then we obtain the vector expression for angular velocity:

. (45)

Angular acceleration

The rate of change in the angular velocity of a body over time is called angular acceleration body and is designated (epsilon):

. (46)

Angular acceleration is a vector quantity, and this vector built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the direction of rotation of the epsilon counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (46). Application point on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action.

If we denote the orth-vector of the rotation axis by , then we obtain the vector expression for angular acceleration:

. (47)

If the angular velocity and acceleration are of the same sign, then the body rotates expedited, and if different - slowly. An example of slow rotation is shown in Fig. 58.

Let us consider special cases of rotational motion.

1. Uniform rotation:

,
.

,
,
,

,
. (48)

2. Equal rotation:

.

,
,
,
,
,
,
,
,

,
,
.(49)

Relationship between linear and angular parameters

Consider the movement of an arbitrary point
rotating body. In this case, the trajectory of the point will be a circle with radius
, located in a plane perpendicular to the axis of rotation (Fig. 59, A).

Let us assume that at the moment of time the point is in position
. Let us assume that the body rotates in a positive direction, i.e. in the direction of increasing angle . At a moment in time
the point will take position
. Let's denote the arc
. Therefore, over a period of time
the point has passed the way
. Her average speed , and when
,
. But, from Fig. 59, b, it is clear that
. Then. Finally we get

. (50)

Here - linear speed of the point
. As was obtained earlier, this speed is directed tangentially to the trajectory at a given point, i.e. tangent to the circle.

Thus, the module of the linear (circumferential) velocity of a point of a rotating body is equal to the product of the absolute value of the angular velocity and the distance from this point to the axis of rotation.

Now let's connect the linear components of the acceleration of a point with the angular parameters.

,
. (51)

The modulus of the tangential acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the angular acceleration of the body and the distance from this point to the axis of rotation.

,
. (52)

The modulus of normal acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the square of the angular velocity of the body and the distance from this point to the axis of rotation.

Then the expression for the total acceleration of the point takes the form

. (53)

Vector directions ,,shown in Figure 59, V.

Flat motion A rigid body is called such a movement in which all points of the body move parallel to some fixed plane. Examples of such movement:

The motion of any body whose base slides along a given fixed plane;

Rolling of a wheel along a straight section of track (rail).

We obtain the equations of plane motion. To do this, consider a flat figure moving in the plane of the sheet (Fig. 60). Let us relate this movement to a fixed coordinate system
, and with the figure itself we connect the moving coordinate system
, which moves with it.

Obviously, the position of a moving figure on a stationary plane is determined by the position of the moving axes
relative to fixed axes
. This position is determined by the position of the moving origin , i.e. coordinates ,and rotation angle , a moving coordinate system, relatively fixed, which we will count from the axis in the direction opposite to the clockwise movement.

Therefore, the movement flat figure in its plane will be completely defined if for each moment of time the values ​​are known ,,, i.e. equations of the form:

,
,
. (54)

Equations (54) are equations of plane motion of a rigid body, since if these functions are known, then for each moment of time it is possible to find from these equations, respectively ,,, i.e. determine the position of a moving figure at a given moment in time.

Let's consider special cases:

1.

, then the movement of the body will be translational, since the moving axes move while remaining parallel to their initial position.

2.

,

. With this movement, only the angle of rotation changes , i.e. the body will rotate about an axis passing perpendicular to the drawing plane through the point .

Decomposition of the motion of a flat figure into translational and rotational

Consider two consecutive positions And
occupied by the body at moments of time And
(Fig. 61). Body from position to position
can be transferred as follows. Let's move the body first progressively. In this case, the segment
will move parallel to itself to position
and then let's turn body around a point (pole) at an angle
until the points coincide And .

Hence, any plane motion can be represented as the sum of translational motion together with the selected pole and rotational motion, relative to this pole.

Let's consider methods that can be used to determine the velocities of points of a body performing plane motion.

1. Pole method. This method is based on the resulting decomposition of plane motion into translational and rotational. The speed of any point of a flat figure can be represented in the form of two components: translational, with a speed equal to the speed of an arbitrarily chosen point -poles , and rotational around this pole.

Let's consider a flat body (Fig. 62). The equations of motion are:
,
,
.

From these equations we determine the speed of the point (as in coordinate method assignments)

,
,
.

Thus, the speed of the point - the quantity is known. We take this point as a pole and determine the speed of an arbitrary point
bodies.

Speed
will consist of a translational component , when moving along with the point , and rotational
, when rotating the point
relative to the point . Point speed move to point
parallel to itself, since during translational motion the velocities of all points are equal both in magnitude and direction. Speed
will be determined by dependence (50)
, and this vector is directed perpendicular to the radius
in the direction of rotation
. Vector
will be directed along the diagonal of a parallelogram built on vectors And
, and its module is determined by the dependency:

, .(55)

2. Theorem on the projections of velocities of two points of a body.

The projections of the velocities of two points of a rigid body onto a straight line connecting these points are equal to each other.

Consider two points of the body And (Fig. 63). Taking a point beyond the pole, we determine the direction depending on (55):
. We project this vector equality onto the line
and considering that
perpendicular
, we get

3. Instantaneous velocity center.

Instantaneous velocity center(MCS) is a point whose speed at a given time is zero.

Let us show that if a body does not move translationally, then such a point exists at every moment of time and, moreover, is unique. Let at a moment in time points And bodies lying in section , have speeds And , not parallel to each other (Fig. 64). Then point
, lying at the intersection of perpendiculars to the vectors And , and there will be an MCS, since
.

Indeed, if we assume that
, then according to Theorem (56), the vector
must be perpendicular at the same time
And
, which is impossible. From the same theorem it is clear that no other section point at this moment in time cannot have a speed equal to zero.

Using the pole method
- pole, determine the speed of the point (55): because
,
. (57)

A similar result can be obtained for any other point of the body. Therefore, the speed of any point on the body is equal to its rotational speed relative to the MCS:

,
,
, i.e. the velocities of body points are proportional to their distances to the MCS.

From the three considered methods for determining the velocities of points of a flat figure, it is clear that the MCS is preferable, since here the speed is immediately determined both in magnitude and in the direction of one component. However, this method can be used if we know or can determine the position of the MCS for the body.

Determining the position of the MCS

1. If we know for a given position of the body the directions of the velocities of two points of the body, then the MCS will be the point of intersection of the perpendiculars to these velocity vectors.

2. The velocities of two points of the body are antiparallel (Fig. 65, A). In this case, the perpendicular to the velocities will be common, i.e. The MCS is located somewhere on this perpendicular. To determine the position of the MCS, it is necessary to connect the ends of the velocity vectors. The point of intersection of this line with the perpendicular will be the desired MCS. In this case, the MCS is located between these two points.

3. The velocities of two points of the body are parallel, but not equal in magnitude (Fig. 65, b). The procedure for obtaining the MDS is similar to that described in paragraph 2.

d) The velocities of two points are equal in both magnitude and direction (Fig. 65, V). We obtain the case of instantaneous translational motion, in which the velocities of all points of the body are equal. Consequently, the angular velocity of the body in this position is zero:

4. Let us determine the MCS for a wheel rolling without sliding on a stationary surface (Fig. 65, G). Since the movement occurs without sliding, at the point of contact of the wheel with the surface the speed will be the same and equal to zero, since the surface is stationary. Consequently, the point of contact of the wheel with a stationary surface will be the MCS.

Determination of accelerations of points of a plane figure

When determining the accelerations of points of a flat figure, there is an analogy with methods for determining velocities.

1. Pole method. Just as when determining velocities, we take as a pole an arbitrary point of the body whose acceleration we know or we can determine. Then the acceleration of any point of a flat figure is equal to the sum of the accelerations of the pole and the acceleration in rotational motion around this pole:

In this case, the component
determines the acceleration of a point as it rotates around the pole . When rotating, the trajectory of the point will be curvilinear, which means
(Fig. 66).

Then dependence (58) takes the form
. (59)

Taking into account dependencies (51) and (52), we obtain
,
.

2. Instant acceleration center.

Instant acceleration center(MCU) is a point whose acceleration at a given time is zero.

Let us show that at any given moment of time such a point exists. We take a point as a pole , whose acceleration
we know. Finding the angle , lying within
, and satisfying the condition
. If
, That
and vice versa, i.e. corner delayed in direction . Let's postpone from the point at an angle to vector
segment
(Fig. 67). The point obtained by such constructions
there will be an MCU.

Indeed, the acceleration of the point
equal to the sum of accelerations
poles and acceleration
in rotational motion around the pole :
.

,
. Then
. On the other hand, acceleration
forms with the direction of the segment
corner
, which satisfies the condition
. A minus sign is placed in front of the tangent of the angle , since rotation
relative to the pole counterclockwise, and the angle
is deposited clockwise. Then
.

Hence,
and then
.

Special cases of determining the MCU

1.
. Then
, and, therefore, the MCU does not exist. In this case, the body moves translationally, i.e. the velocities and accelerations of all points of the body are equal.

2.
. Then
,
. This means that the MCU lies at the intersection of the lines of action of the accelerations of the points of the body (Fig. 68, A).

3.
. Then,
,
. This means that the MCU lies at the intersection of perpendiculars to the accelerations of points of the body (Fig. 68, b).

4.
. Then
,

. This means that the MCU lies at the intersection of rays drawn to the accelerations of points of the body at an angle (Fig. 68, V).

From the considered special cases we can conclude: if we accept the point
beyond the pole, then the acceleration of any point of a flat figure is determined by the acceleration in rotational motion around the MCU:

. (60)

Complex point movement a movement in which a point simultaneously participates in two or more movements is called. With such movement, the position of the point is determined relative to the moving and relatively stationary reference systems.

The movement of a point relative to a moving reference frame is called relative motion of a point . We agree to denote the parameters of relative motion
.

The movement of that point of the moving reference system with which the moving point relative to the stationary reference system currently coincides is called portable movement of the point . We agree to denote the parameters of portable motion
.

The movement of a point relative to a fixed frame of reference is called absolute (complex) point movement . We agree to denote the parameters of absolute motion
.

As an example of complex movement, we can consider the movement of a person in a moving vehicle (tram). In this case, the human movement is related to the moving coordinate system - the tram and to the fixed coordinate system - the earth (road). Then, based on the definitions given above, the movement of a person relative to the tram is relative, the movement together with the tram relative to the ground is portable, and the movement of a person relative to the ground is absolute.

We will determine the position of the point
radii - vectors relative to the moving
and motionless
coordinate systems (Fig. 69). Let us introduce the following notation: - radius vector defining the position of the point
relative to the moving coordinate system
,
;- radius vector that determines the position of the beginning of the moving coordinate system (point ) (dots );- radius – a vector that determines the position of a point
relative to a fixed coordinate system
;
,.

Let us obtain conditions (constraints) corresponding to relative, portable and absolute motions.

1. When considering relative motion, we will assume that the point
moves relative to the moving coordinate system
, and the moving coordinate system itself
relative to a fixed coordinate system
doesn't move.

Then the coordinates of the point
will change in relative motion, but the orth-vectors of the moving coordinate system will not change in direction:

,

,

.

2. When considering portable motion, we will assume that the coordinates of the point
relative to the moving coordinate system are fixed, and the point moves along with the moving coordinate system
relatively stationary
:

,

,

,.

3. With absolute motion, the point also moves relatively
and together with the coordinate system
relatively stationary
:

Then the expressions for the velocities, taking into account (27), have the form

,
,

Comparing these dependencies, we obtain an expression for absolute speed:
. (61)

We obtained a theorem on the addition of the velocities of a point in complex motion: the absolute speed of a point is equal to the geometric sum of the relative and portable speed components.

Using dependence (31), we obtain expressions for accelerations:

,

Comparing these dependencies, we obtain an expression for absolute acceleration:
.

We found that the absolute acceleration of a point is not equal to the geometric sum of the relative and portable acceleration components. Let us determine the absolute acceleration component in parentheses for special cases.

1. Portable translational movement of the point
. In this case, the axes of the moving coordinate system
move all the time parallel to themselves, then.

,

,

,
,
,
, Then
. Finally we get

. (62)

If the portable motion of a point is translational, then the absolute acceleration of the point is equal to the geometric sum of the relative and portable components of the acceleration.

2. The portable movement of the point is non-translational. This means that in this case the moving coordinate system
rotates around the instantaneous axis of rotation with angular velocity (Fig. 70). Let us denote the point at the end of the vector through . Then, using the vector method of specifying (15), we obtain the velocity vector of this point
.

On the other side,
. Equating the right-hand sides of these vector equalities, we obtain:
. Proceeding similarly for the remaining unit vectors, we obtain:
,
.

In the general case, the absolute acceleration of a point is equal to the geometric sum of the relative and portable acceleration components plus the doubled vector product of the angular velocity vector of the portable motion and the linear velocity vector of the relative motion.

The double vector product of the angular velocity vector of the portable motion and the linear velocity vector of the relative motion is called Coriolis acceleration and is designated

. (64)

Coriolis acceleration characterizes the change in relative speed in portable movement and change in transfer speed in relative motion.

Headed
according to the vector product rule. The Coriolis acceleration vector is always directed perpendicular to the plane formed by the vectors And , in such a way that, looking from the end of the vector
, see the turn To , through the smallest angle, counterclockwise.

The Coriolis acceleration modulus is equal to.

Rotation of a rigid body around a fixed axis (axis of rotation) It is called such a movement in which the points of the body lying on the axis of rotation remain motionless during the entire time of movement.

Let the axis of rotation be the axis, which can have any direction in space. One direction of the axis is taken as positive (Fig. 28).

Through the axis of rotation we draw a fixed plane and a movable plane connected to the rotating body. Let at the initial moment of time both planes coincide. Then, at the moment of time, the position of the moving plane and the rotating body itself can be determined by the dihedral angle between the planes and the corresponding linear angle between straight lines located in these planes and perpendicular to the axis of rotation. The angle is called body rotation angle.

The position of the body relative to the selected reference system is completely determined at any time if the equation is given

where is any twice differentiable function of time. This equation is called equation of rotation of a rigid body around a fixed axis.

A body rotating around a fixed axis has one degree of freedom, since its position is determined by specifying only one parameter - the angle.

An angle is considered positive if it is plotted counterclockwise, and negative in the opposite direction when viewed from positive direction axes The trajectories of points of a body during its rotation around a fixed axis are circles located in planes perpendicular to the axis of rotation.

To characterize the rotational motion of a rigid body around a fixed axis, we introduce the concepts of angular velocity and angular acceleration. Algebraic angular velocity of the body at any moment in time is called the first derivative with respect to time of the angle of rotation at this moment, i.e. . It is a positive quantity when the body rotates counterclockwise, since the angle of rotation increases with time, and negative when the body rotates clockwise, because the angle of rotation decreases.

The angular velocity module is denoted by . Then

Algebraic angular acceleration of the body is called the first derivative with respect to time of the algebraic speed, i.e. second derivative of the rotation angle. We denote the module of angular acceleration by , then

If at , then the algebraic angular velocity increases with time and, therefore, the body rotates accelerated at the considered moment in time in positive side(counterclockwise). At and , the body rotates rapidly in the negative direction. If at , then we have slow rotation in the positive direction. When and slow rotation occurs in the negative direction.