Axis of rotation of a rigid body. Rotational motion of a rigid body around a fixed axis

31.05.2020

This article describes an important section of physics - "Kinematics and dynamics of rotational motion".

Basic concepts of kinematics of rotational motion

The rotational movement of a material point around a fixed axis is such a movement, the trajectory of which is a circle located in a plane perpendicular to the axis, and its center lies on the axis of rotation.

The rotational motion of a rigid body is a motion in which all points of the body move along concentric (the centers of which lie on the same axis) circles in accordance with the rule for the rotational motion of a material point.

Let an arbitrary rigid body T perform rotations around the axis O, which is perpendicular to the plane of the figure. Let us choose a point M on the given body. During rotation, this point will describe a circle around the O axis with a radius r.

After some time, the radius will rotate relative to its original position by an angle Δφ.

The direction of the right screw (clockwise) is taken as the positive direction of rotation. The change in the angle of rotation with time is called the equation of rotational motion of a rigid body:

φ = φ(t).

If φ is measured in radians (1 rad is the angle corresponding to an arc with a length equal to its radius), then the length of the circular arc ΔS, which the material point M will pass in time Δt, is equal to:

∆S = ∆φr.

The main elements of the kinematics of uniform rotational motion

A measure of the movement of a material point in a short period of time dt serves as an elementary rotation vector dφ.

The angular velocity of a material point or body is physical quantity, which is determined by the ratio of the elementary turn vector to the duration of this turn. The direction of the vector can be determined by the rule of the right screw along the O axis. In scalar form:

ω = dφ/dt.

If ω = dφ/dt = const, then such a motion is called uniform rotational motion. With it, the angular velocity is determined by the formula

ω = φ/t.

According to the preliminary formula, the dimension of the angular velocity

[ω] = 1 rad/s.

The uniform rotational motion of a body can be described by a period of rotation. The rotation period T is a physical quantity that determines the time during which the body around the axis of rotation performs one complete revolution ([T] = 1 s). If in the formula for the angular velocity we take t = T, φ = 2 π (full one revolution of radius r), then

ω = 2π/T,

Therefore, the rotation period is defined as follows:

T = 2π/ω.

The number of revolutions that a body makes per unit time is called the rotation frequency ν, which is equal to:

ν = 1/T.

Frequency units: [ν] \u003d 1 / c \u003d 1 c -1 \u003d 1 Hz.

Comparing the formulas for the angular velocity and rotation frequency, we obtain an expression relating these quantities:

ω = 2πν.

The main elements of the kinematics of non-uniform rotational motion

The uneven rotational motion of a rigid body or a material point around a fixed axis characterizes its angular velocity, which changes with time.

Vector ε characterizing the rate of change of the angular velocity is called the angular acceleration vector:

ε = dω/dt.

If the body rotates, accelerating, that is dω/dt > 0, the vector has a direction along the axis in the same direction as ω.

If the rotational movement is slowed down - dω/dt< 0 , then the vectors ε and ω are oppositely directed.

Comment. When an uneven rotational motion occurs, the vector ω can change not only in magnitude, but also in direction (when the rotation axis is rotated).

Relationship between quantities characterizing translational and rotational motion

It is known that the length of the arc with the angle of rotation of the radius and its value is related by the relation

∆S = ∆φr.

Then the linear velocity of a material point performing a rotational motion

υ = ΔS/Δt = Δφr/Δt = ωr.

The normal acceleration of a material point that performs rotational translational motion is defined as follows:

a = υ 2 /r = ω 2 r 2 /r.

So, in scalar form

a = ω 2 r.

Tangential accelerated material point that performs rotational motion

a = εr.

Angular moment of a material point

The vector product of the radius-vector of the trajectory of a material point with mass m i and its momentum is called the angular momentum of this point about the axis of rotation. The direction of the vector can be determined using the right screw rule.

Angular moment of a material point ( L i) is directed perpendicular to the plane drawn through r i and υ i , and forms with them the right triple of vectors (that is, when moving from the end of the vector r i To υ i the right screw will show the direction of the vector L i).

In scalar form

L = m i υ i r i sin(υ i , r i).

Considering that when moving in a circle, the radius vector and the vector linear speed for i-th material points are mutually perpendicular,

sin(υ i , r i) = 1.

So the angular momentum of a material point for rotational motion will take the form

L = m i υ i r i .

Moment of force acting on the i-th material point

The vector product of the radius-vector, which is drawn to the point of application of the force, by this force is called the moment of the force acting on i-th material point about the axis of rotation.

In scalar form

M i = r i F i sin(r i , F i).

Considering that r i sinα = l i ,M i = l i F i .

Value l i , equal to the length of the perpendicular dropped from the point of rotation to the direction of the force, is called the arm of the force F i.

Rotational dynamics

The equation for the dynamics of rotational motion is written as follows:

M = dL/dt.

The formulation of the law is as follows: the rate of change of the angular momentum of a body that rotates around a fixed axis is equal to the resulting moment about this axis of all external forces applied to the body.

Moment of momentum and moment of inertia

It is known that for the i-th material point the angular momentum in scalar form is given by the formula

L i = m i υ i r i .

If instead of the linear velocity we substitute its expression in terms of the angular one:

υ i = ωr i ,

then the expression for the angular momentum will take the form

L i = m i r i 2 ω.

Value I i = m i r i 2 is called the moment of inertia about axes i-th material point of an absolutely rigid body passing through its center of mass. Then we write the angular momentum of the material point:

L i = I i ω.

We write the angular momentum of an absolutely rigid body as the sum of the angular momentum material points that make up this body:

L = Iω.

Moment of force and moment of inertia

The law of rotation says:

M = dL/dt.

It is known that the angular momentum of a body can be represented in terms of the moment of inertia:

L = Iω.

M = Idω/dt.

Considering that the angular acceleration is determined by the expression

ε = dω/dt,

we obtain the formula for the moment of force, represented through the moment of inertia:

M = Ie.

Comment. The moment of force is considered positive if the angular acceleration by which it is caused is greater than zero, and vice versa.

Steiner's theorem. The law of addition of moments of inertia

If the body's axis of rotation does not pass through its center of mass, then its moment of inertia can be found relative to this axis using Steiner's theorem:
I \u003d I 0 + ma 2,

where I 0- the initial moment of inertia of the body; m- body mass; a- distance between axles.

If the system that rotates around the fixed axis consists of n bodies, then the total moment of inertia of this type of system will be is equal to the sum moments, its components (the law of addition of moments of inertia).

rotational they call such a movement in which two points connected with the body, therefore, and the straight line passing through these points, remain motionless during the movement (Fig. 2.16). Fixed line A B called axis of rotation.

Rice. 2.1B. To the definition of the rotational motion of the body

The position of the body during rotational motion determines the angle of rotation φ, rad (see Fig. 2.16). When moving, the angle of rotation changes with time, i.e. the law of rotational motion of a body is defined as the law of change in time of the value of the dihedral angle Φ = φ(/) between the fixed half-plane TO () , passing through the axis of rotation, and movable p 1 a half-plane associated with the body and also passing through the axis of rotation.

The trajectories of all points of the body during rotational motion are concentric circles located in parallel planes with centers on the axis of rotation.

Kinematic characteristics of rotational motion of the body. Similarly to how the kinematic characteristics were introduced for a point, a kinematic concept is introduced that characterizes the rate of change of the function f(c), which determines the position of the body during rotational motion, i.e. angular velocity ω = φ = s/f/s//, angular velocity dimension [ω] = rad /With.

In technical calculations, the expression of angular velocity is often used with a different dimension - through the number of revolutions per minute: [i] = rpm, and the relationship between P and w can be represented as: w = 27sh/60 = 7sh/30.

In general, the angular velocity changes with time. The measure of the rate of change of the angular velocity is the angular acceleration e = c/co/c//= co = f, the dimension of the angular acceleration is [e] = rad/s 2 .

The introduced angular kinematic characteristics are completely determined by setting one function - the angle of rotation from time.

Kinematic characteristics of body points during rotational motion. Consider a point M body located at a distance p from the axis of rotation. This point moves along a circle of radius p (Fig. 2.17).

Rice. 2.17.

points of the body during its rotation

Arc length M Q M circle of radius p is defined as s= ptp, where φ is the angle of rotation, rad. If the law of motion of the body is given as φ = φ(r), then the law of motion of the point M along the trajectory defines the formula S= rf(7).

Using the expressions for kinematic characteristics with the natural way of specifying the movement of a point, we obtain kinematic characteristics for points of a rotating body: speed according to the formula (2.6)

V= 5 = rf = pco; (2.22)

tangential acceleration according to expression (2.12)

i t \u003d K \u003d wor \u003d ep; (2.23)

normal acceleration according to formula (2.13)

a„ = And 2 / p \u003d co 2 p 2 / p \u003d ogr; (2.24)

full acceleration using expression (2.15)

a = -]a + a] = px/e 2 + co 4 . (2.25)

As a characteristic of the direction of full acceleration, p is taken - the angle of deviation of the full acceleration vector from the radius of the circle described by the point (Fig. 2.18).

From fig. 2.18 we get

tgjLi = ajan\u003d re / pco 2 \u003d g / (o 2. (2.26)

Rice. 2.18.

Note that all kinematic characteristics of the points of a rotating body are proportional to the distances to the axis of rotation. Ve-

Their masks are determined through the derivatives of the same function - the angle of rotation.

Vector expressions for angular and linear kinematic characteristics. For an analytical description of the angular kinematic characteristics of a rotating body, together with the axis of rotation, the concept is introduced rotation angle vector(Fig. 2.19): φ = φ(/)A:, where To- go

vector of rotation axis

1; To= con51 .

The vector φ is directed along this axis so that it can be seen from the “end”

counter-clockwise rotation.

Rice. 2.19.

characteristics in vector form

If the vector φ(/) is known, then all other angular characteristics of rotational motion can be represented in vector form:

angular velocity vector ω = φ = φ To. The direction of the angular velocity vector determines the sign of the derivative of the angle of rotation;
angular acceleration vector є = ω = φ To. The direction of this vector determines the sign of the derivative of the angular velocity.

The introduced vectors co and є make it possible to obtain vector expressions for the kinematic characteristics of points (see Fig. 2.19).

Note that the modulus of the point velocity vector is the same as the modulus vector product angular velocity vector and radius vector: |cox G= sogvipa = sor. Given the directions of the vectors ω and r and the rule for the direction of the cross product, we can write an expression for the velocity vector:

V= co xg.

Similarly, it is easy to show that

? X
- egBipa= єр = a t and

Sosor = co p = i.

(Besides this, the vectors of these kinematic characteristics coincide in direction with the corresponding vector products.

Therefore, the vectors of tangential and normal accelerations can be represented as vector products:

(2.28)
(2.29)

a x = z X G

a= co x v.

Translational called such a motion of a rigid body in which any straight line, invariably associated with this body, remains parallel to its initial position.

Theorem. In the translational motion of a rigid body, all of its points describe the same trajectories and at any given moment have equal velocities and accelerations in magnitude and direction.

Proof. Pass through two points and , translationally moving body segment
and consider the motion of this segment in the position
. At the same time, the point describes the trajectory
, and the 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. Hence, the first part of the theorem is proved. We will determine the position of the points and in a vector way with respect to the fixed origin . At the same time, these radii - vectors are dependent on
. Because. neither the length nor the direction of the segment
does not change when the body moves, then the vector

. We proceed to the determination of velocities according to dependence (24):

, we get
.

We proceed to the determination of accelerations according to dependence (26):

, we get
.

It follows from the proved theorem that the translational motion of a body will be completely determined if the motion of only one of some points is known. Therefore, the study forward movement of a rigid body is reduced to the study of the motion of one of its points, i.e. to the problem of point kinematics.

Topic 11. Rotational motion of a rigid body

rotational is such a motion of a rigid body in which two of its points remain motionless for the entire time of motion. The line passing through these two fixed points is called axis of rotation.

Each point of the body that does not lie on the axis of rotation, during such a movement, 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 with the body and rotating with it (Fig. 57). The position of plane II, and, accordingly, of the entire body, with respect to plane I in space, is completely determined by the angle . When a body rotates around an axis this angle is a continuous and single-valued 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 body rotation. (43)

In this case, we will assume that the angle counted from the fixed plane in the counter-clockwise direction 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, it is said that such a body has one degree of freedom.

Angular velocity

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

.(44)

Angular velocity, 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 translated along its line of action. If we denote the ortho-vector of the rotation axis through , then we get the vector expression of the angular velocity:

. (45)

Angular acceleration

The rate of change in the angular velocity of a body over time is called angular acceleration bodies and is denoted (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 translated along its line of action.

If we denote the ortho-vector of the rotation axis through , then we get the vector expression of the angular acceleration:

. (47)

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

Consider special cases of rotational motion.

1. Uniform rotation:

,
.

,
,
,

,
. (48)

2. Equal-variable 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, radius
, located in a plane perpendicular to the axis of rotation (Fig. 59, a).

Let's assume that at the time point is in position
. Let us assume that the body rotates in the positive direction, i.e. in the direction of increasing angle . At the point in time
point will take position
. 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's clear that
. Then. Finally we get

. (50)

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

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

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

,
. (51)

The module of 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 by the distance from this point to the axis of rotation.

,
. (52)

The module 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 such a movement in which all points of the body move parallel to some fixed plane. Examples of such movement:

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

Wheel rolling along a straight section of the 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). We refer this motion to a fixed coordinate system
, and with the figure itself we will associate a moving coordinate system
, which moves with it.

Obviously, the position of a moving figure on a fixed 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 angle of rotation , a moving coordinate system relative to the fixed one, which will be counted from the axis in a counter-clockwise direction.

Consequently, the motion of a flat figure in its plane will be completely determined if the values are known for each moment of time ,,, 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 the moving figure at a given time.

Consider special cases:

1.

, then the movement of the body will be translational, since the movable axes move, 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 plane of the figure 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 times and
(Fig. 61). body out of position into position
can be transferred as follows. Let's move the body first progressively. At the same time, the segment
moves parallel to itself to the position
, and then let's turn body around a point (pole) on the corner
until the points match and .

Hence, any planar motion can be represented as the sum of translational motion along with the chosen pole and rotational motion, about this pole.

Let us consider the methods by which it is possible to determine the velocities of the points of a body making a 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 as two components: translational, with a speed equal to the speed of an arbitrarily chosen point -poles , and rotational around this pole.

Consider a flat body (Fig. 62). The equations of motion are:
,
,
.

We determine from these equations the speed of the point (as with the coordinate method of setting)

,
,
.

So the point speed - the value is known. We take this point as a pole and determine the speed of an arbitrary point
body.

Speed
will be made up of the translational component , when moving along with the point , and rotational
, when the point is rotated
relative to the point . Point speed move to point
parallel to itself, since in translational motion the velocities of all points are equal both in magnitude and in direction. Speed
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 the vectors and
, and its module is determined by the dependency:

, .(55)

2. The theorem on the projections of the velocities of two points of the body.

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

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

3. Instantaneous center of speeds.

Instant center of speeds(MCS) is a point whose speed at a given time is zero.

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

Indeed, if we assume that
, then by theorem (56), the vector
must be perpendicular
and
, which is impossible. It can be seen from the same theorem that no other section point at this point in time cannot have a speed equal to zero.

Applying the pole method
- pole, determine the speed of the point (55): since
,
. (57)

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

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

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

Determination of 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 MCC will be the point of intersection of the perpendiculars to these velocity vectors.

2. The speeds 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 MCC is located somewhere on this perpendicular. To determine the position of the MCC, 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 speeds of two points of the body are parallel, but not equal in magnitude (Fig. 65, b). The procedure for obtaining an MDS is similar to that described in paragraph 2.

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

4. We define the MCC for a wheel rolling without slipping on a fixed surface (Fig. 65, G). Since the movement occurs without sliding, then 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. Therefore, the point of contact of the wheel with a fixed surface will be the MCC.

Determination of accelerations of points of a plane figure

When determining the accelerations of points of a flat figure, an analogy can be traced with methods for determining velocities.

1. Pole method. Just as in determining velocities, we take as a pole an arbitrary point of the body, the acceleration of which we know, or we can determine it. 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:

At the same time, 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 dependences (51) and (52), we obtain
,
.

2. Instantaneous center of acceleration.

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

Let us show that such a point exists at any given moment of time. We take a point as a pole , whose acceleration
we know. Finding an angle , lying within
, and satisfying the condition
. If
, then
and vice versa, i.e. injection is deposited in the direction . Set aside from the point at an angle to the vector
section
(Fig. 67). The point obtained by such constructions
will be MCU.

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

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

Hence,
and then
.

Special cases of determining the MCC

1.
. Then
, and, therefore, the MCU does not exist. In this case, the body moves forward, 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 MCC lies at the intersection of perpendiculars to the accelerations of the points of the body (Fig. 68, b).

4.
. Then
,

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

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

. (60)

Complicated point movement is called such a movement in which the point simultaneously participates in two or more movements. With such a movement, the position of the point is determined relative to the mobile and relative to the fixed reference systems.

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

The movement of that point of the moving frame of reference, with which the moving point coincides at a given moment with respect to the fixed frame of reference, is called point movement . Let us denote the parameters of the portable motion
.

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

As an example of a complex movement, we can consider the movement of a person in a moving vehicle (tram). In this case, the movement of a person is related to a moving coordinate system - a tram and to a fixed coordinate system - the earth (road). Then, based on the above definitions, the movement of a person relative to the tram is relative, the movement together with the tram relative to the ground is figurative, 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 notation: - radius vector defining the position of the point
relative to the moving coordinate system
,
;- radius vector that determines the position of the origin of the moving coordinate system (points ) (points );- radius - a vector that defines the position of a point
relative to the fixed coordinate system
;
,.

Let's get conditions (restrictions) corresponding to relative, figurative and absolute motions.

1. When considering the relative motion, we will assume that the point
moves relative to the moving coordinate system
, and the moving coordinate system itself
relative to the fixed coordinate system
does not move.

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

,

,

.

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

,

,

,.

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

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

,
,

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

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

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

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

It was found that the absolute acceleration of a point is not equal to the geometric sum of the relative and portable components of the accelerations. Let us define the component of absolute acceleration, which is in parentheses, for particular cases.

1. 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 the 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. So, in this case, the moving coordinate system
rotates around the instantaneous axis of rotation with angular velocity (Fig. 70). Denote the point at the end of the vector across . Then, using the vector method of specifying (15), we obtain the velocity vector of this point
.

On the other side,
. Equating the right parts of these vector equalities, we get:
. Proceeding similarly, for the rest of the vector vectors, we get:
,
.

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

The doubled vector product of the angular velocity vector of the portable motion by the vector of the linear velocity of the relative motion is called Coriolis acceleration and denoted

. (64)

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

Forwarded
according to the vector product rule. The Coriolis acceleration vector is always directed perpendicular to the plane formed by the vectors and , so that, looking from the end of the vector
, see turn To , through the smallest angle, counterclockwise.

The Coriolis acceleration modulus is equal to.

The 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 motion.(Fig. 2.2) .

Figure 2.2

passing through fixed points A and V 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 fixed, and all the rest will describe circles whose planes 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 will call body angle. We will consider the angle φ positive if delayed from a fixed plane in a counterclockwise direction (for an observer looking from the positive end of the axis Az), but negative if clockwise. measure angle φ will be in radians. To know the position of the body at any time, you need to know the dependence of the angle φ from time t, i.e.

This equation expresses the law of rotation 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 value characterizing the rate of change of the angle of rotation φ over time is called the angular velocity.

If for a period of time
body makes a turn
, then the numerically average angular velocity of the body for this period of time will be
. In the limit at
we get

In this way, the numerical value of the angular velocity of the body at a given moment of time is equal to the first derivative of the angle of rotation with respect to time.

Rule of signs: when the rotation is counterclockwise, ω> 0, and when clockwise, then ω< 0.

or, since the 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 module of the angular velocity, and the axis of rotation, and the direction of rotation around this axis.

The quantity characterizing the rate of change of the angular velocity over time is called the angular acceleration of the body.

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

When striving
we get the value of the angular acceleration at the moment t:

In this way, the numerical value of the angular acceleration of the body at a given moment of 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 measure is usually 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 quantities ω and ε have the same signs, then the rotation will be accelerated, when different - slowed down. By analogy with angular velocity, angular acceleration can also be represented as a vector directed along the axis of rotation. Wherein

If the body rotates with an accelerated direction coincides with , and opposite during slow rotation.

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

From
we have
. Hence, assuming that at the initial moment of time
injection
, and taking integrals to the left of before , and on the right from 0 to t, we 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 as 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 revolutions per 2π n; this turn is done in 1 min, i.e. t= 1min=60s. It follows that

If the angular acceleration of the body remains constant throughout the 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 before , and the right one from 0 to t, find

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

Rotational motion of a rigid body. Rotational is the movement 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 - fixed and half-plane II - connected with a solid 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 the 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 in time D t= t1 + t the body makes a turn by Dj = j1 –j, then the average angular velocity of the body over this period of time will be equal to

(1.16)

To determine the value of the angular velocity of the body at a given time t find the limit of the ratio of the rotation angle increment 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 moment of 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 for j > 0, and vice versa, if j < 0. then w < 0. The unit of angular velocity is usually 1/s, so the radian is a dimensionless quantity.

The 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 is seen to occur counterclockwise.

The change in the angular velocity of the body over time characterizes the angular acceleration e. By analogy with finding the average value of the angular velocity, we 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 moment of 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 in the opposite direction during slow motion.

Having established the characteristics of the motion of a rigid body as a whole, let us proceed to the study of the motion of its individual points. Consider some point M a rigid 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 circumferential 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 turn of the body occurs at an angle dj , then point M at the same time, it performs an elementary displacement along its trajectory dS = h * dj ,. Then the speed of the point M was determined from the expression

(2.20)

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

Thus, the linear velocity of a point of 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 it follows from the formula for the linear velocity that the linear velocities 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.

White is the distance from the axis of rotation of the solid ash to a certain point M considered as the radius vector h of the point M, then the linear velocity vector of the 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 of the vector product (2.21) is a vector equal in absolute value 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 occurring counterclockwise , i.e. tangential to the trajectory of the point M.

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

Rice. 2.5

To find expression for acceleration a point M we perform time differentiation of expression (2.21) for the speed of the point

(2.22)

Considering that dj/dt=e, and dh/dt = v, we write expression (2.22) as

where a r and an, respectively, are the tangential and normal components of the total acceleration of the point of the body during rotational motion, determined from the expressions

The tangential component of the full acceleration of the body point (tangential acceleration) at characterizes the change in the velocity vector modulo 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 modulus of the tangential acceleration vector of a point of a body during the rotational motion of a rigid body is determined by the expression

(2,25)

Normal component of full acceleration (normal acceleration) a" arises due to a change in the direction of the velocity vector of a point during dyeing of 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 vector of normal acceleration of a point during the rotational motion of a rigid body is determined, taking into account (2.20), by the expression