Kinematics of the rotational motion of a body around a fixed axis. Rotational motion of a rigid body. The main elements of the kinematics of uneven rotational motion

In nature and technology, we often encounter the manifestation of the rotational motion of rigid bodies, for example, shafts and gears. How this type of motion is described in physics, what formulas and equations are used for this, these and other issues are covered in this article.

What is rotation?

Each of us intuitively represents which movement will be discussed. Rotation is a process in which a body or material point moves along a circular path around a certain axis. From a geometric point of view, a rigid body is a straight line, the distance to which remains unchanged during the movement. This distance is called the radius of rotation. In what follows, we will denote it by the letter r. If the axis of rotation passes through the center of mass of the body, then it is called its own axis. An example of rotation around its own axis is the corresponding motion of the planets Solar system.

For the rotation to take place, there must be a centripetal acceleration, which occurs due to the centripetal force. This force is directed from the center of mass of the body to the axis of rotation. The nature of centripetal force can be very different. So, on a cosmic scale, gravity plays its role, if the body is fixed with a thread, then the tension force of the latter will be centripetal. When the body rotates around its own axis, the role of the centripetal force is played by the internal electrochemical interaction between the elements that make up the body (molecules, atoms).

It is necessary to understand that without the presence of centripetal force, the body will move in a straight line.

Physical quantities describing rotation

Firstly, these are dynamic characteristics. These include:

• moment of impulse L;
• moment of inertia I;
• moment of force M.

Secondly, these are kinematic characteristics. Let's list them:

• angle of rotation θ;
• angular velocity ω;
• angular acceleration α.

Let us briefly describe each of the named quantities.

The moment of impulse is determined by the formula:

Where p is a linear impulse, m is the mass of a material point, v is its linear velocity.

The moment of inertia of a material point is calculated using the expression:

For any body of complex shape, the value of I is calculated as the integral sum of the moments of inertia material points.

The moment of force M is calculated as follows:

Here F is the external force, d is the distance from the point of its application to the axis of rotation.

The physical meaning of all quantities in the name of which the word "moment" is present is analogous to the meaning of the corresponding linear quantities... For example, a moment of force shows the ability of an applied force to impart a system of rotating bodies.

The kinematic characteristics are mathematically determined by the following formulas:

As can be seen from these expressions, the angular characteristics are similar in their meaning to linear ones (velocities v and acceleration a), only they are applicable for a circular path.

Rotation dynamics

In physics, the study of the rotational motion of a rigid body is carried out using two sections of mechanics: dynamics and kinematics. Let's start with the dynamics.

Dynamics studies external forces acting on a system of rotating bodies. We will immediately write down the equation of the rotational motion of a rigid body, and then we will analyze its constituent parts. So, this equation looks like:

Which acts on the system, which has the moment of inertia I, causes the appearance of the angular acceleration α. The smaller the value of I, the easier it is with the help of a certain moment M to spin the system up to high speeds in short periods of time. For example, a metal rod is easier to rotate along its axis than perpendicular to it. However, the same rod is easier to rotate around an axis perpendicular to it and passing through the center of mass than through its end.

The conservation law for the quantity L

This value was introduced above, it is called the angular momentum. The equation of the rotational motion of a rigid body, presented in the previous paragraph, is often written in a different form:

If the moment of external forces M acts on the system during the time dt, then it causes a change in the angular momentum of the system by the value dL. Accordingly, if the moment of forces is zero, then L = const. This is the law of conservation of the quantity L. For it, using the relationship between the linear and angular velocity, we can write:

L = m * v * r = m * ω * r 2 = I * ω.

Thus, in the absence of a moment of forces, the product of the angular velocity and the moment of inertia is constant. This physical law is used by skaters in their performances or artificial satellites, which must be rotated around its own axis in open space.

Centripetal acceleration

Above, when studying the rotational motion of a rigid body, this value has already been described. The nature of centripetal forces was also noted. Here we will only supplement this information and present the corresponding formulas for calculating this acceleration. Let's denote it by a c.

Since the centripetal force is directed perpendicular to the axis and passes through it, it does not create a moment. That is, this force has absolutely no effect on the kinematic characteristics of rotation. However, it creates centripetal acceleration. Here are two formulas for its definition:

Thus, the greater the angular velocity and radius, the more force must be applied to keep the body on a circular path. A prime example of this physical process is a car skidding while cornering. Skidding occurs when the centripetal force, the role of which is played by the friction force, becomes less than the centrifugal force (inertial characteristic).

Three main kinematic characteristics were listed above in the article. a solid body is described by the following formulas:

θ = ω * t => ω = const., α = 0;

θ = ω 0 * t + α * t 2/2 => ω = ω 0 + α * t, α = const.

The first line contains formulas for uniform rotation, which assumes the absence of an external moment of forces acting on the system. The second line contains formulas for uniformly accelerated motion along a circle.

Note that rotation can occur not only with a positive acceleration, but also with a negative one. In this case, a minus sign should be placed in front of the second term in the formulas of the second line.

An example of solving the problem

A torque of 1000 N * m acted on the metal shaft for 10 seconds. Knowing that the moment of inertia of the shaft is 50 kg * m 2, it is necessary to determine the angular velocity that the mentioned moment of force gave to the shaft.

Using the basic equation of rotation, we calculate the acceleration of the shaft:

Since this angular acceleration acted on the shaft for a time t = 10 seconds, then to calculate the angular velocity we use the formula for uniformly accelerated motion:

ω = ω 0 + α * t = M / I * t.

Here ω 0 = 0 (the shaft did not rotate before the action of the moment of forces M).

Substituting the numerical values ​​of the quantities into the equality, we get:

ω = 1000/50 * 10 = 200 rad / s.

To convert this number into the usual revolutions per second, you must divide it by 2 * pi. Having performed this action, we find that the shaft will rotate at a frequency of 31.8 rpm. / S.

The rotation of a rigid body around a fixed axis is called such a movement in which two points of the body remain motionless during the entire time of movement. In this case, all points of the body, located on a straight line passing through its fixed points, also remain motionless. This line is called the axis of rotation of the body .

Let points A and B be fixed. Direct the axis along the axis of rotation. Through the axis of rotation we draw a fixed plane and a movable one, fastened to a rotating body (at).

The position of the plane and the body itself is determined by the dihedral angle between the planes and. Let's designate it. The angle is called body rotation angle .

The position of the body relative to the selected frame of reference is uniquely determined at any moment of time if the equation is given, where is any twice differentiable function of time. This equation is called the equation of rotation of a rigid body about a fixed axis .

A body that rotates 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. The trajectories of points of a body when it rotates 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 body at any moment in time is called the first time derivative of the angle of rotation at that moment, that is.

The angular velocity is a positive value 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 dimension of the angular velocity by definition:

In engineering, angular velocity is the rotational speed expressed in revolutions per minute. In one minute, the body will turn through an angle, where n is the number of revolutions per minute. Dividing this angle by the number of seconds in a minute, we get

Algebraic Angular Acceleration of a Body the first time derivative of the angular velocity is called, that is, the second derivative of the rotation angle, i.e.

Dimension of angular acceleration by definition:

Let us introduce the concepts of vectors of angular velocity and angular acceleration of a body.

And, where is the unit vector of the rotation axis. Vectors and can be drawn at any points of the axis of rotation, they are sliding vectors.

Algebraic angular velocity is the projection of the angular velocity vector onto the axis of rotation. Algebraic angular acceleration is the projection of the angular velocity vector onto the axis of rotation.

If at, then the algebraic angular velocity increases with time and, therefore, the body rotates at an accelerated rate at the moment in question in the positive direction. The directions of the vectors and coincide, both of them are directed towards the positive side of the axis of rotation.

When and, the body rotates at an accelerated rate in negative side... The directions of the vectors and coincide, both of them are directed towards the negative side of the axis of rotation.

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

Basic concepts of rotary motion kinematics

The rotational motion of a material point around a fixed axis is called such a motion, 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 movement in which all points of the body move along concentric (whose centers lie on the same axis) circles in accordance with the rule for rotational motion of a material point.

Let an arbitrary rigid body T rotate around the O axis, which is perpendicular to the plane of the drawing. Let's select point M on this body. When rotating, this point will describe a circle around the O axis with a radius r.

After some time, the radius will rotate relative to the initial 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 over time is called the equation of the 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 during the time Δt, is equal to:

ΔS = Δφr.

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

ω = dφ / dt.

If ω = dφ / dt = const, then this movement is called uniform rotary movement. 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.

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

ω = 2π / T,

therefore, the rotation period is defined as follows:

T = 2π / ω.

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

ν = 1 / T.

Frequency units: [ν] = 1 / c = 1 s -1 = 1 Hz.

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

ω = 2πν.

The main elements of the kinematics of uneven rotational motion

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

Vector ε , which characterizes the rate of change in 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 vectors ε and ω are oppositely directed.

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

Relationship between the quantities characterizing the translational and rotational motion

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

ΔS = Δφ r.

Then the linear velocity of a material point performing 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 = ε r.

Moment of momentum of a material point

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

Moment of momentum of a material point ( L i) is directed perpendicular to the plane drawn through r i and υ i, and forms with them a right triplet 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 of linear velocity 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.

The moment of force that acts on the i-th material point

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

The magnitude l i, equal to the length of the perpendicular dropped from the point of rotation to the direction of action of the force, is called the shoulder 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 in the angular momentum of a body that rotates around a fixed axis is equal to the resulting moment relative to this axis of all external forces applied to the body.

Moment of impulse 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 velocity:

υ i = ωr i,

then the expression for the angular momentum takes the form

L i = m i r i 2 ω.

The magnitude I i = m i r i 2 called the moment of inertia with respect to i-th axis material point of an absolutely rigid body passing through its center of mass. Then we write the angular momentum of a material point:

L i = I i ω.

We write the moment of momentum of an absolutely rigid body as the sum of the moments of momentum of material points that make up a given body:

L = Iω.

Moment of force and moment of inertia

The law of rotational motion states:

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 in terms of the moment of inertia:

M = Iε.

Comment. A 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 axis of rotation of a body does not pass through its center of mass, then relative to this axis one can find its moment of inertia by Steiner's theorem:
I = I 0 + ma 2,

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

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 equal to the sum of the moments that make up it (the law of addition of the moments of inertia).

Translational is called a motion 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 the same trajectories and at any given moment have the same velocity and acceleration in magnitude and direction.

Proof. Let's draw through two points and , a translationally moving body segment
and consider the movement of this segment in position
... In this case, 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 the points and will be the same. Hence, the first part of the theorem is proved. We will determine the position of the points and vector method with respect to the fixed origin ... Moreover, these radii - vectors are in dependence
... Because. neither the length nor the direction of the line
does not change when the body moves, then the vector

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

, we get
.

We turn to the definition of accelerations by dependence (26):

, we get
.

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

Topic 11. Rotational motion of a rigid body

Rotational is called a motion of a rigid body in which two of its points remain motionless for the entire time of motion. Moreover, the straight 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 describes a circle during this movement, the plane of which is perpendicular to the axis of rotation, and its center lies on this axis.

Draw through the axis of rotation a fixed plane I and a movable plane II, which is invariably connected to the body and rotates with it (Fig. 57). The position of plane II, and, accordingly, of the whole body, in relation to plane I in space, is quite determined by the angle ... When the body rotates around the axis this angle is a continuous and single-valued function of time. Therefore, knowing the law of the change in this angle over time, we will be able to determine the position of the body in space:

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

In this case, we will assume that the angle measured from a fixed plane in the opposite direction to the clockwise movement, as 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 the angular body speed and denoted
(omega):

.(44)

The angular velocity, just like the linear velocity, is a vector quantity, and this vector plot 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 at its beginning, you 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 ort-vector of the rotation axis through , then we get a vector expression for the angular velocity:

. (45)

Angular acceleration

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

. (46)

The angular acceleration is a vector quantity, and this vector plot on the axis of rotation of the body. It is directed along the axis of rotation in the direction that, looking from its end at its beginning, to see the direction of rotation of 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 ort-vector of the rotation axis through , then we get a vector expression for the angular acceleration:

. (47)

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

Consider special cases of rotational motion.

1. Uniform rotation:

,
.

,
,
,

,
. (48)

2. Equidistant rotation:

.

,
,
,
,
,
,
,
,

,
,
.(49)

Relationship between linear and angular parameters

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

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

. (50)

Here - line speed of a 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 modulus 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 by the distance from this point to the axis of rotation.

Now let's connect the linear components of the point acceleration to 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 by 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 by the distance from this point to the axis of rotation.

Then the expression for full acceleration points takes the form

. (53)

Direction vectors ,,are shown in Figure 59, v.

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

The movement of any body, the base of which slides on a given fixed plane;

Rolling a wheel along a straight 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 associate the moving coordinate system
that moves with it.

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

Consequently, the motion of a flat figure in its plane will be quite definite if the values ​​of ,,, i.e. equations of the form:

,
,
. (54)

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

Let's 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 drawing plane through a point .

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

Consider two consecutive positions and
, which the body occupies at the moments of time and
(fig. 61). Body out of position into position
can be transferred as follows. Move the body first progressively... In this case, the segment
will move parallel to itself into position
, and then turn body around a point (pole) at the corner
until the points match 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.

Consider the methods by which it is possible to determine the speeds of points of a body performing a plane motion.

1. Pole method. This method is based on the obtained decomposition of the 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.

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

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

,
,
.

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
body.

Speed
will consist of a translational component , when moving together with the point , and rotational
, when rotating the point
relative to point ... Point speed move to point
parallel to itself, since during translational motion the speeds of all points are equal both in magnitude and in 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 the parallelogram built on the vectors and
, and its module is determined by the dependency:

, .(55)

2. 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 onto a straight line connecting these points are equal to each other.

Consider two points of the body and (fig. 63). Taking the point beyond the pole, define the direction by dependence (55):
... We project this vector equality onto the line
and considering that
perpendicular
, we get

3. Instantaneous center of speeds.

Instantaneous speed center(MCS) is called a point, the speed of which at a given time is equal to zero.

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

Indeed, if we assume that
, then by 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 point of the section at this moment in time it cannot have a speed equal to zero.

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

A similar result can be obtained for any other point on the body. Consequently, 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 three considered methods for determining the velocities of the points of a flat figure, it can be seen that the MCS is preferable, since here the velocity 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 MDC

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 vectors of velocities.

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 MDC is located somewhere on this perpendicular. To determine the position of the MDC, 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 MDS. In this case, the MCC is 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 the MDC is similar to that described in clause 2.

d) The speeds of the two points are equal both in magnitude and in direction (Fig. 65, v). We get 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 a given position is zero:

4. Define the MDC for a wheel rolling without sliding on a stationary 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. Consequently, the point of contact of the wheel with a fixed surface will be the MCC.

Determining the Acceleration of Points of a Planar Shape

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

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 plane figure is equal to the sum of the pole accelerations and the acceleration in rotational motion around this pole:

In this case, the component
determines point acceleration when it rotates around the pole ... When rotating, the trajectory of the point will be curved, which means
(fig. 66).

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

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

2. Instant acceleration center.

Instant acceleration center(MCU) is called a point, the acceleration of which at a given time is equal to zero.

Let us show that such a point exists at any given time. Take a point for the pole whose acceleration
we know. Find the angle lying within
, and satisfying the condition
... If
, then
and vice versa, i.e. injection postponed towards ... Set aside from the point at an angle to vector
section
(fig. 67). The point obtained by such constructions
will be the ICU.

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

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

Hence,
and then
.

Particular cases of determining the MCU

1.
... Then
, and therefore the MCU does not exist. In this case, the body moves translationally, i.e. the speeds 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 acceleration points of the body (Fig. 68, a).

3.
... Then,
,
... This means that the MCC lies at the intersection of the 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 you accept the point
beyond the pole, then the acceleration of any point of the plane figure is determined by the acceleration in rotational motion around the MCC:

. (60)

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

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

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

The movement of a point relative to a fixed frame of reference is called absolute (complex) point movement ... We will agree to designate 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 stationary coordinate system - the earth (road). Then, based on the above definitions, the movement of a person relative to the tram is relative, the movement 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 with respect to the movable
and motionless
coordinate systems (Fig. 69). Let us introduce the notation: - the radius vector defining the position of the point
relative to the moving coordinate system
,
;is the radius vector defining the position of the origin of the moving coordinate system (points ) (points );- radius - a vector defining the position of the point
with respect to a fixed coordinate system
;
,.

We will obtain conditions (restrictions) corresponding to the relative, figurative and absolute movements.

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
with respect to a fixed coordinate system
does not move.

Then the coordinates of the point
will change in relative motion, and the ort-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
are fixed in relation to the moving coordinate system, and the point moves with the moving coordinate system
relatively motionless
:

,

,

,.

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

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

,
,

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

We 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 transport components of the speed.

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

,

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

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

1. Translational movement of a point translational
... 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 translational motion of a point is translational, then the absolute acceleration of the point is equal to the geometric sum of the relative and translational components of acceleration.

2. The transferable movement of a point is non-translational. Hence, in this case, the moving coordinate system
rotates around the instantaneous axis of rotation with angular velocity (fig. 70). We denote a point at the end of the vector across ... Then, using the vector method of setting (15), we obtain the velocity vector of this point
.

On the other side,
... Equating the right-hand sides of these vector equalities, we get:
... Proceeding in a similar way, for the remaining vector vectors, we obtain:
,
.

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

The doubled vector product of the vector of the angular velocity 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 velocity in portable movement and changing the portable speed in relative motion.

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

Coriolis acceleration modulus is.

Steering angle, angular velocity and angular acceleration

Rotation of a rigid body around a fixed axis its movement is called, in which two points of the body remain motionless during the entire time of movement. In this case, all points of the body, located on a straight line passing through its fixed points, also remain motionless. This line is called the axis of rotation of the body.

If A and V- fixed points of the body (Fig. 15 ), then the axis of rotation is the axis Oz, which can have any direction in space, not necessarily vertical. One axis direction Oz taken for positive.

Draw a fixed plane through the axis of rotation By and mobile P, fastened to a rotating body. Let both planes coincide at the initial moment of time. Then at the moment of time t the position of the movable plane and the rotating body itself can be determined by the dihedral angle between the planes and the corresponding linear angle φ between straight lines located in these planes and perpendicular to the axis of rotation. Injection φ called the angle of rotation of the body.

The position of the body relative to the selected frame of reference is completely determined in any

time moment, if the equation is given φ =f (t) (5)

where f (t)- any, twice differentiable function of time. This equation is called the equation of rotation of a rigid body about 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 φ .

Injection φ considered positive if it is laid counterclockwise, and negative - in the opposite direction, when viewed from the positive direction of the axis Oz. The trajectories of points of a body when it rotates 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 a body at any moment in time, the first time derivative of the angle of rotation at this moment is called, i.e. dφ / dt = φ. It is a positive value 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 ω. Then ω= ׀dφ / dt׀= ׀φ ׀ (6)

The dimension of the angular velocity is set in accordance with (6)

[ω] = angle / time = rad / s = s -1.

In engineering, angular velocity is the speed expressed in revolutions per minute. In 1 minute, the body will turn at an angle 2πп, if P- the number of revolutions per minute. Dividing this angle by the number of seconds in a minute, we get: (7)

Algebraic Angular Acceleration of a Body the first time derivative of the algebraic velocity is called, i.e. second derivative of the angle of rotation d 2 φ / dt 2 = ω... The angular acceleration modulus is denoted by ε , then ε=|φ| (8)

The dimension of the angular acceleration is obtained from (8):

[ε ] = angular velocity / time = rad / s 2 = s -2

If φ’’>0 at φ’>0 , then the algebraic angular velocity increases with time and, therefore, the body rotates at an accelerated rate at the moment in question in the positive direction (counterclockwise). At φ’’<0 and φ’<0 the body rotates rapidly in the negative direction. If φ’’<0 at φ’>0 , then we have a slower rotation in the positive direction. At φ’’>0 and φ’<0 , i.e. slower rotation is in the negative direction. The angular velocity and angular acceleration in the figures are depicted by arc arrows around the axis of rotation. The arc arrow for the angular velocity indicates the direction of rotation of the bodies;

For accelerated rotation, arc arrows for angular velocity and angular acceleration have the same directions for decelerated rotation - their directions are opposite.

Special cases of rotation of a rigid body

Rotation is called uniform if ω = const, φ = φ't

The rotation will be uniformly variable if ε = const. φ '= φ' 0 + φ''t and

In general, if φ’’ not always,

Velocities and accelerations of body points

The equation of rotation of a rigid body about a fixed axis is known φ = f (t)(fig. 16). Distance s points M in the movable plane P along an arc of a circle (trajectory of a point), measured from a point M o, located in a fixed plane, expressed through the angle φ addiction s = hφ, where h is the radius of the circle along which the point moves. It is the shortest distance from the point M to the axis of rotation. It is sometimes called the radius of the point rotation. At each point of the body, the radius of rotation remains unchanged when the body rotates around a fixed axis.

Algebraic point speed M determined by the formula v τ = s ’= hφ Point speed module: v = hω(9)

The velocities of the points of the body when rotating around a fixed axis are proportional to their shortest distances to this axis. The aspect ratio is the angular velocity. The velocities of the points are directed along the tangent to the trajectories and, therefore, are perpendicular to the radii of rotation. Velocities of body points located on a straight line segment OM, in accordance with (9) are distributed according to a linear law. They are mutually parallel, and their ends are located on one straight line passing through the axis of rotation. We decompose the acceleration of a point into tangent and normal components, i.e. a = a τ + a nτ The tangential and normal accelerations are calculated by the formulas (10)

since for a circle the radius of curvature p = h(fig. 17 ). In this way,

Tangent, normal and total accelerations of points, as well as velocities, are also distributed according to a linear law. They linearly depend on the distances of the points to the axis of rotation. Normal acceleration is directed along the radius of the circle to the axis of rotation. The direction of the tangential acceleration depends on the sign of the algebraic angular acceleration. At φ’>0 and φ’’>0 or φ’<0 and φ’<0 we have an accelerated rotation of the body and the direction of the vectors a τ and v match up. If φ’ and φ’" have different signs (slow rotation), then a τ and v directed opposite to each other.

By designating α the angle between the total acceleration of a point and its radius of rotation, we have

tgα = | a τ | / a n = ε / ω 2 (11)

since normal acceleration a n always positive. Injection a for all points of the body the same. It should be postponed from acceleration to the radius of rotation in the direction of the arc arrow of angular acceleration, regardless of the direction of rotation of the rigid body.

Vectors of angular velocity and angular acceleration

Let us introduce the concepts of vectors of angular velocity and angular acceleration of a body. If TO is the unit vector of the axis of rotation directed towards its positive side, then the angular velocity vectors ώ and angular acceleration ε are determined by expressions (12)

Because k is a vector constant in magnitude and direction, then it follows from (12) that

ε = dώ / dt(13)

At φ’>0 and φ’’>0 direction vectors ώ and ε match up. They are both directed towards the positive side of the axis of rotation. Oz(Fig. 18.a) If φ’>0 and φ’’<0 , then they are directed in opposite directions (Fig. 18.b ). The angular acceleration vector coincides in direction with the angular velocity vector during accelerated rotation and opposite to it during decelerated one. Vectors ώ and ε can be drawn at any point on the axis of rotation. They are sliding vectors. This property follows from the vector formulas for the velocities and accelerations of the points of the body.

Complex point movement

Basic concepts

To study some of the more complex types of rigid body movements, it is advisable to consider the simplest complex motion of a point. In many problems, the motion of a point has to be considered relative to two (or more) reference frames moving relative to each other. So, the motion of a spacecraft moving towards the Moon must be considered simultaneously both relative to the Earth and relative to the Moon, which moves relative to the Earth. Any movement of a point can be considered complex, consisting of several movements. For example, the movement of a ship along a river relative to the Earth can be considered complex, consisting of movement along the water and together with the flowing water.

In the simplest case, the complex movement of a point consists of relative and figurative movements. Let's define these movements. Let us have two frames of reference moving relative to each other. If one of these systems O l x 1 y 1 z 1(fig. 19 ) taken for basic or stationary (its motion relative to other frames of reference is not considered), then the second frame of reference Oxyz will move relative to the first. Movement of a point relative to a moving frame of reference Oxyz called relative. The characteristics of this movement, such as trajectory, speed and acceleration, are called relative. They are denoted by the index r; for speed and acceleration v r, a r. The movement of a point relative to the main or stationary system of reference O 1 x 1 y 1 z 1 called absolute(or complex ). It is also sometimes called composite movement. The trajectory, speed and acceleration of this movement are called absolute. The speed and acceleration of absolute motion are indicated by letters v, a no indices.

The transferable motion of a point is called the movement that it performs together with a moving frame of reference, as a point rigidly attached to this system at a given moment in time. Due to the relative motion, the moving point at different times coincides with different points of the body S, with which the movable frame of reference is attached. Transportable speed and transportable acceleration are the speed and acceleration of that point of the body. S, with which the moving point currently coincides. Portable speed and acceleration mean v e, and e.

If the trajectories of all points of the body S, fastened to the moving frame of reference, depicted in the figure (Fig. 20), then we get a family of lines - a family of trajectories of the translational motion of a point M. Due to the relative movement of the point M at each moment of time it is on one of the trajectories of the portable motion. Dot M can coincide with only one point of each of the trajectories of this family of transfer trajectories. In this regard, it is sometimes believed that there are no trajectories of the transferable movement, since it is necessary to consider the trajectories of the transferable movement of the lines, in which only one point is actually a point of the trajectory.

In the kinematics of a point, the movement of a point relative to any frame of reference was studied, regardless of whether this frame of reference moves relative to other frames or not. Let us supplement this study by considering a complex movement, in the simplest case, consisting of relative and figurative. One and the same absolute motion, choosing different moving frames of reference, can be considered as consisting of different figurative and, accordingly, relative motions.

Speed ​​addition

Let us determine the speed of the absolute movement of a point if the speeds of the relative and figurative movements of this point are known. Let the point perform only one, relative motion with respect to the moving frame of reference Oxyz and at time t occupies position M on the trajectory of relative motion (Fig. 20). At the moment of time t + t, due to the relative movement, the point will be in position М 1, having made the movement of ММ 1 along the trajectory of relative movement. Suppose point is involved Oxyz and relative trajectory it will move along some curve on MM 2. If the point participates simultaneously in both the relative and the figurative movements, then during the time A; she will move to MM" along the trajectory of absolute motion and at the moment of time t + At take position M ". If the time At is small and then pass to the limit at At, tending to zero, then small displacements along curves can be replaced by segments of chords and taken as vectors of displacements. Adding vector displacements, we get

In this respect, small quantities of a higher order are discarded, which tend to zero at At, tending to zero. Passing to the limit, we have (14)

Therefore, (14) takes the form (15)

The so-called velocity addition theorem is obtained: the speed of the absolute movement of a point is equal to the vector sum of the speeds of the figurative and relative movements of this point. Since, in the general case, the speeds of the portable and relative movements are not perpendicular, then (15 ')

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