And Savelyeva.
During the forward motion of a body (§ 60 in the textbook by E. M. Nikitin), all its points move along identical trajectories and in each this moment they have equal speeds and equal accelerations.
Therefore, the translational motion of a body is determined by the movement of any one point, usually the movement of the center of gravity.
When considering the movement of a car (problem 147) or a diesel locomotive (problem 141) in any problem, we actually consider the movement of their centers of gravity.
The rotational movement of a body (E.M. Nikitin, § 61) cannot be identified with the movement of any one of its points. The axis of any rotating body (diesel flywheel, electric motor rotor, machine spindle, fan blades, etc.) during movement occupies the same place in space relative to the surrounding stationary bodies.
Movement material point or forward movement bodies are characterized depending on time linear quantities s (path, distance), v (speed) and a (acceleration) with its components a t and a n.
Rotational movement bodies depending on time t characterize angular values: φ (angle of rotation in radians), ω (angular velocity in rad/sec) and ε (angular acceleration in rad/sec 2).
The law of rotational motion of a body is expressed by the equation
φ = f(t).
Angular velocity- a quantity characterizing the speed of rotation of a body is defined in the general case as the derivative of the angle of rotation with respect to time
ω = dφ/dt = f" (t).
Angular acceleration- a quantity characterizing the rate of change angular velocity, is defined as the derivative of angular velocity
ε = dω/dt = f"" (t).
When starting to solve problems on the rotational motion of a body, it is necessary to keep in mind that in technical calculations and problems, as a rule, angular displacement is expressed not in radians φ, but in revolutions φ about.
Therefore, it is necessary to be able to move from the number of revolutions to the radian measurement of angular displacement and vice versa.
Since one full revolution corresponds to 2π rad, then
φ = 2πφ about and φ about = φ/(2π).
Angular velocity in technical calculations is very often measured in revolutions produced per minute (rpm), so it is necessary to clearly understand that ω rad/sec and n rpm express the same concept - the speed of rotation of a body (angular speed) , but in different units - in rad/sec or in rpm.
The transition from one unit of angular velocity to another is made according to the formulas
ω = πn/30 and n = 30ω/π.
When a body rotates, all its points move in circles, the centers of which are located on one fixed straight line (the axis of the rotating body). When solving the problems given in this chapter, it is very important to clearly understand the relationship between the angular quantities φ, ω and ε, which characterize the rotational motion of the body, and the linear quantities s, v, a t and an, characterizing the movement of various points of this body (Fig. 205).
If R is the distance from the geometric axis of a rotating body to any point A (in Fig. 205 R = OA), then the relationship between φ - the angle of rotation of the body and s - the distance traveled by a point of the body during the same time is expressed as follows:
s = φR.
The relationship between the angular velocity of a body and the velocity of a point at each given moment is expressed by the equality
v = ωR.
The tangential acceleration of a point depends on the angular acceleration and is determined by the formula
a t = εR.
The normal acceleration of a point depends on the angular velocity of the body and is determined by the relationship
a n = ω 2 R.
When solving the problem given in this chapter, it is necessary to clearly understand that rotation is the movement solid, not points. A single material point does not rotate, but moves in a circle - it makes a curvilinear movement.
§ 33. Uniform rotational motion
If the angular velocity is ω=const, then the rotational motion is called uniform.
The uniform rotation equation has the form
φ = φ 0 + ωt.
In the particular case when the initial angle of rotation φ 0 =0,
φ = ωt.
Angular velocity of a uniformly rotating body
ω = φ/t
can be expressed like this:
ω = 2π/T,
where T is the period of rotation of the body; φ=2π - angle of rotation for one period.
§ 34. Uniform rotational motion
Rotational motion with variable angular velocity is called uneven (see below § 35). If the angular acceleration ε=const, then the rotational motion is called equally variable. Thus, uniform rotation of the body is special case uneven rotational movement.
Equation of uniform rotation
(1) φ = φ 0 + ω 0 t + εt 2 /2
and the equation expressing the angular velocity of a body at any time,
(2) ω = ω 0 + εt
represent a set of basic formulas for the rotational uniform motion of a body.
These formulas include only six quantities: three constants for a given problem φ 0, ω 0 and ε and three variables φ, ω and t. Consequently, the condition of each problem for uniform rotation must contain at least four specified quantities.
For the convenience of solving some problems, two more auxiliary formulas can be obtained from equations (1) and (2).
Let us exclude angular acceleration ε from (1) and (2):
(3) φ = φ 0 + (ω + ω 0)t/2.
Let us exclude time t from (1) and (2):
(4) φ = φ 0 + (ω 2 - ω 0 2)/(2ε).
In the particular case of uniformly accelerated rotation starting from a state of rest, φ 0 =0 and ω 0 =0. Therefore, the above basic and auxiliary formulas take the following form:
(5) φ = εt 2 /2;
(6) ω = εt;
(7) φ = ωt/2;
(8) φ = ω 2 /(2ε).
§ 35. Uneven rotational motion
Let's consider an example of solving a problem in which non-uniform rotational motion of a body is specified.
The rotation of a rigid body around a fixed axis is 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 body rotation axis .
Let points A and B be stationary. Let's direct the axis along the axis of rotation. Through the axis of rotation we draw a stationary plane and a movable one, attached 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 denote it . The angle is called body rotation angle .
The position of the body relative to the chosen reference system is uniquely determined at any time if the equation is given, where is any twice differentiable function of time. This equation is called equation of rotation of a rigid body around a fixed axis .
A body rotating around a fixed axis has one degree of freedom, since its position is determined by specifying only one parameter - angle.
An angle is considered positive if it is laid counterclockwise, and negative in the opposite direction. The trajectories of points of a body during its rotation around a fixed axis are circles located in planes perpendicular to the axis of rotation.
To characterize the rotational motion of a rigid body around a fixed axis, we introduce the concepts of angular velocity and angular acceleration.
Algebraic angular velocity of a body at any moment in time is called the first derivative with respect to time of the angle of rotation at this moment, that is.
Angular velocity is positive 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 angular velocity by definition:
In engineering, angular velocity is the rotational speed expressed in revolutions per minute. In one minute the body will rotate 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 the body is called the first derivative with respect to time of the angular velocity, that is, the second derivative of the angle of rotation, i.e.
The 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 depicted at any point on the rotation axis; 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 acceleration vector of velocity onto the axis of rotation.
If at , then the algebraic angular velocity increases with time and, therefore, the body rotates accelerated at the moment in time in the positive direction. The directions of the vectors and coincide, they are both directed in the positive direction of the axis of rotation.
When and the body rotates accelerated in negative side. The directions of the vectors and coincide, they are both directed in the negative direction of the axis of rotation.
Progressive is the movement of a rigid body in which any straight line invariably associated with this body remains parallel to its initial position.
Theorem. During the translational motion of a rigid body, all its points describe identical trajectories and at each given moment have equal velocity and acceleration in magnitude and direction.
Proof. Let's draw through two points and 
, a linearly moving body segment 
and consider the movement of this segment in position 
. At the same time, the point 
describes the trajectory 
, and point 
– trajectory 
(Fig. 56).
Considering that the segment 
moves parallel to itself, and its length does not change, it can be established that the trajectories of points 
And 
will be the same. This means that the first part of the theorem is proven. We will determine the position of the points 
And 
vector method relative to a fixed origin 
. Moreover, these radii - vectors are dependent 
. Because. neither the length nor the direction of the segment 
does not change when the body moves, then the vector 
. Let's move on to determining the velocities using dependence (24):
, we get 
.
Let's move on to determining accelerations using dependence (26):
, we get 
.
From the proven theorem it follows that the translational motion of a body will be completely determined if the motion of only one point is known. Therefore, the study of the translational motion of a rigid body comes down to the study of the movement of one of its points, i.e. to the point kinematics problem.
Topic 11. Rotational motion of a rigid body
Rotational This is the movement of a rigid body in which two of its points remain motionless throughout the entire movement. In this case, the straight line passing through these two fixed points is called axis of rotation.
During this movement, each point of the body that does not lie on the axis of rotation describes a circle, the plane of which is perpendicular to the axis of rotation, and its center lies on this axis.
We draw through the axis of rotation a fixed plane I and a movable plane II, invariably connected to the body and rotating with it (Fig. 57). The position of plane II, and accordingly the entire body, in relation to plane I in space, is completely determined by the angle 
. When a body rotates around an axis 
this angle is a continuous and unambiguous function of time. Therefore, knowing the law of change of this angle over time, we can determine the position of the body in space:
-
law of rotational motion of a body.
(43)
In this case, we will assume that the angle 
measured from a fixed plane in the direction opposite to the clockwise movement, when viewed from the positive end of the axis 
. Since the position of a body rotating around a fixed axis is determined by one parameter, such a body is said to have one degree of freedom.
Angular velocity
The change in the angle of rotation of a body over time is called angular body speed
and is designated 
(omega):
.(44)
Angular velocity, just like linear velocity, is a vector quantity, and this vector 
built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the rotation of the body counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (44). Application point 
on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action. If we denote the orth-vector of the rotation axis by 
, then we obtain the vector expression for angular velocity:
.
(45)
Angular acceleration
The rate of change in the angular velocity of a body over time is called angular acceleration
body and is designated 
(epsilon):
.
(46)
Angular acceleration is a vector quantity, and this vector 
built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the direction of rotation of the epsilon counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (46). Application point 
on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action.
If we denote the orth-vector of the rotation axis by 
, then we obtain the vector expression for angular acceleration:
.
(47)
If the angular velocity and acceleration are of the same sign, then the body rotates expedited, and if different – slowly. An example of slow rotation is shown in Fig. 58.
Let us consider special cases of rotational motion.
1. Uniform rotation: 
,
.
,
,
,
,
.
(48)
2. Equal rotation: 
.
,
,
,
,
,
,
,
,
,
,
.(49)
Relationship between linear and angular parameters
Consider the movement of an arbitrary point 
rotating body. In this case, the trajectory of the point will be a circle with radius 
, located in a plane perpendicular to the axis of rotation (Fig. 59, A).
Let us assume that at the moment of time 
the point is in position 
. Let us assume that the body rotates in a positive direction, i.e. in the direction of increasing angle 
. At a moment in time 
the point will take position 
. Let's denote the arc 
. Therefore, over a period of time 
the point has passed the way 
. Her average speed
, and when 
,
. But, from Fig. 59, b, it's clear that 
. Then. Finally we get
.
(50)
Here 
- linear speed of the point 
. As was obtained earlier, this speed is directed tangentially to the trajectory at a given point, i.e. tangent to the circle.
Thus, the module of the linear (circumferential) velocity of a point of a rotating body is equal to the product of the absolute value of the angular velocity and the distance from this point to the axis of rotation.
Now let's connect the linear components of the acceleration of a point with the angular parameters.
,
.
(51)
The modulus of the tangential acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the angular acceleration of the body and the distance from this point to the axis of rotation.
,
.
(52)
The modulus of normal acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the square of the angular velocity of the body and the distance from this point to the axis of rotation.
Then the expression for full acceleration points takes the form
.
(53)
Vector directions 
,
,
shown in Figure 59, V.
Flat movement of a rigid body is a movement in which all points of the body move parallel to some fixed plane. Examples of such movement:
The motion of any body whose base slides along a given fixed plane;
Rolling of a wheel along a straight section of track (rail).
We obtain the equations of plane motion. To do this, consider a flat figure moving in the plane of the sheet (Fig. 60). Let us relate this movement to a fixed coordinate system 
, and with the figure itself we connect the moving coordinate system 
, which moves with it.
Obviously, the position of a moving figure on a stationary plane is determined by the position of the moving axes 
relative to fixed axes 
. This position is determined by the position of the moving origin 
, i.e. coordinates 
,
and rotation angle 
, a moving coordinate system, relatively fixed, which we will count from the axis 
in the direction opposite to the clockwise movement.
Consequently, the movement of a flat figure in its plane will be completely determined if the values of 
,
,
, i.e. equations of the form:
,
,
.
(54)
Equations (54) are equations of plane motion of a rigid body, since if these functions are known, then for each moment of time it is possible to find from these equations, respectively 
,
,
, i.e. determine the position of a moving figure at a given moment in time.
Let's consider special cases:
1.
, then the movement of the body will be translational, since the moving axes move while remaining parallel to their initial position.
2.
,
. With this movement, only the angle of rotation changes 
, i.e. the body will rotate about an axis passing perpendicular to the drawing plane through the point 
.
Decomposition of the motion of a flat figure into translational and rotational
Consider two consecutive positions 
And 
occupied by the body at moments of time 
And 
(Fig. 61). Body from position 
to position 
can be transferred as follows. Let's move the body first progressively. In this case, the segment 
will move parallel to itself to position 
, and then let's turn body around a point (pole) 
at an angle 
until the points coincide 
And 
.
Hence, any plane motion can be represented as the sum of translational motion together with the selected pole and rotational motion, relative to this pole.
Let's consider methods that can be used to determine the velocities of points of a body performing plane motion.
1. Pole method. This method is based on the resulting decomposition of plane motion into translational and rotational. The speed of any point of a flat figure can be represented in the form of two components: translational, with a speed equal to the speed of an arbitrarily chosen point -poles , and rotational around this pole.
Let's consider a flat body (Fig. 62). The equations of motion are: 
,
,
.
From these equations we determine the speed of the point 
(as with the coordinate method of specifying)
,
,
.
Thus, the speed of the point 
- the quantity is known. We take this point as a pole and determine the speed of an arbitrary point 
bodies.
Speed 
will consist of a translational component 
, when moving along with the point 
, and rotational 
, when rotating the point 
relative to the point 
. Point speed 
move to point 
parallel to itself, since during translational motion the velocities of all points are equal both in magnitude and direction. Speed 
will be determined by dependence (50) 
, and this vector is directed perpendicular to the radius 
in the direction of rotation 
. Vector 
will be directed along the diagonal of a parallelogram built on vectors 
And 
, and its module is determined by the dependency:
,
.(55)
2. Theorem on the projections of velocities of two points of a body.
The projections of the velocities of two points of a rigid body onto a straight line connecting these points are equal to each other.
Consider two points of the body 
And 
(Fig. 63). Taking a point 
beyond the pole, we determine the direction 
depending on (55): 
. We project this vector equality onto the line 
and considering that 
perpendicular 
, we get
3. Instantaneous velocity center.
Instantaneous velocity center(MCS) is a point whose speed at a given time is zero.
Let us show that if a body does not move translationally, then such a point exists at every moment of time and, moreover, is unique. Let at a moment in time 
points 
And 
bodies lying in section 
, have speeds 
And 
, not parallel to each other (Fig. 64). Then point 
, lying at the intersection of perpendiculars to the vectors 
And 
, and there will be an MCS, since 
.
Indeed, if we assume that 
, then according to Theorem (56), the vector 
must be perpendicular at the same time 
And 
, which is impossible. From the same theorem it is clear that no other section point 
at this moment in time cannot have a speed equal to zero.
Using the pole method 
- pole, determine the speed of the point 
(55): because 
,
. (57)
A similar result can be obtained for any other point of the body. Therefore, the speed of any point on the body is equal to its rotational speed relative to the MCS:
,
,
, i.e. the velocities of body points are proportional to their distances to the MCS.
From the three considered methods for determining the velocities of points of a flat figure, it is clear that the MCS is preferable, since here the speed is immediately determined both in magnitude and in the direction of one component. However, this method can be used if we know or can determine the position of the MCS for the body.
Determining the position of the MCS
1. If we know for a given position of the body the directions of the velocities of two points of the body, then the MCS will be the point of intersection of the perpendiculars to these velocity vectors.
2. The velocities of two points of the body are antiparallel (Fig. 65, A). In this case, the perpendicular to the velocities will be common, i.e. The MCS is located somewhere on this perpendicular. To determine the position of the MCS, it is necessary to connect the ends of the velocity vectors. The point of intersection of this line with the perpendicular will be the desired MCS. In this case, the MCS is located between these two points.
3. The velocities of two points of the body are parallel, but not equal in magnitude (Fig. 65, b). The procedure for obtaining the MDS is similar to that described in paragraph 2.
d) The velocities of two points are equal in both magnitude and direction (Fig. 65, V). We obtain the case of instantaneous translational motion, in which the velocities of all points of the body are equal. Consequently, the angular velocity of the body in this position is zero:
4. Let us determine the MCS for a wheel rolling without sliding on a stationary surface (Fig. 65, G). Since the movement occurs without sliding, at the point of contact of the wheel with the surface the speed will be the same and equal to zero, since the surface is stationary. Consequently, the point of contact of the wheel with a stationary surface will be the MCS.
Determination of accelerations of points of a plane figure
When determining the accelerations of points of a flat figure, there is an analogy with methods for determining velocities.
1. Pole method. Just as when determining velocities, we take as a pole an arbitrary point of the body whose acceleration we know or we can determine. Then the acceleration of any point of a flat figure is equal to the sum of the accelerations of the pole and the acceleration in rotational motion around this pole:
In this case, the component 
determines the acceleration of a point 
as it rotates around the pole 
. When rotating, the trajectory of the point will be curvilinear, which means 
(Fig. 66).
Then dependence (58) takes the form 
.
(59)
Taking into account dependencies (51) and (52), we obtain 
,
.
2. Instant acceleration center.
Instant acceleration center(MCU) is a point whose acceleration at a given time is zero.
Let us show that at any given moment of time such a point exists. We take a point as a pole 
, whose acceleration 
we know. Finding the angle 
, lying within 
, and satisfying the condition 
. If 
, That 
and vice versa, i.e. corner 
delayed in direction 
. Let's postpone from the point 
at an angle 
to vector 
line segment 
(Fig. 67). The point obtained by such constructions 
there will be an MCU.
Indeed, the acceleration of the point 
equal to the sum of accelerations 
poles 
and acceleration 
in rotational motion around the pole 
:
.
,
. Then 
. On the other hand, acceleration 
forms with the direction of the segment 
corner 
, which satisfies the condition 
. A minus sign is placed in front of the tangent of the angle 
, since rotation 
relative to the pole 
counterclockwise, and the angle 
is deposited clockwise. Then 
.
Hence, 
and then 
.
Special cases of determining the MCU
1.
. Then 
, and, therefore, the MCU does not exist. In this case, the body moves translationally, i.e. the velocities and accelerations of all points of the body are equal.
2.
. Then 
,
. This means that the MCU lies at the intersection of the lines of action of the accelerations of the points of the body (Fig. 68, A).
3.
. Then, 
,
. This means that the MCU lies at the intersection of perpendiculars to the accelerations of points of the body (Fig. 68, b).
4.
. Then 
,
. This means that the MCU lies at the intersection of rays drawn to the accelerations of points of the body at an angle 
(Fig. 68, V).
From the considered special cases we can conclude: if we accept the point 
beyond the pole, then the acceleration of any point of a flat figure is determined by the acceleration in rotational motion around the MCU:
.
(60)
Complex point movement a movement in which a point simultaneously participates in two or more movements is called. With such movement, the position of the point is determined relative to the moving and relatively stationary reference systems.
The movement of a point relative to a moving reference frame is called relative motion of a point
. We agree to denote the parameters of relative motion 
.
The movement of that point of the moving reference system with which the moving point relative to the stationary reference system currently coincides is called portable movement of the point
. We agree to denote the parameters of portable motion 
.
The movement of a point relative to a fixed frame of reference is called absolute (complex)
point movement
. We agree to denote the parameters of absolute motion 
.
As an example of complex movement, we can consider the movement of a person in a moving vehicle (tram). In this case, the human movement is related to the moving coordinate system - the tram and to the fixed coordinate system - the earth (road). Then, based on the definitions given above, the movement of a person relative to the tram is relative, the movement together with the tram relative to the ground is portable, and the movement of a person relative to the ground is absolute.
We will determine the position of the point 
radii - vectors relative to the moving 
and motionless 
coordinate systems (Fig. 69). Let us introduce the following notation: 
- radius vector defining the position of the point 
relative to the moving coordinate system 
,
;
- radius vector that determines the position of the beginning of the moving coordinate system (point 
) (dots 
);
- radius – a vector that determines the position of a point 
relative to a fixed coordinate system 
;
,.
Let us obtain conditions (constraints) corresponding to relative, portable and absolute motions.
1. When considering relative motion, we will assume that the point 
moves relative to the moving coordinate system 
, and the moving coordinate system itself 
relative to a fixed coordinate system 
doesn't move.
Then the coordinates of the point 
will change in relative motion, but the orth-vectors of the moving coordinate system will not change in direction:
,
,
.
2. When considering portable motion, we will assume that the coordinates of the point 
relative to the moving coordinate system are fixed, and the point moves along with the moving coordinate system 
relatively stationary 
:
,
,
,.
3. With absolute motion, the point also moves relatively 
and together with the coordinate system 
relatively stationary 
:
Then the expressions for the velocities, taking into account (27), have the form
,
,
Comparing these dependencies, we obtain the expression for absolute speed: 
.
(61)
We obtained a theorem on the addition of the velocities of a point in complex motion: the absolute speed of a point is equal to the geometric sum of the relative and portable speed components.
Using dependence (31), we obtain expressions for accelerations:
,
Comparing these dependencies, we obtain an expression for absolute acceleration: 
.
We found that the absolute acceleration of a point is not equal to the geometric sum of the relative and portable acceleration components. Let us determine the absolute acceleration component in parentheses for special cases.
1. Portable translational movement of the point 
. In this case, the axes of the moving coordinate system 
move all the time parallel to themselves, then. 
,
,
,
,
,
, Then 
. Finally we get
.
(62)
If the portable motion of a point is translational, then the absolute acceleration of the point is equal to the geometric sum of the relative and portable components of the acceleration.
2. The portable movement of the point is non-translational. This means that in this case the moving coordinate system 
rotates around the instantaneous axis of rotation with angular velocity 
(Fig. 70). Let us denote the point at the end of the vector 
through 
. Then, using the vector method of specifying (15), we obtain the velocity vector of this point 
.
On the other side, 
. Equating the right-hand sides of these vector equalities, we obtain: 
. Proceeding similarly for the remaining unit vectors, we obtain: 
,
.
In the general case, the absolute acceleration of a point is equal to the geometric sum of the relative and translational components of the acceleration plus the doubled vector product of the angular velocity vector of the translational motion and the linear velocity vector of the relative motion.
The double vector product of the angular velocity vector of the portable motion and the linear velocity vector of the relative motion is called Coriolis acceleration and is designated
.
(64)
Coriolis acceleration characterizes the change in relative speed in portable movement and change in transfer speed in relative motion.
Headed 
according to the vector product rule. The Coriolis acceleration vector is always directed perpendicular to the plane formed by the vectors 
And 
, in such a way that, looking from the end of the vector 
, see the turn 
To 
, through the smallest angle, counterclockwise.
The Coriolis acceleration modulus is equal to.
Rigid body kinematics
In contrast to the kinematics of a point, the kinematics of rigid bodies solves two main problems:
Specifying movement and determining the kinematic characteristics of the body as a whole;
Determination of kinematic characteristics of body points.
Methods for specifying and determining kinematic characteristics depend on the types of body motion.
This manual discusses three types of motion: translational, rotational around a fixed axis and plane-parallel motion of a rigid body.
Translational motion of a rigid body
Translational is a movement in which a straight line drawn through two points of the body remains parallel to its original position (Fig. 2.8).
The theorem has been proven: during translational motion, all points of the body move along the same trajectories and at each moment of time have the same magnitude and direction of speed and acceleration (Fig. 2.8).
Conclusion: The translational motion of a rigid body is determined by the movement of any of its points, and therefore, the task and study of its motion is reduced to the kinematics of the point.
Rice. 2.8 Fig. 2.9
Rotational motion of a rigid body around a fixed axis.
Rotational motion around a fixed axis is the motion of a rigid body in which two points belonging to the body remain motionless during the entire time of motion.
The position of the body is determined by the angle of rotation (Fig. 2.9). The unit of measurement for angle is radian. (A radian is the central angle of a circle whose arc length is equal to the radius; the full angle of the circle contains 2 radians.)
The law of rotational motion of a body around a fixed axis = (t). We determine the angular velocity and angular acceleration of the body by the differentiation method
Angular velocity, rad/s; (2.10)
Angular acceleration, rad/s 2 (2.11)
When a body rotates around a fixed axis, its points that do not lie on the axis of rotation move in circles with the center on the axis of rotation.
If you dissect the body with a plane perpendicular to the axis, select a point on the axis of rotation WITH and an arbitrary point M, then point M will describe around a point WITH circle radius R(Fig. 2.9). During dt an elementary rotation occurs through an angle, and the point M will move along the trajectory for a distance. Let us determine the linear velocity module:
Point acceleration M with a known trajectory, it is determined by its components, see (2.8)
Substituting expression (2.12) into the formulas we get:
where: - tangential acceleration,
Normal acceleration.
Plane - parallel motion of a rigid body
Plane-parallel motion is the motion of a rigid body in which all its points move in planes parallel to one fixed plane (Fig. 2.10). To study the motion of a body, it is enough to study the motion of one section S of this body by a plane parallel to the fixed plane. Section movement S in its plane can be considered as complex, consisting of two elementary movements: a) translational and rotational; b) rotational relative to the moving (instantaneous) center.
In the first version the movement of the section can be specified by the equations of motion of one of its points (poles) and the rotation of the section around the pole (Fig. 2.11). Any section point can be taken as a pole.
Rice. 2.10 Fig. 2.11
The equations of motion will be written in the form:
X A = X A (t)
Y A = Y A (t) (2.14)
A = A (t)
The kinematic characteristics of the pole are determined from the equations of its motion.
The speed of any point of a flat figure moving in its plane is composed of the speed of the pole (arbitrarily chosen in the section of the point A) and the speed of rotation around the pole (rotation of the point IN around the point A).
The acceleration of a point of a moving flat figure consists of the acceleration of the pole relative to a stationary reference frame and the acceleration due to rotational motion around the pole.
In the second option the movement of the section is considered as rotational around a moving (instantaneous) center P(Fig. 1.12). In this case, the speed of any point B of the section will be determined by the formula for rotational motion
Angular velocity around the instantaneous center R can be determined if the speed of any section point, for example point A, is known.
Fig.2.12
The position of the instantaneous center of rotation can be determined based on the following properties:
The point's velocity vector is perpendicular to the radius;
The absolute velocity of a point is proportional to the distance from the point to the center of rotation ( V= R) ;
The speed at the center of rotation is zero.
Let's consider some cases of determining the position of the instantaneous center.
1. The directions of the velocities of two points of a flat figure are known (Fig. 2.13). Let's draw radius lines. The instantaneous center of rotation P is located at the intersection of perpendiculars drawn to the velocity vectors.
2. The velocities of points A and B are known, and the vectors and are parallel to each other, and the line AB perpendicular (Fig. 2. 14). In this case, the instantaneous center of rotation lies on the line AB. To find it, we draw a line of proportionality of speeds based on the dependence V= R.
3. A body rolls without sliding on the stationary surface of another body (Fig. 2.15). The point of contact of the bodies at the moment has zero velocity, while the velocities of other points of the body are not zero. The tangent point P will be the instantaneous center of rotation.
Rice. 2.13 Fig. 2.14 Fig. 2.15
In addition to the options considered, the velocity of a section point can be determined based on the theorem on the projections of the velocities of two points of a rigid body.
Theorem: the projections of the velocities of two points of a rigid body onto a straight line drawn through these points are equal to each other and equally directed.
Proof: distance AB cannot change, therefore
V And cos cannot be more or less V In cos (Fig. 2.16).
Rice. 2.16
Output: V A cos = V IN cos. (2.19)
Complex point movement
In the previous paragraphs, we considered the movement of a point relative to a fixed frame of reference, the so-called absolute movement. In practice, there are problems in which the motion of a point relative to a coordinate system is known, which moves relative to a fixed system. In this case, it is necessary to determine the kinematic characteristics of the point relative to the stationary system.
It is commonly called: the movement of a point relative to a moving system - relative, the movement of a point together with a moving system - portable, the movement of a point relative to a stationary system - absolute. Velocities and accelerations are called accordingly:
Relative; - figurative; -absolute.
According to the theorem on the addition of velocities, the absolute speed of a point is equal to the vector sum of the relative and portable velocities (Fig.).
The absolute value of the speed is determined by the cosine theorem
Fig.2.17
Acceleration according to the parallelogram rule is determined only with translational movement
With non-translational translational motion, a third component of acceleration appears, called rotational or Coriolis.
The Coriolis acceleration is numerically equal to
where is the angle between the vectors and
The direction of the Coriolis acceleration vector is conveniently determined by the rule of N.E. Zhukovsky: project the vector onto a plane perpendicular to the axis of portable rotation, rotate the projection 90 degrees in the direction of portable rotation. The resulting direction will correspond to the direction of the Coriolis acceleration.
Questions for self-control on the section
1. What are the main tasks of kinematics? Name the kinematic characteristics.
2. Name the methods for specifying the movement of a point and determining kinematic characteristics.
3. Give the definition of translational, rotational around a fixed axis, plane-parallel motion of a body.
4. How is the motion of a rigid body determined during translational, rotational around a fixed axis and plane-parallel motion of the body, and how is the speed and acceleration of a point determined during these body movements?
Absolutely rigid body - body mutual arrangement parts of which do not change during movement.
Translational motion of a rigid body - this is its movement in which any straight line rigidly connected to the body moves while remaining parallel to its original direction.
During the translational motion of a rigid body, all its points move equally in a short time dt, the radius vector of these points changes by the same amount. Accordingly, at each moment of time the velocities of all its points are the same and equal. Therefore, the kinematics of the considered translational motion of a rigid body comes down to the study of the movement of any of its points. Usually we consider the movement of the center of inertia of a rigid body moving freely in space.
Rotational motion of a rigid body - this is a movement in which all its points move in circles, the centers of which are located outside the body . The straight line is called the axis of rotation of the body.
Angular velocity– vector quantity characterizing the speed of rotation of the body; the ratio of the angle of rotation to the time during which this rotation occurred; a vector determined by the first derivative of the angle of rotation of a body with respect to time. The angular velocity vector is directed along the axis of rotation according to the right screw rule. ω=φ/t=2π/T=2πn, where T is the rotation period, n is the rotation frequency. ω=lim Δt → 0 Δφ/Δt=dφ/dt.
Angular acceleration– vector determined by the first derivative of the angular velocity with respect to time. When a body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of angular velocity. Second derivative of the rotation angle with respect to time. When a body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of angular velocity. When the motion is accelerated, the vector ε is codirectional to the vector φ, and when it is slow, it is opposite to it. ε=dω/dt.
If dω/dt> 0, then εω
If dω/dt< 0, то ε ↓ω
4. The principle of inertia (Newton's first law). Inertial reference systems. The principle of relativity.
Newton's first law (law of inertia): every material point (body) maintains a state of rest or uniform linear motion until the influence of other bodies forces it to change this state
The desire of a body to maintain a state of rest or uniform rectilinear motion is called inertia. Therefore, Newton's first law is called the law of inertia.
Newton's first law states the existence of inertial frames of reference.
Inertial reference frame– this is a reference system relative to which a free material point, unaffected by other bodies, moves uniformly in a straight line; this is a system that is either at rest or moving uniformly and rectilinearly relative to some other inertial system.
The principle of relativity- a fundamental physical law, according to which any process proceeds identically in an isolated material system, which is at rest, and in the same system in a state of uniform rectilinear motion. States of motion or rest are defined with respect to an arbitrarily chosen inertial reference frame. The principle of relativity underlies Einstein's special theory of relativity.
5. Galilean transformations.
Principle of relativity (Galilee): no experiments (mechanical, electrical, optical) carried out inside a given inertial reference system make it possible to detect whether this system is at rest or moving uniformly and rectilinearly; all laws of nature are invariant with respect to the transition from one inertial frame of reference to another.
Let us consider two reference systems: the inertial frame K (with coordinates x,y,z), which we will conventionally consider stationary and the system K’ (with coordinates x’,y’,z’), moving relative to K uniformly and rectilinearly with speed U (U = const). Let's find the connection between the coordinates of an arbitrary point A in both systems. r = r’+r0=r’+Ut. (1.)
Equation (1.) can be written in projections on the coordinate axes:
y=y’+Uyt; (2.)
z=z’+Uzt; Equations (1.) and (2.) are called Galilean coordinate transformations.
Relationship between potential energy and force
Each point of the potential field corresponds, on the one hand, to a certain value of the force vector acting on the body, and, on the other hand, to a certain value of potential energy. Therefore, there must be a certain relationship between force and potential energy.
To establish this connection, let us calculate the elementary work performed by field forces during a small displacement of the body occurring along an arbitrarily chosen direction in space, which we denote by the letter . This work is equal to
where is the projection of the force onto the direction.
Since in this case the work is done due to the reserve of potential energy, it is equal to the loss of potential energy on the axis segment:
From the last two expressions we get
This formula determines the projection of the force vector onto the coordinate axes. If these projections are known, the force vector itself turns out to be determined:
in mathematics vector 
,
where a - scalar function x, y, z, is called the gradient of this scalar, denoted by the symbol . Therefore, the force is equal to the potential energy gradient taken with the opposite sign
