Elementary angle of rotation, angular velocity
Figure 9. Elementary angle of rotation ()
Elementary (infinitely small) rotations are treated as vectors. The module of the vector is equal to the angle of rotation, and its direction coincides with the direction of the translational movement of the tip of the screw, the head of which rotates in the direction of movement of the point along the circle, that is, it obeys the rule of the right screw.
Angular velocity
The vector is directed along the axis of rotation according to the right screw rule, i.e., in the same way as the vector (see Figure 10).
Figure 10.
Figure 11
The vector value determined by the first derivative of the angle of rotation of the body with respect to time.
Relationship between the modules of linear and angular velocities
Figure 12
Relationship between linear and angular velocity vectors
The position of the point under consideration is given by the radius vector (drawn from the origin 0 lying on the axis of rotation). The vector product coincides in direction with the vector and has a modulus equal to
The unit of angular velocity is .
Pseudovectors (axial vectors) are vectors whose directions are associated with the direction of rotation (for example,). These vectors do not have specific application points: they can be drawn from any point on the axis of rotation.
Uniform motion of a material point along a circle
Uniform motion in a circle - a motion in which a material point (body) for equal periods of time passes the arcs of a circle equal in length.
Angular velocity
: (-- angle of rotation).
The rotation period T is the time during which the material point makes one complete revolution around the circumference, i.e., rotates through an angle.
Since it corresponds to the time interval, then.
Frequency of rotation - the number of complete revolutions made by a material point with its uniform movement along a circle, per unit time.
Figure 13
A characteristic feature of uniform motion in a circle
Uniform circular motion is a special case of curvilinear motion. Movement along a circle with a speed constant modulo () is accelerated. This is due to the fact that at a constant modulus, the direction of the velocity changes all the time.
Acceleration of a material point uniformly moving in a circle
The tangential component of acceleration in the uniform motion of a point along a circle is equal to zero.
The normal component of acceleration (centripetal acceleration) is directed along the radius towards the center of the circle (see Figure 13). At any point on the circle, the normal acceleration vector is perpendicular to the velocity vector. The acceleration of a material point moving uniformly along a circle at any of its points is centripetal.
angular acceleration. Relationship between linear and angular quantities
Angular acceleration is a vector quantity determined by the first derivative of the angular velocity with respect to time.
Direction of the angular acceleration vector
When the body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of the angular velocity.
With accelerated motion, the vector is aligned with the vector, with slow motion, it is opposite to it. A vector is a pseudo-vector.
The unit of angular acceleration is .
Relationship between linear and angular quantities
(-- radius of the circle; -- linear velocity; -- tangential acceleration; -- normal acceleration; -- angular velocity).
with linear values.
Angular movement- a vector quantity characterizing the change in the angular coordinate in the process of its movement.
Angular velocity- vector physical quantity characterizing the speed of rotation of the body. The angular velocity vector is equal in magnitude to the angle of rotation of the body per unit time:
and is directed along the axis of rotation according to the rule of the gimlet, that is, in the direction in which the gimlet with a right-hand thread would be screwed in if it rotated in the same direction.
The unit of measurement of angular velocity adopted in the SI and CGS systems) is radians per second. (Note: the radian, like any unit of angle measurement, is physically dimensionless, so the physical dimension of angular velocity is simply ). The technique also uses revolutions per second, much less often - degrees per second, degrees per second. Perhaps, revolutions per minute are used most often in technology - this has been going on since the times when the rotational speed of low-speed steam engines was determined by simply “manually” counting the number of revolutions per unit of time.
The (instantaneous) velocity vector of any point of an (absolutely) rigid body rotating at an angular velocity is given by:
where is the radius vector to the given point from the origin located on the axis of rotation of the body, and square brackets denote the vector product. The linear speed (coinciding with the modulus of the velocity vector) of a point at a certain distance (radius) r from the axis of rotation can be considered as follows: v = rω. If other units of angles are used instead of radians, then in the last two formulas a multiplier will appear that is not equal to one.
In the case of plane rotation, i.e. when all the velocity vectors of the body's points lie (always) in the same plane ("plane of rotation"), the angular velocity of the body is always perpendicular to this plane, and in fact - if the plane of rotation is known in advance - can be replaced by a scalar - projection onto an axis orthogonal to the plane of rotation. In this case, the kinematics of rotation is greatly simplified, however, in the general case, the angular velocity can change direction over time in three-dimensional space, and such a simplified picture does not work.
The derivative of the angular velocity with respect to time is the angular acceleration.
Motion with a constant angular velocity vector is called uniform rotational motion (in this case, the angular acceleration is zero).
The angular velocity (considered as a free vector) is the same in all inertial frames of reference, however, in different inertial frames of reference, the axis or center of rotation of the same specific body at the same moment of time may differ (that is, there will be a different “point of application” of the angular speed).
In the case of the movement of a single point in three-dimensional space, you can write an expression for the angular velocity of this point relative to the selected origin:
Where is the radius vector of the point (from the origin), is the speed of this point. - vector product, - scalar product of vectors. However, this formula does not uniquely determine the angular velocity (in the case of a single point, you can choose other vectors that are suitable by definition, otherwise - arbitrarily - choosing the direction of the axis of rotation), but for the general case (when the body includes more than one material point) - this formula is not true for the angular velocity of the entire body (because it gives different values for each point, and during the rotation of an absolutely rigid body, by definition, the angular velocity of its rotation is the only vector). With all this, in the two-dimensional case (the case of plane rotation) this formula is quite sufficient, unambiguous and correct, since in this particular case the direction of the axis of rotation is definitely uniquely determined.
In the case of uniform rotational motion (that is, motion with a constant angular velocity vector), the Cartesian coordinates of the points of a body rotating in this way perform harmonic oscillations with an angular (cyclic) frequency equal to the modulus of the angular velocity vector.
When measuring angular velocity in revolutions per second (r/s), the modulus of angular velocity of uniform rotational motion is the same as the rotational speed f, measured in hertz (Hz)
(that is, in such units).
In the case of using the usual physical unit of angular velocity - radians per second - the modulus of angular velocity is related to the rotational speed as follows:
Finally, when using degrees per second, the relation to RPM would be:
Angular acceleration- pseudovector physical quantity characterizing the rate of change of the angular velocity of a rigid body.
When a body rotates around a fixed axis, the angular acceleration modulo is:
The angular acceleration vector α is directed along the axis of rotation (to the side with accelerated rotation and oppositely - with slow rotation).
When rotating around a fixed point, the angular acceleration vector is defined as the first derivative of the angular velocity vector ω with respect to time, that is
and is directed tangentially to the hodograph of the vector at its corresponding point.
There is a relationship between tangential and angular accelerations:
where R is the radius of curvature of the point trajectory at a given time. So, the angular acceleration is equal to the second derivative of the angle of rotation with respect to time or the first derivative of the angular velocity with respect to time. Angular acceleration is measured in rad/sec2.
Angular Velocity and Angular Acceleration
Consider a rigid body that rotates around a fixed axis. Then individual points of this body will describe circles of different radii, the centers of which lie on the axis of rotation. Let some point move along a circle of radius R(Fig. 6). Its position after a period of time D t set the angle D. Elementary (infinitely small) rotations can be considered as vectors (they are denoted by or ) . The module of the vector is equal to the angle of rotation, and its direction coincides with the direction of the translational movement of the tip of the screw, the head of which rotates in the direction of movement of the point along the circle, i.e. obeys right screw rule(Fig. 6). Vectors whose directions are associated with the direction of rotation are called pseudovectors or axial vectors. These vectors do not have specific application points: they can be drawn from any point on the rotation axis.
angular velocity is called a vector quantity equal to the first derivative of the angle of rotation of the body with respect to time:
The vector is directed along the rotation axis according to the right screw rule, i.e. the same as the vector (Fig. 7). Dimension of angular velocity dim w =T - 1 , and its unit is radian per second (rad/s).
Point Linear Speed (See Fig. 6)
In vector form, the formula for linear velocity can be written as a cross product:
In this case, the module of the vector product, by definition, is equal, and the direction coincides with the direction of the translational movement of the right screw when it rotates from to R.
If ( = const, then the rotation is uniform and can be characterized rotation period T - the time for which the point makes one complete revolution, i.e. rotates through an angle of 2p. Since the time interval D t= T corresponds to = 2p, then = 2p/ T, where
The number of complete revolutions made by the body during its uniform motion in a circle per unit time is called the frequency of rotation:
Angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect to time:
When the body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of the angular velocity. With accelerated motion, the vector is co-directed to the vector (Fig. 8), with slow motion, it is opposite to it (Fig. 9).
Tangential component of acceleration
Normal component of acceleration
Thus, the relationship between linear (path length s passed by the point along the arc of a circle of radius R, linear speed v, tangential acceleration , normal acceleration ) and angular quantities (angle of rotation j, angular velocity w, angular acceleration e) is expressed by the following formulas:
In the case of uniformly variable motion of a point along a circle (e=const)
where w 0 is the initial angular velocity.
Newton's laws.
Newton's first law. Weight. Force
Dynamics is the main branch of mechanics, it is based on Newton's three laws, formulated by him in 1687. Newton's laws play an exceptional role in mechanics and are (like all physical laws) a generalization of the results of vast human experience. They are considered as system of interconnected laws and not every single law is subjected to experimental verification, but the whole system as a whole.
Newton's first law: any material point (body) retains a state of rest or uniform rectilinear motion until the impact from other bodies makes it 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 also called law of inertia.
Mechanical motion is relative, and its nature depends on the frame of reference. Newton's first law is not valid in any frame of reference, and those systems in relation to which it is performed are called inertial reference systems. An inertial frame of reference is such a frame of reference, relative to which a material point, free from external influences, either at rest or moving uniformly and in a straight line. Newton's first law states the existence of inertial frames of reference.
It has been experimentally established that the heliocentric (stellar) frame of reference can be considered inertial (the origin of coordinates is in the center of the Sun, and the axes are drawn in the direction of certain stars). The reference frame associated with the Earth, strictly speaking, is non-inertial, but the effects due to its non-inertiality (the Earth rotates around its own axis and around the Sun) are negligible in solving many problems, and in these cases it can be considered inertial.
From experience it is known that under the same influences, different bodies change the speed of their movement unequally, i.e., in other words, acquire different accelerations. Acceleration depends not only on the magnitude of the impact, but also on the properties of the body itself (on its mass).
Weight bodies - a physical quantity, which is one of the main characteristics of matter, which determines its inertial ( inertial mass) and gravitational ( gravitational mass) properties. At present, it can be considered proven that the inertial and gravitational masses are equal to each other (with an accuracy not less than 10–12 of their values).
To describe the effects mentioned in Newton's first law, the concept of force is introduced. Under the action of forces, bodies either change their speed of movement, i.e., acquire accelerations (dynamic manifestation of forces), or deform, i.e., change their shape and dimensions (static manifestation of forces). At each moment of time, the force is characterized by a numerical value, a direction in space, and a point of application. So, force- this is a vector quantity, which is a measure of the mechanical impact on the body from other bodies or fields, as a result of which the body acquires acceleration or changes its shape and size.
Newton's second law
Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the mechanical motion of a material point (body) changes under the action of forces applied to it.
If we consider the action of different forces on the same body, it turns out that the acceleration acquired by the body is always directly proportional to the resultant of the applied forces:
a~f(t=const). (6.1)
Under the action of the same force on bodies with different masses, their accelerations turn out to be different, namely
a ~ 1 /t (F= const). (6.2)
Using expressions (6.1) and (6.2) and taking into account that force and acceleration are vector quantities, we can write
a = kF/m. (6.3)
Relation (6.3) expresses Newton's second law: the acceleration acquired by a material point (body), proportional to the force causing it, coincides with it in direction and is inversely proportional to the mass of the material point (body).
In SI, the proportionality factor k= 1. Then
(6.4)
Considering that the mass of a material point (body) in classical mechanics is a constant value, in expression (6.4) it can be brought under the sign of the derivative:
Vector quantity
numerically equal to the product of the mass of a material point and its speed and having the direction of speed, is called momentum (momentum) this material point.
Substituting (6.6) into (6.5), we obtain
This expression - more general formulation of Newton's second law: the rate of change of momentum of a material point is equal to the force acting on it. Expression (6.7) is called the equation of motion of a material point.
Unit of force in SI - newton(N): 1 N is a force that imparts an acceleration of 1 m / s 2 to a mass of 1 kg in the direction of the force:
1 N \u003d 1 kg × m / s 2.
Newton's second law is valid only in inertial frames of reference. Newton's first law can be derived from the second. Indeed, if the resultant force is equal to zero (in the absence of influence on the body from other bodies), the acceleration (see (6.3)) is also equal to zero. However Newton's first law regarded as independent law(and not as a consequence of the second law), since it is he who asserts the existence of inertial frames of reference, in which only equation (6.7) is satisfied.
In mechanics, it is of great importance principle of independence of action of forces: if several forces act simultaneously on a material point, then each of these forces imparts an acceleration to the material point according to Newton's second law, as if there were no other forces. According to this principle, forces and accelerations can be decomposed into components, the use of which leads to a significant simplification of problem solving. For example, in fig. 10 acting force F= m a is decomposed into two components: tangential force F t , (directed tangentially to the trajectory) and normal force F n(directed along the normal to the center of curvature). Using expressions and , as well as , you can write:
If several forces act simultaneously on a material point, then, according to the principle of independence of the action of forces, F in Newton's second law is understood as the resulting force.
Newton's third law
The interaction between material points (bodies) is determined by Newton's third law: any action of material points (bodies) on each other has the character of interaction; the forces with which material points act on each other are always equal in absolute value, oppositely directed and act along the straight line connecting these points:
F 12 = - F 21, (7.1)
where F 12 is the force acting on the first material point from the second;
F 21 - force acting on the second material point from the first. These forces are applied to different material points (bodies), always act in pairs and are the forces one nature.
Newton's third law allows the transition from dynamics separate material point to dynamics systems material points. This follows from the fact that for a system of material points, the interaction is reduced to the forces of pair interaction between material points.
The motion of an extended body whose dimensions cannot be neglected under the conditions of the problem under consideration. The body will be considered non-deformable, in other words, absolutely rigid.
The movement in which any a straight line connected with a moving body remains parallel to itself, is called progressive.
A straight line "rigidly connected with a body" is understood as such a straight line, the distance from any point of which to any point of the body remains constant during its movement.
The translational motion of an absolutely rigid body can be characterized by the motion of any point of this body, since in translational motion all points of the body move with the same speeds and accelerations, and the trajectories of their motion are congruent. Having determined the motion of any of the points of a rigid body, we will at the same time determine the motion of all its other points. Therefore, when describing the translational motion, no new problems arise in comparison with the kinematics of a material point. An example of translational movement is shown in fig. 2.20.
Fig.2.20. Translational movement of the body
An example of translational motion is shown in the following figure:
Fig.2.21. Planar body movement
Another important particular case of the motion of a rigid body is the motion in which two points of the body remain stationary.
A movement in which two points of the body remain stationary is called rotation around a fixed axis.
The line connecting these points is also fixed and is called axis of rotation.
Fig.2.22. Rotation of a rigid body
With such a movement, all points of the body move along circles located in planes perpendicular to the axis of rotation. The centers of the circles lie on the axis of rotation. In this case, the axis of rotation can also be located outside the body.
Video 2.4. Translational and rotational movements.
Angular velocity, angular acceleration. When a body rotates around an axis, all its points describe circles of different radii and, therefore, have different displacements, velocities, and accelerations. However, it is possible to describe the rotational motion of all points of the body in the same way. For this, other (compared to a material point) kinematic characteristics of movement are used - angle of rotation, angular velocity, angular acceleration.
Rice. 2.23. Acceleration vectors of a point moving in a circle
The role of displacement in rotational motion is played by small turn vector, around the axis of rotation 00" (Fig. 2.24.). It will be the same for any point absolutely rigid body(for example, dots 1, 2, 3 ).
Rice. 2.24. Rotation of a perfectly rigid body about a fixed axis
The module of the rotation vector is equal to the value of the angle of rotation, and angle is measured in radians.
The vector of an infinitesimal rotation along the axis of rotation is directed towards the movement of the right screw (gimlet) rotated in the same direction as the body.
Video 2.5. The final angular displacements are not vectors, since they do not add up according to the parallelogram rule. Infinitely small angular displacements are vectors.
Vectors whose directions are associated with the gimlet rule are called axial(from English. axis- axis) as opposed to polar. vectors that we used earlier. Polar vectors are, for example, the radius vector, the velocity vector, the acceleration vector, and the force vector. Axial vectors are also called pseudovectors, since they differ from true (polar) vectors in their behavior during the reflection operation in the mirror (inversion or, which is the same, the transition from the right to the left coordinate system). It can be shown (this will be done later) that the addition of vectors of infinitesimal rotations occurs in the same way as the addition of true vectors, that is, according to the parallelogram (triangle) rule. Therefore, if the operation of reflection in a mirror is not considered, then the difference between pseudovectors and true vectors does not manifest itself in any way and it is possible and necessary to treat them as with ordinary (true) vectors.
The ratio of the vector of an infinitesimal rotation to the time during which this rotation took place
called angular speed of rotation.
The basic unit for measuring the magnitude of the angular velocity is rad/s. In printed publications, for reasons that have nothing to do with physics, they often write 1/s or from -1 which, strictly speaking, is false. Angle is a dimensionless quantity, but its units of measurement are different (degrees, rhumbs, grads ...) and they must be indicated, at least to avoid misunderstandings.
Video 2.6. Stroboscopic effect and its use for remote measurement of the angular velocity of rotation.
The angular velocity, like the vector to which it is proportional, is an axial vector. When spinning around motionless axis angular velocity does not change its direction. With uniform rotation, its value also remains constant, so that the vector . In the case of sufficient constancy in time of the value of the angular velocity, the rotation can be conveniently characterized by its period T :
Rotation period- this is the time for which the body makes one revolution (rotation through an angle of 2π) around the axis of rotation.
The words "sufficient constancy" obviously mean that during the period (the time of one revolution) the module of the angular velocity changes insignificantly.
Also often used number of revolutions per unit time
At the same time, in technical applications (first of all, all kinds of engines), it is customary to take not a second, but a minute as a unit of time. That is, the angular velocity of rotation is indicated in revolutions per minute. As you can easily see, the relationship between (in radians per second) and (in revolutions per minute) is as follows
The direction of the angular velocity vector is shown in fig. 2.25.
By analogy with linear acceleration, angular acceleration is introduced as the rate of change of the angular velocity vector. Angular acceleration is also an axial vector (pseudovector).
Angular acceleration - axial vector defined as the time derivative of angular velocity
When rotating about a fixed axis, more generally when rotating about an axis that remains parallel to itself, the angular velocity vector is also directed parallel to the axis of rotation. With an increase in the value of the angular velocity || angular acceleration coincides with it in direction, while decreasing - it is directed in the opposite direction. We emphasize that this is only a special case of the invariance of the direction of the axis of rotation, in the general case (rotation around a point) the axis of rotation itself rotates and then the above is not true.
Connection of angular and linear velocities and accelerations. Each of the points of the rotating body moves with a certain linear velocity directed tangentially to the corresponding circle (see Fig. 19). Let the material point rotate around the axis 00" around a circle with a radius R. For a small period of time, it will pass the path corresponding to the angle of rotation. Then
Passing to the limit , we obtain an expression for the modulus of the linear velocity of a point of a rotating body.
Recall here R- distance from the considered point of the body to the axis of rotation.
Rice. 2.26.
Since the normal acceleration is
then, taking into account the relationship for the angular and linear speeds, we obtain
The normal acceleration of points in a rotating rigid body is often referred to as centripetal acceleration.
Differentiating with respect to time the expression for , we find
where is the tangential acceleration of a point moving along a circle with a radius R.
Thus, both tangential and normal accelerations grow linearly with increasing radius R- distance from the axis of rotation. The total acceleration also depends linearly on R :
Example. Let us find the linear velocity and centripetal acceleration of points lying on the earth's surface at the equator and at the latitude of Moscow ( = 56°). We know the period of rotation of the Earth around its own axis T \u003d 24 hours \u003d 24x60x60 \u003d 86,400 s. From here is the angular velocity of rotation
Earth mean radius
The distance to the axis of rotation at latitude is
From here we find the linear velocity
and centripetal acceleration
At the equator = 0, cos = 1, therefore,
At the latitude of Moscow cos = cos 56° = 0.559 and we get:
We see that the influence of the Earth's rotation is not so great: the ratio of the centripetal acceleration at the equator to the free fall acceleration is
However, as we shall see later, the effects of the Earth's rotation are quite observable.
Relationship between linear and angular velocity vectors. The relations between the angular and linear speeds obtained above are written for the modules of the vectors and . To write these relationships in vector form, we use the concept of a vector product.
Let be 0z- the axis of rotation of an absolutely rigid body (Fig. 2.28).
Rice. 2.28. Relationship between linear and angular velocity vectors
Dot BUT revolves around a circle with a radius R. R- distance from the axis of rotation to the considered point of the body. Let's take a point 0 for the origin of coordinates. Then
and since
then by definition of the vector product, for all points of the body
Here is the radius vector of the point of the body, starting at the point O, lying in an arbitrary fixed place, necessarily on the axis of rotation
But on the other side
The first term is equal to zero, since the vector product of collinear vectors is equal to zero. Hence,
where vector R is perpendicular to the axis of rotation and directed away from it, and its modulus is equal to the radius of the circle along which the material point moves and this vector starts at the center of this circle.
Rice. 2.29. To the definition of the instantaneous axis of rotation
Normal (centripetal) acceleration can also be written in vector form:
and the sign "-" shows that it is directed to the axis of rotation. Differentiating the relation for the linear and angular velocity with respect to time, we find the expression for the total acceleration
The first term is directed tangentially to the trajectory of a point on a rotating body and its modulus is , since
Comparing with the expression for tangential acceleration, we conclude that this is the tangential acceleration vector
Therefore, the second term is the normal acceleration of the same point:
Indeed, it is directed along the radius R to the axis of rotation and its modulus is equal to
Therefore, this relation for normal acceleration is another form of writing the previously obtained formula.
Additional Information
http://www.plib.ru/library/book/14978.html - Sivukhin D.V. General course of physics, volume 1, Mechanics Ed. Science 1979 - pp. 242–243 (§46, p. 7): a rather difficult to understand question about the vector nature of the angular rotations of a rigid body is discussed;
http://www.plib.ru/library/book/14978.html - Sivukhin D.V. General course of physics, volume 1, Mechanics Ed. Science 1979 - pp. 233–242 (§45, §46 pp. 1–6): instantaneous axis of rotation of a rigid body, addition of rotations;
http://kvant.mirror1.mccme.ru/1990/02/kinematika_basketbolnogo_brosk.html - Kvant magazine - basketball throw kinematics (R. Vinokur);
http://kvant.mirror1.mccme.ru/ - Kvant magazine, 2003, No. 6, - pp. 5–11, field of instantaneous velocities of a rigid body (S. Krotov);
Euler angles, aircraft (ship) angles.
Traditionally, Euler angles are introduced as follows. The transition from the reference position to the actual one is carried out by three turns (Fig. 4.3):
1. Rotate around the corner precession At the same time, it goes to the position, (c) .
2. Rotate around the corner nutation. Wherein, . (4.10)
4. Rotate around the corner own (pure) rotation
For a better understanding, Fig. 4.4 shows a top and Euler angles describing it
The transition from the reference position to the actual one can be carried out by three turns (turn it yourself!) (Fig. 4.5):
1. Rotate around the corner yaw, wherein
2. Rotate around by the pitch angle, while (4.12)
3.Roll angle around
The expression "can be done" is not accidental; it is not difficult to understand that other options are possible, for example, rotations around fixed axes
1. Rotate around the corner roll(at the risk of breaking wings)
2. Rotate around the corner pitch(lifting the "nose") (4.13)
3. Rotate around at an angle yaw
However, the identity of (4.12) and (4.13) also needs to be proved.
Let's write an obvious vector formula for the position vector of any point (Fig. 4.6) in matrix form. Find the coordinates of the vector relative to the reference basis. Let us expand the vector according to the actual basis and introduce a “transferred” vector whose coordinates in the reference basis are equal to the coordinates of the vector in the actual one; in other words, - a vector “rotated” together with the body (Fig.4.6).
| Rice. 4.6. |
Expanding the vectors according to the reference basis, we obtain
We introduce a rotation matrix and columns,
The vector formula in matrix notation has the form
1. The rotation matrix is orthogonal, i.e.
The proof of this statement is the formula (4.9)
Calculating the determinant of the product (4.15), we obtain and since in the reference position, then (orthogonal matrices with determinant equal to (+1) are called actually orthogonal or rotation matrices). The rotation matrix, when multiplied by vectors, does not change either the lengths of the vectors or the angles between them, i.e. really them turns.
2. The rotation matrix has one eigenvector (fixed) that defines the axis of rotation. In other words, it is necessary to show that the system of equations where has a unique solution. We write the system in the form (. The determinant of this homogeneous system is equal to zero, since
hence the system has a non-zero solution. Assuming that there are two solutions, we immediately come to the conclusion that the one perpendicular to them is also a solution (the angles between the vectors do not change), which means that i.e. no turn..
| Fig.4.7 |
Matrix in an orthonormal basis
has a look.
2. Differentiating (4.15), we obtain or, denoting - matrix back (eng. to spin - twirl). Thus, the spin matrix is skew-symmetric: . Multiplying from the right by, we obtain the Poisson formula for the rotation matrix:
We have come to the most difficult moment within the framework of the matrix description - the determination of the angular velocity vector.
You can, of course, act in a standard way (see, for example, the method and write: “ we introduce the notation for the elements of the skew-symmetric matrix S according to the formula
If we make a vector , then the result of multiplying a matrix by a vector can be represented as a cross product". In the above quote - the vector of angular velocity.
Differentiating (4.14), we obtain the matrix representation of the basic formula for the kinematics of a rigid body :
The matrix approach, being convenient for calculations, is very little suitable for analyzing and deriving relationships; any formula written in a vector and tensor language can be easily written in a matrix form, but it is difficult to obtain a compact and expressive formula for describing any physical phenomenon in a matrix form.
In addition, one should not forget that the matrix elements are the coordinates (components) of the tensor in some basis. The tensor itself does not depend on the choice of basis, but its components do. For error-free writing in matrix form, it is necessary that all vectors and tensors included in the expression be written in the same basis, and this is not always convenient, since different tensors have a “simple” form in different bases, so you need to recalculate matrices using transition matrices .
