Angular displacement, angular velocity, angular acceleration, their connection. Angular displacement, angular velocity, angular acceleration, their relationship. What is a rotation angle vector

Euler angles, airplane (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 at an angle precession In this case it goes to position (c) .

2. Rotate around at an angle nutation. Wherein, . (4.10)

4. Rotate around at an angle own (pure) rotation

For a better understanding, Fig. 4.4 shows a top and the Euler angles describing it

1. Rotate around at an angle yaw, wherein

2. Rotate around by a pitch angle, while (4.12)

3.Rotate by roll angle around

The expression “can be accomplished” is not accidental; it is easy to understand that other options are possible, for example, rotations around fixed axes

1. Rotate around at an angle roll(at the risk of breaking his wings)

2. Rotate around at an angle pitch(nose lift) (4.13)

3. Rotate around at an angle yaw

However, the identity of (4.12) and (4.13) also needs to be proven.

Let's write the obvious vector formula for the position vector of a point (Fig. 4.6) in matrix form. Let's 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, the coordinates of which 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).

Let us introduce the 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 formula (4.9)

Calculating the determinant of the product (4.15), we obtain and since in the reference position, then (orthogonal matrices with a determinant equal to (+1) are called actually orthogonal or rotation matrices). When multiplied by vectors, the rotation matrix 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 eigen(fixed) vector, which specifies the rotation axis. In other words, it is necessary to show that the system of equations has a unique solution. Let us write the system in the form (. The determinant of this homogeneous system is equal to zero, since

therefore, 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..

Matrix in an orthonormal basis

has a look.

2. Differentiating (4.15), we obtain or, denoting – matrix spin (English: 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 - determining the angular velocity vector.

You can, of course, do the standard thing (see, for example, the method and write: “ Let us introduce notation for the elements of the skew-symmetric matrix S according to the formula

If you make a vector , then the result of multiplying a matrix by a vector can be represented as a vector product" In the above quote - the angular velocity vector.

Differentiating (4.14), we obtain a matrix representation of the basic formula for the kinematics of a rigid body :

The matrix approach, while convenient for calculations, is very unsuitable for analyzing and deriving relationships; Any formula written in vector and tensor language can be easily written in matrix form, but it is difficult to obtain a compact and expressive formula to describe any physical phenomenon in matrix form.

In addition, we should not forget that the elements of the matrix 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 recording in matrix form, it is necessary that all vectors and tensors included in the expression be written in one basis, and this is not always convenient, since different tensors have a “simple” form in different bases, so you need to recalculate the matrices using transition matrices .

On a circle it is determined by the radius vector $ \overrightarrow (r)$ drawn from the center of the circle. The modulus of the radius vector is equal to the radius of the circle R (Fig. 1).

Figure 1. Radius vector, displacement, path and angle of rotation when a point moves around a circle

In this case, the motion of a body in a circle can be unambiguously described using such kinematic characteristics as the angle of rotation, angular velocity and angular acceleration.

During time ∆t, the body, moving from point A to point B, makes a displacement $\triangle r$ equal to the chord AB, and covers a path equal to the length of the arc l. The radius vector rotates through the angle ∆$ \varphi $.

The angle of rotation can be characterized by the vector of angular displacement $d\overrightarrow((\mathbf \varphi ))$, the magnitude of which is equal to the angle of rotation ∆$ \varphi $, and the direction coincides with the axis of rotation, and so that the direction of rotation corresponds to the right-hand screw rule according to relative to the direction of the vector $d\overrightarrow((\mathbf \varphi ))$.

The vector $d\overrightarrow((\mathbf \varphi ))$ is called an axial vector (or pseudo-vector), while the displacement vector $\triangle \overrightarrow(r)$ is a polar vector (these also include velocity and acceleration vectors) . They differ in that the polar vector, in addition to length and direction, has an application point (pole), and the axial vector has only length and direction (axis - axis in Latin), but does not have an application point. Vectors of this type are often used in physics. These, for example, include all vectors that are the vector product of two polar vectors.

A scalar physical quantity, numerically equal to the ratio of the angle of rotation of the radius vector to the period of time during which this rotation occurred, is called the average angular velocity: $\left\langle \omega \right\rangle =\frac(\triangle \varphi )(\triangle t)$. The SI unit of angular velocity is radian per second $(\frac (rad) (c))$.

Definition

The angular velocity of rotation is a vector that is numerically equal to the first derivative of the angle of rotation of the body with respect to time and directed along the axis of rotation according to the right-hand screw rule:

\[\overrightarrow((\mathbf \omega ))\left(t\right)=(\mathop(lim)_(\triangle t\to 0) \frac(\triangle (\mathbf \varphi ))(\triangle t)=\frac(d\overrightarrow((\mathbf \varphi )))(dt)\ )\]

With uniform motion in a circle, the angular velocity and the magnitude of the linear velocity are constant quantities: $(\mathbf \omega )=const$; $v=const$.

Considering that $\triangle \varphi =\frac(l)(R)$, we obtain the formula for the relationship between linear and angular velocity: $\omega =\frac(l)(R\triangle t)=\frac(v)( R)$. Angular velocity is also related to normal acceleration: $a_n=\frac(v^2)(R)=(\omega )^2R$

With non-uniform motion in a circle, the angular velocity vector is a vector function of time $\overrightarrow(\omega )\left(t\right)=(\overrightarrow(\omega ))_0+\overrightarrow(\varepsilon )\left(t\right) t$, where $(\overrightarrow((\mathbf \omega )))_0$ is the initial angular velocity, $\overrightarrow((\mathbf \varepsilon ))\left(t\right)$ is the angular acceleration. In the case of uniformly variable motion, $\left|\overrightarrow((\mathbf \varepsilon ))\left(t\right)\right|=\varepsilon =const$, and $\left|\overrightarrow((\mathbf \omega ) )\left(t\right)\right|=\omega \left(t\right)=(\omega )_0+\varepsilon t$.

Describe the motion of a rotating rigid body in cases where the angular velocity changes according to graphs 1 and 2, shown in Fig. 2.

Figure 2.

Rotation occurs in two directions - clockwise and counterclockwise. The direction of rotation is associated with the pseudovector of the rotation angle and angular velocity. Let us consider the direction of rotation clockwise to be positive.

For motion 1, the angular velocity increases, but the angular acceleration $\varepsilon $=d$\omega $/dt (derivative) decreases, remaining positive. Consequently, this movement is accelerated clockwise with decreasing acceleration.

For motion 2, the angular velocity decreases, then reaches zero at the point of intersection with the abscissa axis, and then becomes negative and increases in absolute value. Angular acceleration is negative and decreases in magnitude. Thus, at first the point moved clockwise slowly with an angular acceleration decreasing in absolute value, stopped and began to rotate rapidly with an acceleration decreasing in absolute value.

Find the radius R of a rotating wheel if it is known that the linear speed $v_1$ of a point lying on the rim is 2.5 times greater than the linear speed $v_2$ of a point lying at a distance $r = 5 cm$ closer to the wheel axis.

Figure 3.

$$R_2 = R_1 - 5$$ $$v_1 = 2.5v_2$$ $$R_1 = ?$$

The points move along concentric circles, the vectors of their angular velocities are equal, $\left|(\overrightarrow(\omega ))_1\right|=\left|(\overrightarrow(\omega ))_2\right|=\omega $ , therefore , can be written in scalar form:

Answer: wheel radius R = 8.3 cm

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Movements of an extended body, the dimensions of which cannot be neglected under the conditions of the problem under consideration. We will consider the body to be non-deformable, in other words, absolutely solid.

The movement in which any a straight line associated with a moving body remains parallel to itself is called progressive.

By a straight line “rigidly connected with a body” we mean 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 movement of any point of this body, since during 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 point of a rigid body, we will at the same time determine the motion of all its other points. Therefore, when describing translational motion, no new problems arise in comparison with the kinematics of a material point. An example of translational motion is shown in Fig. 2.20.

Fig.2.20. Forward movement of the body

An example of translational motion is shown in the following figure:

Fig.2.21. Flat body movement

Another important special case of motion of a rigid body is motion in which two points of the body remain motionless.

A movement in which two points of the body remain motionless is called rotation around a fixed axis.

The straight line connecting these points is also stationary and is called axis of rotation.

Fig.2.22. Rigid body rotation

With this movement, all points of the body move in 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 be located outside the body.

Video 2.4. Translational and rotational movements.

Angular velocity, angular acceleration. When a body rotates around any 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. To do this, they use other (compared to the material point) kinematic characteristics of motion - angle of rotation, angular velocity, angular acceleration.

Rice. 2.23. Acceleration vector of a point moving in a circle

The role of displacement in rotational motion is played by small rotation vector, around the axis of rotation 00" (Fig. 2.24.). It will be the same for any point absolutely rigid body(for example, points 1, 2, 3 ).

Rice. 2.24. Rotation of an absolutely rigid body around a fixed axis

The magnitude of the rotation vector is equal to the magnitude of the rotation angle and angle is measured in radians.

The vector of 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. Finite angular displacements are not vectors, since they do not add up according to the parallelogram rule. Infinitesimal angular displacements are vectors.

Vectors whose directions are related to the gimlet rule are called axial(from English axis- axis) in contrast 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 operation of reflection in a mirror (inversion or, what is the same, transition from a right-handed to a left-handed 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 pseudo-vectors and true vectors does not manifest itself in any way and they can and should be treated 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 rotation speed.

The basic unit of measurement for angular velocity is rad/s. In printed publications, for reasons that have nothing to do with physics, they often write 1/s or s -1, which, strictly speaking, is not true. Angle is a dimensionless quantity, but its units of measurement are different (degrees, points, grads...) and they must be indicated, at least to avoid misunderstandings.

Video 2.6. The stroboscopic effect and its use for remote measurement of angular velocity.

Angular velocity, like the vector to which it is proportional, is an axial vector. When rotating around motionless axis, the angular velocity does not change its direction. With uniform rotation, its magnitude also remains constant, so the vector . In the case of sufficient constancy in time of the angular velocity, it is convenient to characterize the rotation by its period T :

Rotation period- this is the time during 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

Moreover, in technical applications (primarily all kinds of engines), it is customary to take not a second, but a minute as a unit of time. That is, the angular speed 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 is an axial vector defined as the time derivative of angular velocity

When rotating about a fixed axis, or 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 increasing angular velocity || the angular acceleration coincides with it in direction; when 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 rotation axis; in the general case (rotation around a point), the rotation axis itself rotates and then the above is incorrect.

Relationship between angular and linear velocities and accelerations. Each of the points of the rotating body moves with a certain linear speed, directed tangentially to the corresponding circle (see Fig. 19). Let the material point rotate around an axis 00" along a circle with radius R. In a short period of time, it will travel a path corresponding to the angle of rotation. Then

Moving to the limit, we obtain an expression for the modulus of the linear velocity of a point of a rotating body.

We remind you here R- the 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 angular and linear speed, we obtain

The normal acceleration of points on a rotating rigid body is often called centripetal acceleration.

Differentiating the expression for with respect to time, we find

where is the tangential acceleration of a point moving in a circle with radius R.

Thus, both tangential and normal accelerations increase linearly with increasing radius R- distance from the axis of rotation. The total acceleration also depends linearly on R :

Example. Let's 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 = 24 hours = 24x60x60 = 86,400 s. From here we find the angular velocity of rotation

Average radius of the Earth

The distance to the axis of rotation at latitude is equal to

From here we find the linear speed

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 centripetal acceleration at the equator to the acceleration of free fall is equal to

Nevertheless, as we will see later, the effects of the Earth's rotation are quite observable.

Relationship between linear and angular velocity vectors. The relationships between angular and linear speed obtained above are written for the modules of vectors and . To write these relations in vector form, we use the concept of a vector product.

Let 0z- axis of rotation of an absolutely rigid body (Fig. 2.28).

Rice. 2.28. Relationship between linear and angular velocity vectors

Dot A rotates in a circle with 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. Then

and since

then by definition of the vector product, for all points of the body

Here is the radius vector of a point of the body, starting at point O, lying in an arbitrary fixed location, necessarily on the axis of rotation

But in other way

The first term is equal to zero, since the vector product of collinear vectors is equal to zero. Hence,

where is the vector R is perpendicular to the axis of rotation and directed away from it, and its module 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. Towards the determination of the instantaneous axis of rotation

Normal (centripetal) acceleration can also be written in vector form:

and the “–” sign indicates that it is directed towards the axis of rotation. Differentiating the relationship for linear and angular velocity with respect to time, we find the expression for total acceleration

The first term is directed tangent to the trajectory of a point on a rotating body and its module is equal to , since

Comparing with the expression for tangential acceleration, we come to the conclusion that this is the tangential acceleration vector

Therefore, the second term represents the normal acceleration of the same point:

Indeed, it is directed along the radius R to the axis of rotation and its module is equal to

Therefore, this relationship 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, paragraph 7): the rather difficult to understand question of 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 – kinematics of a basketball throw (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);

Elementary rotation angle, angular velocity

Figure 9. Elementary rotation angle ()

Elementary (infinitesimal) rotations are considered as vectors. The magnitude of the vector is equal to the angle of rotation, and its direction coincides with the direction of translational movement of the tip of the screw, the head of which rotates in the direction of the point's movement along the circle, i.e., it obeys the right-hand screw rule.

Angular velocity

The vector is directed along the axis of rotation according to the rule of the right screw, i.e. the same as the vector (see Figure 10).

Figure 10.

Figure 11

A vector quantity determined by the first derivative of the angle of rotation of a body with respect to time.

Communication between linear and angular velocity modules

Figure 12

Relationship between linear and angular velocity vectors

The position of the point under consideration is specified by the radius vector (drawn from the origin 0 lying on the rotation axis). The cross 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 points of application: they can be plotted from any point on the rotation axis.

Uniform motion of a material point around a circle

Uniform motion along a circle is a motion in which a material point (body) passes an arc of a circle equal in length in equal intervals of time.

Angular velocity

: (-- angle of rotation).

The period of rotation T is the time during which a material point makes one complete revolution around a circle, i.e., rotates through an angle.

Since it corresponds to the period of time, then.

Rotation frequency is the number of full revolutions made by a material point during its uniform motion around a circle, per unit time.

Figure 13

A characteristic feature of uniform circular motion

Uniform circular motion is a special case of curvilinear motion. Circular motion with a speed constant in absolute value () is accelerated. This is due to the fact that with a constant modulus the direction of velocity changes all the time.

Acceleration of a material point moving uniformly in a circle

The tangential component of acceleration when a point moves uniformly around a circle is zero.

The normal component of acceleration (centripetal acceleration) is directed radially 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 around a circle at any point is centripetal.

Angular acceleration. Relationship between linear and angular quantities

Angular acceleration is a vector quantity determined by the first derivative of angular velocity with respect to time.

Angular acceleration vector direction

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 movement is accelerated, the vector is codirectional to the vector, when it is slow, it is opposite to it. 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).