Rotation (physics)

Rotation , also rotational movement , rotation , rotational movement or gyral movement , is in classical physics a movement of a body around an axis of rotation . The term is used both for a single rotation through a certain angle and for a continuous movement with a certain angular velocity . The axis of rotation can, but does not have to go through the center of mass of the body. A distinction must be made between the rotation discussed here and the circular movement , in which a body revolves around a circle without changing its orientation and the points of the body all move on circles of the same size and offset from one another. The two forms of movement only coincide with the movement of a point mass .

Rotating rings

In physics , the term belongs to the sub-areas of mechanics and kinematics . In astronomy it occurs, among other things, in connection with changes in the rotation of the earth and the movements of other objects up to galaxies . Everyday applications and examples that are often used to clearly explain the phenomena associated with rotation are the top and the carousel .

During the rotation, all points of the axis of rotation stay in place ( fixed points ), while all other points move around it at a fixed distance from the axis on a circle perpendicular to the axis at the same angle or at the same angular speed. Therefore, the lengths of the connecting lines for two points on the object and the angles between them also remain the same.

Rotation parameters

A finite rotation is clearly characterized by the specification of a fixed point and a vector which lies parallel to the axis of rotation and which specifies the angle of rotation through its length. In the case of a progressive rotational movement, this vector is the angular velocity. The rotation around a certain point of a fixed reference system can therefore be described by the three components of the associated vector. Another possibility is to specify the three Euler angles .

Comparison with translational motion

The following table compares the characteristic quantities and the equations of motion for a translational movement with those for a rotational movement . Because of the analogies, every sentence about translation can be converted into a sentence about rotation by replacing the corresponding quantities.

Translational movement	Rotational movement
Position vector : ${\ displaystyle {\ vec {r}}}$	Rotation angle or rotation matrix : ${\ displaystyle \ varphi}$ ${\ displaystyle A}$
Speed : ⁽¹⁾ ${\ displaystyle {\ vec {v}} = {\ dot {\ vec {r}}}}$	Angular velocity : ⁽³⁾ ${\ displaystyle {\ vec {\ omega}} = {\ dot {\ psi}} {\ vec {\ mathbf {u}}} _ {1} + {\ dot {\ theta}} {\ vec {\ mathbf {u}}} _ {2} + {\ dot {\ phi}} {\ vec {\ mathbf {u}}} _ {3}}$
Acceleration : ${\ displaystyle {\ vec {a}} = {\ dot {\ vec {v}}} = {\ ddot {\ vec {r}}}}$	Angular acceleration : ${\ displaystyle {\ vec {\ alpha}} = {\ dot {\ vec {\ omega}}}}$
Mass : ( scalar ) ${\ displaystyle \ m}$	Inertial tensor : ( second order tensor , in special cases scalar ) ⁽²⁾ ${\ displaystyle \ mathbf {\ Theta}}$ ${\ displaystyle I}$
Power : ${\ displaystyle {\ vec {F}} = m \, {\ vec {a}}}$	Torque : ${\ displaystyle {\ vec {M}} = {\ vec {r}} \ times {\ vec {F}}}$
Impulse : ${\ displaystyle {\ vec {p}} = m \, {\ vec {v}}}$	Angular momentum ⁽²⁾ : ${\ displaystyle {\ vec {L}} = \ mathbf {\ Theta} {\ vec {\ omega}}}$
Drive (linear) / impulse : ${\ displaystyle \ Delta {\ vec {p}} = \ int {\ vec {F}} \ mathrm {d} t}$	Drive (rotation) / rotary shock : ${\ displaystyle \ Delta {\ vec {L}} = \ int {\ vec {M}} \ mathrm {dt}}$
Kinetic energy : ${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} m \, {\ vec {v}} ^ {2} \ equiv {\ frac {1} {2}} {\ vec {v}} \ cdot {\ vec {p}}}$	Rotational energy : ${\ displaystyle E _ {\ mathrm {red}} = {\ frac {1} {2}} {\ vec {\ omega}} \ cdot \ mathbf {\ Theta} {\ vec {\ omega}}}$
Work : ${\ displaystyle W = \ int {\ vec {F}} \ cdot \ mathrm {d} {\ vec {s}} = \ int {\ vec {F}} \ cdot {\ vec {v}} \ \ mathrm {d} t}$	Work with rotary motion (turning work): ${\ displaystyle W = \ int {\ vec {M}} \ cdot \ mathrm {d} {\ vec {\ varphi}} = \ int {\ vec {M}} \ cdot {\ vec {\ omega}} \ \ mathrm {d} t}$
Performance : ${\ displaystyle P = {\ dot {W}} = {\ vec {F}} \ cdot {\ frac {\ mathrm {d} {\ vec {s}}} {\ mathrm {d} t}} = { \ vec {F}} \ cdot {\ vec {v}}}$	Rotational power (rotary power): ${\ displaystyle P = {\ dot {W}} = {\ vec {M}} \ cdot {\ vec {\ omega}}}$
Equations of motion
General: Force is linked to a change in momentum (momentum set ): ${\ displaystyle {\ dot {\ vec {p}}} = {\ vec {F}}}$	General: Torque is linked to a change in angular momentum (principle of twist ): ${\ displaystyle {\ dot {\ vec {L}}} = {\ vec {M}}}$
In the case of constant mass ( Newton's second axiom ): ${\ displaystyle m}$ ${\ displaystyle m \, {\ vec {a}} = {\ vec {F}}}$	In the case of constant moment of inertia : ⁽²⁾ ${\ displaystyle I}$ ${\ displaystyle I {\ vec {\ alpha}} = {\ vec {M}}}$

⁽¹⁾The point above a quantity means that this is a change in time ( derivative ). The point between two vectors means the scalar product.

{\ displaystyle {\ tfrac {\ mathrm {d}} {\ mathrm {d} t}}}

⁽²⁾In general, and not in the same direction (a rotating body “wobbles” or out of balance ), so the moment of inertia is generally not constant. The equivalent to the mass of the translational motion is therefore a 2nd order tensor - the inertia tensor . A constant moment of inertia occurs precisely when the body rotates around one of its main axes of inertia .

{\ displaystyle {\ vec {\ omega}}}

{\ displaystyle {\ vec {L}} = \ mathbf {\ Theta} {\ vec {\ omega}}}

⁽³⁾expressed in the derivatives of the Euler angles . Axes of rotation (unit vectors).

{\ displaystyle {\ vec {\ mathbf {u}}} _ {i}}

Rigid Body Rotation

In order to clearly describe the orientation of a rigid body in space, three scalar (angle) specifications are necessary. Two of them only indicate the direction of its axis of rotation, the third how far the body has been rotated around this axis.

The rotational movement of a rigid body has at least two stable rotational axes (moment-free axis) through the center of mass with free rotational movement: the main axis of inertia with the smallest or the largest moment of inertia is stable. If all three main moments of inertia are different, the rotation around the main axis of inertia with the mean main moment of inertia is in an unstable state, because the smallest disturbances lead to strong staggering movements (see e.g. Dschanibekow effect ).

If you try to rotate a rigid body around another axis than one of its main axes of inertia, moments arise that want to make it change its current axis of rotation. If the axis is not held in place by bearings that exert torques on it, the body will wobble.

In the case of a force-free free rotation, the angular momentum is retained, which is generally not collinear with the angular velocity. Thus, the axis of rotation changes continuously, which is colloquially known as "staggering" or "egg", technically and scientifically - depending on the type of axis movement - as wobbling of the axis of rotation or as secondary axis error , precession or nutation .

Regardless of other influences, every top is quasi-integrable, with either very little or a lot of energy (compared to the potential energy difference between bottom and top dead center) in the rotation. The most chaotic movements in the non-integrable types occur, regardless of the shape, when the kinetic energy of the top is just sufficient to reach top dead center. The more precise treatment is carried out with the help of Euler's gyroscopic equations , for more detailed explanations see the main article or there.

In the following special cases, Euler's gyroscopic equations can be solved analytically. The trajectories of the system, in particular the angular velocities, have a periodic course here.

Case of Euler

Euler's case describes a top that is suspended exactly in its center of gravity . Regardless of the shape of the top, the case is integrable , since there are more conserved quantities than degrees of freedom : the energy and the angular momentum with respect to all three spatial directions in the inertial system.

Is the mass of the rotating body around the axis of rotation symmetrically distributed, so act on the axis of the springs from any rotational forces , since the inertia ( centrifugal force ) of each Massenteilchens by an equal and opposite is canceled; such an axis is called a free axis or principal axis of inertia. However, if the rotation does not take place around a free axis, then - even in the symmetrical body - moments of centrifugal forces arise that are in dynamic equilibrium with moments of the Euler forces , which are an expression of the movement of the axis of rotation.

The Euler gyro finds z. B. Technical application in gyroscopic compasses and gyroscopic control systems.

Fall of Lagrange

In the case of Lagrange , it is assumed that the moments of inertia correspond to two main axes. This is fulfilled , for example, by radially symmetrical bodies. In this case there are three physical conservation quantities: the energy, the total angular momentum and the angular momentum with respect to the z-axis (in the direction of the force field). Relative to the rotating body, the direction of the force field changes continuously, but the direction vector always has the same length: This defines a fourth, purely geometrical conservation quantity that occurs when describing the movement in the force field.

Since every mass particle rotating around a free axis tends to remain in its plane of rotation perpendicular to the axis, following inertia , the free axis itself must also show the tendency to maintain its direction in space and thus becomes a force that wants to bring it out of this direction , the greater the moment of inertia and the angular velocity of the rotating body , the greater the resistance . This is why a top that rotates sufficiently quickly does not fall over, even if its axis is crooked, just as wheels , coins, etc. do not fall over when they are rolled on their edge or "dance" around the vertical diameter .

The effect of the disturbing force on the top is expressed in that its axis deviates in a direction perpendicular to the direction of the disturbing force and describes the surface of a cone in slow motion without the axis changing its inclination to the horizontal plane. This movement is known as nutation .

The fall of Lagrange is realized by a typical toy spinning top when its touchdown point is fixed on the ground. The wheels of bicycles and motorcycles also behave like gyroscopes in a gravitational field and, in addition to guiding the vehicle, help to stabilize the vehicle by attempting to adjust the angular momentum to the moment of weight . See also: cycling .

Case of Kovalevskaya

The Kovalevskaya gyroscope , named after Sofja Kovalevskaya , has the same moments of inertia with regard to two of its main axes and exactly half as large with regard to the third main axis. The physical conservation quantities are the energy, the total angular momentum and a complex mathematical expression for which there is no generally understandable equivalent.

Case of Goryachew-Chaplygin

The case of Dmitri Nikanorowitsch Goryachev (Goryachev) and Tschaplygin (Chaplygin) is a modification of the Kovalevskaya case, which instead of half the third moment of inertia calls for a quarter as large. In this case, however, there is only a third physical conservation quantity if the angular momentum in the direction of the force field initially disappears. This angular momentum component is a conservation quantity and in this case therefore permanently zero.

Individual evidence

^ Hans Schmiedel, Johannes Süss: Physics - for technical professions . 16th edition, Büchner, Hamburg 1963, p. 74.
^ Theoretical investigation of the case of Goryachew-Chaplygin .

literature

Peter Brosche, Helmut Lenhardt: The pole movement from the observations of FW Bessel 1842–1844 . In: zfv , magazine for geodesy, geoinformation and land management , issue 6/2011, pp. 329–337, DVW e. V. (editor), Wißner-Verlag, Augsburg 2011, ISSN 1618-8950 , on earth rotation.

Web links

Wiktionary: Rotation - explanations of meanings, word origins, synonyms, translations

Wiktionary: rotate - explanations of meanings, word origins, synonyms, translations

[1] Hans Schmiedel, Johannes Süss: Physics - for technical professions . 16th edition, Büchner, Hamburg 1963, p. 74.

[2] Theoretical investigation of the case of Goryachew-Chaplygin .