How to reverse the rotation of the exhaust fan
synonyms - spinning top
A Gyroscope is a (rigid) body that rotates around an axis. Otherwise it can move freely, but it can also be held more or less at one end of the axis or both. In physics, gyroscopes do not necessarily have to be rotationally symmetrical.
As Gyroscope or Gyro (Greek.) are also used to describe measuring instruments that perform tasks similar to those of a gyroscope, even if they do not contain a rotating top. Examples are laser gyroscopes, fiber optic gyroscopes or vibration gyroscopes.
Spinning tops as children's toys
Spinning tops also serve as children's toys, which are rotated around a vertically held axis on a base and then roughly maintain the axial direction for a while, with the top moving around on the base.
The spinning top is one of the oldest toys that can be found at archaeological sites. In addition to being toys, spinning tops have historically been used for games of chance and fortune-telling. In technology, gyroscopes are used, for example, for stabilization and navigation, since the direction of the angular momentum remains the same when there is no force acting on it. The reason for this is the conservation of angular momentum. If the axis of rotation coincides with the direction of the angular momentum, this does not change either.
In the physical sense, a rotating rigid body is a top. From a mathematical point of view, tops are solid rotating bodies.
The behavior of the toy top is based on the gyroscopic effect. Usually the top wobbles first until the interaction of the top and bottom forces the top into an upright position. After a longer period of upright rotation, the angular momentum and thus the gyroscopic effect gradually decrease. This leads to an increasing precession, which ultimately ends up dragging in a strong tendency to tip.
Equations of motion
The movement of the top is created by the conservation of angular momentum. The angular momentum is a product of the inertia tensor and the speed of rotation of the top. Like the mass, the moment of inertia indicates how “difficult” it is to change the rotational movement. If one calculates the change in the angular momentum L by deriving it with respect to time, we get:
That is, the change in angular momentum is proportional to the change in angular velocity and equal to the torque
If you look at the individual components of the vectors, you get a system of differential equations that are named after their discoverer, Leonhard Euler. See also the main article Euler's equations. The sign I. stands in the formula for the inertia tensor, a "collection" of the moments of inertia when rotating around different axes in a matrix. There are two axes with respect to which the body's moment of inertia is maximal and minimal, a rotation around these axes is stable. These axes are always perpendicular to each other and together with a third axis, which is again perpendicular to both, form the main axes of inertia of the body. In a coordinate system spanned by the main axes of inertia, the inertia tensor is diagonal, which greatly simplifies calculations. The moments of inertia belonging to the main axes of inertia are therefore the eigenvalues of the inertia tensor, they are called main moments of inertia.
Classification and characteristics
- At a symmetrical top at least two main moments of inertia are equal. This group includes rotationally symmetrical toy tops or cuboids with two sides of equal length. One of its main axes coincides with the figure axis, perpendicular to it, through its center of gravity, it has an infinite number of equal equatorial main axes. The ellipsoid of inertia of a symmetrical top is always rotationally symmetrical. If there are three different main moments of inertia, one speaks of one asymmetrical top.
- At a spherical top all three main moments of inertia are the same. Examples are cubes and balls. A sphere also has an infinite number of identical main axes.
- A top is force-freeif no external torques act on it, i.e. all external forces outside the center of gravity must cancel each other out. For this purpose, the support point of the gyroscope must be, for example, just below the center of gravity of the gyro, as a cardanic suspension (center of gravity in the center of the gimbal frame), or as a Kleinscher gyroscope (support directly in the center of gravity of a gyro body hollowed out below). The equations of motion for a force-free symmetrical top are easier to solve than for a heavy top. Its movement generally consists of actual rotation and nutation. In nutation, the angular momentum and the momentary axis of rotation have different directions. This happens when the moment of inertia is direction-dependent (the moments of inertia are not all the same) and the rotation does not take place around the axis of a main moment of inertia. If the top is not symmetrical, it can lead to more complicated movements.
- The opposite is that heavy tops: For example, if a toy top is tilted, gravity tries to tip it over. Since the resulting torque is perpendicular to the angular momentum, the angular momentum only changes its direction. The top rotates around an axis that goes straight up through its point of support. This rotation is called precession. The equations of motion can only be solved approximately here (especially for fast gyroscopes).
Examples of gyroscopes
The toy is available in numerous designs and variants:
Technical applications are:
- gyroscope (Measurement of the axis) and Gyrometer (Measurement of the rotation speed)
- Gyro as Energy storage (see also rotating mass storage).
- Inertia wheels for attitude control of missiles
- The Levitron is a spinning magnet that hovers in the air above an oppositely polarized, ring-shaped magnetic field. Its gyratory motion prevents it from tipping over and being attracted to the magnetic ring. Wikipedia video
- The gyro compass aligns itself towards the North Pole due to the rotation of the earth.
At the beginning of the 20th century, gyroscopes - e.g. B. the Musil color spinning top - used to study human color perception.
In the 2010 film Inception, the protagonist Cobb uses a top as a “totem”.
- Felix Klein, Arnold Sommerfeld: About the theory of the gyro. Stuttgart: Teubner, 1965
- Heinz Parkus: Mechanics of solid bodies. 2nd Edition. Springer textbook, Vienna-Berlin 1966
- H. Westphal: Physics. A textbook. 24th edition, Springer-Verlag Berlin-Heidelberg 1963, chap. I and III
- Renée Holler: Spinning top. Hugendubel, Munich 1989; ISBN 3-88034-401-9
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