Figure axis

In gyro theory , the figure axis of a symmetrical top is that of its main axes of inertia , the main moment of inertia of which is different from the other two of its three main moments of inertia. The figure axis is therefore often referred to as the third main axis , 3-axis , ê ₃ or similar. It forms the axis of symmetry of the symmetrical top because it is symmetrical with respect to all points on it. The main moment of inertia belonging to the figure axis is also called the axial main moment of inertia.

Because a symmetric top is not necessarily the geometric sense symmetrically must be, must also figure axis is not the axis of symmetry in the geometric sense be. Especially with inhomogeneous mass distribution , the figure axis cannot be easily recognized, such as the 3-axis in the Kovalevskaya gyroscope .

On the other hand, in the case of homogeneous rigid bodies , their symmetry can often be used to infer a possible figure axis, with respect to which they give off symmetrical tops. Homogeneous bodies of revolution , in which the axis of rotation is the axis of the figure, are an important special case of symmetrical tops. In addition, the axis around which a body has n-fold rotational symmetry is also the figure axis from n = 3, e.g. B. the axis of symmetry of straight prisms or regular pyramids , the base of which is a regular polygon (see ellipsoids of inertia of special bodies ).

All axes perpendicular to the figure axis that intersect the figure axis have the same moment of inertia and lie in the equatorial plane perpendicular to the figure axis . This plane has a circular section with the ellipsoid of inertia , which is therefore rotationally symmetrical , like all other ellipsoids connected to the top ( energy ellipsoid , MacCullagh ellipsoid , mass ellipsoid ).

Web links

Commons : Gyroscope  - collection of images, videos and audio files

Commons : Solid of revolution  - collection of images, videos and audio files

literature

1. ↑ K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 20 ( limited preview in Google Book Search [accessed February 20, 2018]).
2. ↑ R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, DNB 451641280 , p.   31 ( archive.org - "swing" means angular momentum, "torsional shock" torque and "torsional balance" rotational energy).
3. ↑ F. Klein , A. Sommerfeld : About the theory of the gyro . Teubner, Leipzig 1897, p.  1 ( archive.org [accessed December 8, 2019]).
4. ^ Figure axis - Lexicon of Physics. Spektrum Verlag, accessed January 30, 2019 .