Spherical top

The spherical top or spherical gyroscope is in the centrifugal theory a roundabout with three equal principal moments of inertia . As a result, all axes are the main axis of inertia , and the moments of inertia  Θ about all axes through the reference point are the same.

The spherical top is not suitable for practical applications, see # Stability considerations . Its importance lies in the fact that for every Lagrange top there is a homologous spherical top whose axis of rotation moves like the figure axis of the Lagrange top ; this simplifies the analytical treatment of the Lagrange gyro.

The term spherical top was coined by Felix Klein and Arnold Sommerfeld .

Realizations

Although the ellipsoid of inertia is a sphere, the outer shape of the top does not have to be spherical. For example, all homogeneous , solid or thin-walled Platonic solids are spherical tops with regard to their center of mass , see list of inertia tensors .

A spherical top cannot be easily recognized, especially with inhomogeneous mass distribution . For example, three mass points can always be distributed on three mutually perpendicular axes by a support point in such a way that the rigid body consisting of the mass points is a spherical top with respect to the support point. The product of the mass and the square of the distance from the support point only has to be the same for all three mass points. This example shows that the center of mass does not have to be in the support point.

If the ellipsoid of inertia with respect to the center of mass flattened rotationally symmetrical ( lens -shaped), then there is after the Steiner's theorem to the figure axis on both sides of the center of mass a point with respect to which the symmetric top is a spherical top.

Stability considerations

Each support axis is a stable permanent axis only with the ideal spherical top . With the force-free Euler gyro , rotations around the two main axes of inertia with the largest or smallest main moment of inertia are stable and those around those with the mean main moment of inertia are unstable.

In reality, all main moments of inertia can never be exactly the same, be it due to inhomogeneities, thermal expansions or deformations due to acceleration ; the stability of the rotation cannot therefore be ensured. Therefore, the spherical top is unsuitable for practical use.

Analytical description

The inertia tensor of the spherical top is proportional to the unit tensor 1 :

${\displaystyle \mathbf {\Theta } =\Theta \mathbf {1} }$

As a result, all axes are the main axis of inertia and the moments of inertia about all axes through the reference point are equal (Θ). The spherical top is isotropic to rotary movements because the angular momentum and angular velocity are always parallel in the same direction:

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

see also ellipsoid of inertia . With the spherical top, the centrifugal forces in the body are always in mechanical equilibrium , because the top effect consists exclusively of the moment of the Euler forces : ${\displaystyle {\vec {K}}}$

${\displaystyle {\vec {M}}={\dot {\vec {L}}}=-{\vec {K}}:=(\mathbf {\Theta } \cdot {\vec {\omega }}{\dot {)}}=\Theta {\dot {\vec {\omega }}}}$

The components of the vector equation are Euler's gyro equations , which are particularly simple for the spherical top:

${\displaystyle M_{i}=\Theta {\dot {\omega }}_{i}\,,\qquad i=1,2,3}$

A comparison with Newton's second law for a translational movement shows that the spherical top is the exact analog of the mass point for rotational movements . ${\displaystyle {\vec {F}}=m{\ddot {\vec {x}}}}$

For every self-rotation ν there is only one regular precession with precession velocity μ, which according to

Θμν = mgs

the faster it occurs, the greater the support point moment from weight mg and center of gravity distance s.

Individual evidence

1. ↑ Magnus (1971), p. 19.
2. ↑ ^{a b} Grammel (1920), p. 31.
3. ↑ ^{a b} Grammel (1920), p. 43.
4. ^ Grammel (1950), p. 57.
5. ↑ Magnus (1971), p. 83.
6. ↑ Grammel (1920), p. 92.

literature

• K. Magnus : Kreisel: Theory and Applications . Springer, 1971, ISBN 978-3-642-52163-8 , pp. 19, 83 ( limited preview in Google Book Search [accessed November 9, 2019]).
• R. Grammel : The top . Its theory and its applications. Vieweg Verlag, Braunschweig 1920, DNB  451641280 , p. 31, 43 ( archive.org - "swing" means angular momentum and "torsional balance" means rotational energy).
• R. Grammel : The top . Theory of the gyro. 2. revised Edition volume 1 . Springer, Berlin, Göttingen, Heidelberg 1950, DNB  451641299 .