- #Calculate moment of inertia from precession period series
- #Calculate moment of inertia from precession period free
We only investigate the pure precession (steady nutation angle) of symmetric tops exposed to a torque (“heavy” tops), where dissipation is neglected.ĭepending on the initial conditions and parameters, torque-induced (gyroscopic) precession can show dynamics with various complexity.
Presence of dissipation can lead to exotic dynamics like the inversion of the tippe top. It is important to mention that asymmetric tops will typically show chaotic behavior. Here the smooth change of the precession angle of the spinning (and symmetry) axis, that is, the precession, can be accompanied by nutation, a regular “nodding" of the nutation angle. Tops with an axis having a cylindrical or at least a third order discrete rotational symmetry, spinning around this axis and exposed to a torque follow a regular trajectory (see Supplementary Section II.E). Torque-induced precession ( Figure 1) takes place if a spinning top is exposed to a couple of forces (also referred to as moment or torque). This phenomenon has a major importance in planetary physics (see Supplementary Section II.D).
#Calculate moment of inertia from precession period free
Note that different points of a freely precessing body are subjected to acceleration which varies in space and time, therefore mechanical tensions will arise within precessing free tops. The mainstream physics literature refers to the rotation around the precessing symmetry axis as spinning while to the rotation around the steady axis defined by the angular momentum as precession. Torque-free precession occurs when no external couple of forces is acting on the top its symmetry axis will sweep around the mantle of a cone with opening angle ϑ, hereinafter nutation angle, while the total angular momentum L of the system, which determines the axis of the cone, is conserved (see Supplementary Section II.B). Conversely, in different fields of science and engineering, the very same event may be called precession, wobbling or nutation. This term is used to refer to several phenomena having different physical origins. One of the characteristic phenomena exhibited by spinning tops is precession. The symmetry axis coincides with the principal axle having the smallest or highest moment of inertia, while the two further principal moments of inertia associated with the perpendicular principal axes ( e ⊥) have the same magnitude. Spinning symmetric tops are prolonged or oblated bodies with a cylindrical or discrete rotational symmetry, whirling around their symmetry axis e n, having one point fixed, or at least restricted to move on a surface. More importantly, the square-shaped tubing facilitates an intuitive and elementary quantification of the external couple of forces (in the inertial, laboratory frame) or the couple of inertial forces (in two types of rotating frames) which counteract the gravitational torque and prevent the top from tumbling down. This flow generates an angular momentum enabling the comparison of the novel configuration to the classical heavy tops which are symmetric, spinning objects. Our model relies on a square-shaped rigid tube with a heavy fluid or chain in its interior, which is allowed to circulate smoothly and without resistance in the closed loop determined by the rigid tube.
After a short recapitulation of the underlaying concepts, we introduce and quantify the square wheel model. Here we provide an intuitive and simple explanation of the torque-induced precession. This similarity is referred to as the Kirchhoff kinetic analogy (see Supplementary Section III). Moreover, the mathematical frameworks describing the motion of spinning tops and that for the equilibrium shape of idealized bent and twisted rods are analogous.
#Calculate moment of inertia from precession period series
Deeper understanding of torque-induced precession would be particularly important for teaching a series of its applications, like the gyroscope, electron spin resonance (ESR) and nuclear magnetic resonance (NMR). Such a description would be of pivotal importance, since forces, in contrast to conservation laws, are closer to our everyday experience. However, none of these reveals the interplay of the forces responsible for these counterintuitive behaviors. Some mainstream textbooks provide simplified experimental configurations (e.g., dumbbells) to offer a more comprehensible explanation. Although the mathematical treatment of these phenomena is known, their intuitive understanding is not trivial. Interpreting the wobble of a torque-free spinning plate, the rise of a tippe top, as well as the precession of a fast spinning wheel suspended at the extremity of its axle and exposed to gravitational or electromagnetic torque, inspired the brightest researchers during the past centuries.
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