Adoption of the heavy symmetric top branch: axisymmetric rigid material, a fixed pivot material point, a uniform gravitational field and the balance of angular momentum about the pivot

Adopt as physical assumptions of the framework the following heavy symmetric top branch of the rigid-material model, in the setting of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope.

(i) Setting. A model of the branch carries the data of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: a reference mass distribution $\nu$ with total mass $m>0$ and centred reference coordinates, a time window $I_0$, an inertial frame with adapted coordinates $(t,x)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames), whose origin is chosen at the pivot point of clause (iii), and a motion $x(t,\rho )=c(t)+R(t)\rho$ with configuration path $(c,R)$ of class $C^2$ (Configuration, configuration space and orientation of a rigid-material model in an inertial frame). Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only through the integrals of the model's inertial quantities. Fix once and for all an orthonormal basis $(e_1,e_2,e_3)$ of the reference coordinates.

(ii) Axisymmetry and the declared parameters. The body inertia matrix of the model (Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model clause (a)) is $I=A_c(e_1{e}_1^{\mathsf{T}}+e_2{e}_2^{\mathsf{T}})+C_c{e}_3{e}_3^{\mathsf{T}},A_c>0,C_c>0,$ so that $e_3$ is the symmetry axis; the numbers $A_c$, $C_c$ are declared model parameters of dimension $\mu {\mathrm{\ell }}^2$, read in kilogram metres squared, and are not measured within the framework. The branch also declares a pivot distance $d>0$ of dimension $\mathrm{\ell }$ and a field strength $g>0$ of dimension $\mathrm{\ell }{\tau }^{-2}$, read in metres per second squared (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

(iii) The pivot constraint. The material point of reference coordinate ${\rho }_p:=-d{e}_3$ is the pivot material point; it lies on the symmetry axis at distance $d$ from the centre of mass. The motions of the branch are those satisfying $c(t)+R(t){\rho }_p=0(t\in I_0),$ that is $c(t)=dR(t)e_3$: the pivot material point is held fixed at the origin of the inertial frame for the whole time window, and the centre of mass lies on the sphere of radius $d$ about the origin. The constraint is maintained by a mechanism (a bearing, a hinge or a fine support) whose force system is ideal in the sense adopted for frictionless hinges in Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits clause (iii) together with the virtual-work law of its clause (i): the mechanism exerts on the body a force system equivalent to a single force applied at the pivot point, so that its total torque about the origin vanishes. The holonomic-constraint vocabulary of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction — constrained curve families, reaction fields and the declaration that the constraint is maintained by an additional force contribution — is the framework's account for finitely many particle labels; the present model is a continuum, and this clause adopts the analogous idealisation for it. No friction, damping or driving torque at the pivot is asserted; such torques are excluded and would have to be declared as part of a torque datum with their own model.

(iv) The uniform gravitational field. The model declares a uniform gravitational acceleration $G:=-g{e}_3$ of dimension $\mathrm{\ell }{\tau }^{-2}$: every material point of the body is subject to the external force $Gd\nu$, that is, the force per unit mass is the constant vector $-g{e}_3$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The cross product in ${\mathbb{R}}^3$ for the spatial vectors). The field is a model idealisation: it asserts neither a particular direction in any absolute sense (the direction $e_3$ is a declared frame direction) nor a measured value of $g$, and it is not asserted to represent the Earth's or any planet's field. The resultant of this force system and its total torque about a point are computed in the reduction theorem of this page; in particular the resultant is the total weight $-mg{e}_3$ applied at the centre of mass.

(v) The adopted dynamics. Along every motion of the branch adopt the balance of angular momentum about the pivot: with $J_p(t):=\int (x(t,\rho )-0)\times {\mathrm{\partial }}_{t}x(t,\rho )d\nu (\rho )$ the angular momentum of the model about the origin, and with ${\tau }_p(t)$ the total external torque about the origin — the sum of the gravitational torque and of the pivot reaction torque, the latter vanishing by clause (iii) — adopt $J_p^{\mathrm{\prime }}(t)={\tau }_p(t)(t\in I_0)\mathrm{.}$ This is an adopted dynamical law of the branch, analogous to but distinct from the balance of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (iii) about the centre of mass; it acquires content only together with the kinematic constraint of clause (iii) and the external force system of clause (iv). No equation of motion for the pivot reaction is asserted: the reaction is whatever force the ideal mechanism supplies, and its torque about the origin is zero by clause (iii).

(vi) Scope and exclusions. The branch excludes non-axisymmetric bodies ($A_c$ and $C_c$ are the two transverse and the axial moment and no other inertia data are used), bodies whose mass distribution fails to produce this inertia matrix, non-uniform gravitational fields, friction and driving torques at the pivot, rolling or sliding contacts, impacts, deformable bodies and variable mass, and every relativistic or quantum description. It asserts no force law for the pivot reaction, no conservation law and no measured value of $A_c$, $C_c$, $m$, $d$ or $g$. Whether the branch is adequate for a given apparatus — a top on a bearing, a gyroscope in a gimbal, a spinning shell — is decided by comparing the model's conditional predictions with reported observations of that apparatus under its stated regime, and agreement is evidence about the combined model and apparatus, not a proof of the idealisation.

(vii) Relation to the other balances. The balance of clause (v) is not claimed to be a consequence of the centre-of-mass balance of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (iii), nor conversely; the two are separate adopted laws about different points, and neither is deduced from the framework's particle balances. A model may use both simultaneously, and the reduction theorem of this page uses only clause (v) together with clauses (ii)-(iv).