Moment of Inertia Lookup & Calculator mechanism

Model scope

The listed equations are correct textbook results under the stated idealizations. The instrument should expose the standard axis for every shape, explicitly label the sphere as solid or hollow, define accepted units and conversions, and validate the parallel-axis geometry before it can be approved.

• The formulas are standard only for specific idealized bodies and axes: the disk formula is for a uniform thin disk about its central axis perpendicular to its face; the rod formula is for a uniform slender rod about an axis through its center perpendicular to its length; the sphere formula is for a uniform solid sphere about any diameter; and the hoop formula is for a thin hoop about its central axis.
• The shape specification does not distinguish a solid sphere from a hollow spherical shell, nor a thin disk from a solid cylinder, so the shape-to-formula mapping is ambiguous.
• The parallel-axis theorem requires the two axes to be parallel and separated by perpendicular distance d; this constraint should be stated and enforced.
• The input specification does not define the reference axis for each shape or whether dimensions describe thin bodies, solid bodies, or shells.

Units and conventions

• M is in kilograms, all lengths including R, L, and d are in metres, and the output I is in kg·m².
• The disk is uniform and thin, with its axis perpendicular to the disk through its center.
• The rod is uniform and slender, with the reference axis perpendicular to the rod through its center.
• The sphere is a uniform solid sphere, with the reference axis through its center.
• The hoop is a uniform thin hoop, with the reference axis through its center perpendicular to its plane.
• For the parallel-axis calculation, d is the perpendicular separation between parallel axes.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "shape": "uniform thin disk",
    "M_kg": 2.4,
    "R_m": 0.5,
    "d_m": 0.3,
    "axis": "parallel to central symmetry axis, offset by d"
  },
  "expected_outputs": {
    "I_cm_kg_m2": 0.3,
    "I_about_offset_axis_kg_m2": 0.516
  },
  "working": "I_cm = 1/2 MR² = 1/2 × 2.4 × (0.5)² = 1.2 × 0.25 = 0.300 kg·m². Md² = 2.4 × (0.3)² = 2.4 × 0.09 = 0.216 kg·m². I = I_cm + Md² = 0.300 + 0.216 = 0.516 kg·m²."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The extreme-magnitude disk input produces Infinity in ordinary IEEE-754 JavaScript arithmetic. Without finite-value validation or an overflow warning, the calculator can return a nonphysical non-finite result.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json