The Moment of Inertia Tensor serves as a comprehensive description of an object's mass distribution, capturing how it resists rotation while accounting for the internal "cross-talk" or coupling that typically causes unbalanced objects to wobble. This complex resistance is visualized through the Inertia Ellipsoid, a 3D map where the distance from the center to the surface indicates rotational stiffness; specifically, its narrowest points signify maximum resistance, while its widest points identify the easiest axes to spin. Within this geometric map, the Principal Axes represent unique directions where this internal coupling disappears, allowing the object's angular momentum and rotation to align perfectly without tipping. Finally, the Parallel Axis Theorem demonstrates that this resistance is dynamic, as shifting the rotation point away from the center of mass deforms and shrinks the ellipsoid, thereby increasing overall resistance and reorienting the principal axes.
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%% Condensed Notes
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%% Proof and Derivation
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