Critical Angle of Precession of a re-assembled top

Classical Mechanics Level 5

Jatin and Anish decided to do an experiment. They bought thousands of right conical tops (of the same base radius and uniform mass densities, but with different heights and semi vertical angles) from a store.

Anish cut the tops parallel to their base into infinite thin discs. Jatin began to reassemble these discs in such a way that the centre of all the discs lie on the slant height of the original top and glued them to make a new rigid structure. They both tried spinning all such tops on the ground about an imaginary axis (straight line joining the centre of the discs or slant height of the original tops) with a constant angular velocity. They observed that a particular top had minimum angular velocity of precession.

The tops precesses in a circle parallel to the base about the axis of the original top with the tip of the top not moving. The semi-vertical angle of this top is \(\theta \). Find \( \lfloor \theta \rfloor \)

  • Assume that the new top has a narrow tapered tip.

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