# Rolling Cone

A rigid cone rolls on a very rough, horizontal surface (rough enough to prevent it from sliding) and its center of mass follows a circular trajectory.

Suppose the angular velocity of the center of mass of the cone is $$\Omega$$, and the angular velocity at which the cone rotates as seen by someone in the reference frame of the center of mass is $$\omega$$.

What is the ratio of $$\Omega$$ to $$\omega$$?

Note: The angle between the cone's base and slant height is $$\beta$$

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