# Ring system

The figure shows a system consisting of (i) a ring of outer radius $$3R$$ rolling clockwise without slipping on a horizontal surface with angular speed $$\omega$$ and (ii) an inner disc of radius $$2R$$ rotating anti-clockwise with angular speed $$\omega/2$$. The ring and disc are separated by frictionless ball bearings. The system is in the $$x-z$$ plane. The point $$P$$ on the inner disc is at a distance $$R$$ from the origin, where $$OP$$ makes an angle of $$30°$$ with the horizontal. Then with respect to the horizontal surface,

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