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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