The figure shows a system consisting of (i) a ring of outer radius \( 3 R \) rolling clockwise without slipping on a horizontal surface with angular speed \( \omega \) and (ii) an inner disc of radius
\( \mathrm{P} \) \( 2 R \) rotating anti-clockwise with angular speed \( \omega / 2 \). The ring
W 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 distance \( R \) from the origin, where \( O P \) makes an angle of \( 30^{\circ} \) with the horizontal. Then with respect to the horizontal surface,
(1) The point \( O \) has a linear velocity \( 3 R \omega \hat{i} \)
(2) The point \( P \) has a linear velocity \( \frac{11}{4} R \omega \hat{i}+\frac{\sqrt{3}}{4} R \omega \hat{k} \)
(3) The point \( P \) has a linear velocity \( \frac{13}{4} R \omega \hat{i}+\frac{\sqrt{3}}{4} R \omega \hat{k} \)
(4) The point \( P \) has a linear velocity \( \left(3-\frac{\sqrt{3}}{4}\right) R \omega \hat{i}+\frac{1}{4} R \omega \hat{k} \)
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