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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 \( 2 R \) rotating anti-clockwise with angular speed \( \omega / 2 \). The ring and disc are separated by frictionless ball bearings. The system is in the \( \mathrm{x}- \) z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of \( 30^{\circ} \) with the horizontal. Then with respect to the horizontal surface.
(A) The point \( \mathrm{O} \) has a linear velocity \( 3 \mathrm{R} \omega \hat{\mathrm{i}} \)
(B) The point \( P \) has a linear velocity \( \frac{11}{4} R \omega \hat{i}+\frac{\sqrt{3}}{4} R \omega \hat{k} \)
(C) The point \( P \) has a linear velocity \( \frac{13}{4} R \omega \hat{\mathrm{i}}-\frac{\sqrt{3}}{4} \mathrm{R} \omega \hat{\mathrm{k}} \)
(D) The point \( \mathrm{P} \) has a linear velocity \( \left(3-\frac{\sqrt{3}}{4}\right) \mathrm{R} \omega \hat{\mathrm{i}}+\frac{1}{4} \mathrm{R} \omega \hat{\mathrm{k}} \)
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