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 $ω$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $ω$ /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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The point O has linear velocity $3 Rω \hat{i}$

The point P has velocity $\frac{11}{4} Rω \hat{i} + \frac{\sqrt{3}}{4} Rω \hat{k}$

The point P has velocity $\frac{13}{4} Rω \hat{i} - \frac{\sqrt{3}}{4} Rω \hat{k}$

The point P has velocity $[3 - \frac{\sqrt{3}}{4}] Rω \hat{i} + \frac{1}{4} Rω \hat{k}$

answer is B, A.

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Detailed Solution

For rolling motion, the velocity of the point of contact with respect to the surface should be zero.

$3 R ω (- \hat{i}) + {\vec{v}}_{0} = 0$

Velocity of point O is $v_{0} = (3 Rω) \hat{i}$
$v_{PO} is \frac{R \cdot ω}{2}$ in the direction shown in figure
In vector from
$\begin{array}{l} v_{PO} = - \frac{Rω}{2} \sin 30^{\circ} \hat{i} + \frac{Rω}{2} \cos 30^{\circ} \hat{k} \\ = - \frac{Rω}{4} \hat{i} + \frac{\sqrt{3} Rω}{4} \hat{k} \end{array}$