Velocity of Points on Rolling Outer Ring and Inner Disc System

To determine the correct options, we analyze the motion of the center of mass $O$ and the point $P$ in the given coordinate system.

1. Setup of Coordinate System

The system resides in the $x-z$ plane, where:

• $\hat{i}$ is the unit vector pointing along the horizontal $x$-axis (to the right).
• $\hat{k}$ is the unit vector pointing along the vertical $z$-axis (upwards).

According to the right-handed Cartesian coordinate system ($\hat{i} \times \hat{j} = \hat{k}$), the unit vector $\hat{j}$ points into the plane of the paper (+y-direction).

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2. Linear Velocity of Point $O$ (Option A)

The outer ring has a radius of $3R$ and rolls clockwise without slipping on the horizontal surface with an angular speed $\omega$.

• Clockwise rotation in the $x-z$ plane corresponds to an angular velocity vector directed into the page: $\vec{\omega}_{\text{ring}} = \omega \hat{j}$

• Since the ring undergoes pure rolling, the linear velocity of its center $O$ is: $\vec{v}_O = \omega R_{\text{outer}} \hat{i} = 3R\omega \hat{i}$

Thus, Option A is correct.

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3. Linear Velocity of Point $P$ (Options B, C, D)

Point $P$ is on the inner disc, located at a distance $R$ from the center $O$, and the line $OP$ makes an angle of $30^\circ$ with the horizontal line.

• Position vector of point $P$ relative to $O$: $\vec{r}_{P/O} = R \cos 30^\circ \hat{i} + R \sin 30^\circ \hat{k} = R \left( \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{k} \right)$

• Angular velocity of the inner disc: The inner disc rotates counter-clockwise with an angular speed of $\frac{\omega}{2}$. Counter-clockwise rotation in the $x-z$ plane corresponds to an angular velocity vector directed out of the page: $\vec{\omega}_{\text{disc}} = -\frac{\omega}{2} \hat{j}$

• Velocity of point $P$ relative to $O$: $\vec{v}_{P/O} = \vec{\omega}_{\text{disc}} \times \vec{r}_{P/O}$ $\vec{v}_{P/O} = \left(-\frac{\omega}{2} \hat{j}\right) \times \left( \frac{\sqrt{3}}{2} R \hat{i} + \frac{1}{2} R \hat{k} \right)$

  Using the cross product relations $\hat{j} \times \hat{i} = -\hat{k}$ and $\hat{j} \times \hat{k} = \hat{i}$: $\vec{v}_{P/O} = -\frac{\omega}{2} R \left[ \frac{\sqrt{3}}{2} (-\hat{k}) + \frac{1}{2} (\hat{i}) \right] = -\frac{1}{4} R\omega \hat{i} + \frac{\sqrt{3}}{4} R\omega \hat{k}$

• Total (absolute) velocity of point $P$ with respect to the horizontal surface: $\vec{v}_P = \vec{v}_O + \vec{v}_{P/O}$ $\vec{v}_P = 3R\omega \hat{i} + \left( -\frac{1}{4} R\omega \hat{i} + \frac{\sqrt{3}}{4} R\omega \hat{k} \right)$ $\vec{v}_P = \frac{11}{4} R\omega \hat{i} + \frac{\sqrt{3}}{4} R\omega \hat{k}$

Thus, Option B is correct, while Options C and D are incorrect.

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Conclusion

The correct options are A and B.