The figure shows a system consisting of (i) a ring of outer radius...

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 point

P

on the inner disc is at a distance

R

from the origin, where

O P

makes an angle of

30^{\circ}

with the horizontal. Then with respect to the horizontal surface, The figure shows a system consisting of (i) a ring of outer radius 3 R rolling clockwise without slipping on a

The figure shows a system consisting of (i) a ring of outer radius 3 R rolling clockwise without slipping on a

Options

Athe point $O$ has linear velocity $3 R \omega \hat{i}$
Bthe point $P$ has linear velocity $\frac{11}{4} R \omega \hat{i}+\frac{\sqrt{3}}{4} R \omega \hat{k}$ .
Cthe point $P$ has linear velocity $\frac{13}{4} R \omega \hat{i}-\frac{\sqrt{3}}{4} R \omega \hat{k}$
Dthe point $P$ has linear velocity $\left(3-\frac{\sqrt{3}}{4}\right) R \omega \hat{i}+\frac{1}{4} R \omega \hat{k}$

Correct answer

A. the point $O$ has linear velocity $3 R \omega \hat{i}$

Step-by-step solution

Velocity at centre 'O'

\quad \therefore \quad \vec{v}_{o}=3 R \omega \hat{i}

\vec{V}_{P}=3 R \omega \hat{i}-\frac{R \omega}{2} \sin 30^{\circ} \hat{i}+\frac{R \omega}{2} \cos 30^{\circ} \hat{k}

\therefore \quad \vec{V}_{P}=\left[3 R_{\omega} \hat{i}-\frac{R_{\omega}}{4} \hat{i}\right]+\frac{\sqrt{3} R_{\omega}}{4} \hat{k}

or,

\quad \vec{V}_{P}=\frac{11}{4} R_{\omega} \hat{i}+\frac{\sqrt{3}}{4} R_{\omega} \hat{k}

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