Question

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 $\omega$ and (ii) an inner disc of radius $2R$ 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 $OP$ makes an angle of $30^{\circ }$ with the horizontal. Then with respect to the horizontal surface,

• A. the point $O$ has linear velocity $3R\omega \hat{i}$
• B. the point $P$ has linear velocity $\frac{11}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k}$.
• C. the point $P$ has linear velocity $\frac{13}{4}R\omega \hat{i}-\frac{\sqrt{3}}{4}R\omega \hat{k}$
• D. the point $P$ has linear velocity $\left(3-\frac{\sqrt{3}}{4}\right)R\omega \hat{i}+\frac{1}{4}R\omega \hat{k}$

Answer 1

Answer: (A), (B)

${\overrightarrow{V}}_0(3R)\omega \hat{i}=0$
$\therefore {\overrightarrow{v}}_0=3R\omega \hat{i}$
${\overrightarrow{v}}_{P,O}=\frac{-R\omega }{4}\hat{i}+\frac{R\omega \sqrt{3}}{4}\hat{j}$
$\therefore {\overrightarrow{v}}_p={\overrightarrow{v}}_{p,o}+{\overrightarrow{v}}_o$
$=\frac{11}{4}R\omega \hat{i}+R\omega \frac{\sqrt{3}}{4}\hat{j}$