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 anticlockwise 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)

We have
$\overrightarrow{v_O}=(3R)\omega \hat{i}=0$
$\Rightarrow \overrightarrow{v_O}=3R\omega \hat{i}$
Therefore,
${\overrightarrow{V}}_{P,O}=\frac{-R{\omega }_{\hat{i}}}{4}+\frac{R\omega \sqrt{3}}{4}\hat{k}$
and $\overrightarrow{v_p}={\overrightarrow{V}}_{P,O}+\overrightarrow{v_O}=\frac{11}{4}R{\omega }_i+R\omega \frac{\sqrt{3}}{4}\hat{k}$