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 $2 R$ 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 $3 R \omega \hat{i}$
• 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
$\overrightarrow{v_{O}}=(3 R) \omega \hat{i}=0$
$ \Rightarrow \overrightarrow{v_{O}}=3 R \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}$