Question

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 $OP$ 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)

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