Question

The figure shows in 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-cllockwise with angular speed $\omega /2$. The ring and disc are separated by frictionless ball bearings. The system is in the x-z plane. The point $P$ on the inner disc is at a distance $R$ from the origin, where $OP$ makes an angle of ${30}^o$ with the horizontal. Then with respect to the horizontal surface,

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

Answer 1

Answer: (A), (B)

The correct options are
A the point $O$ has a linear velocity $3R\omega \hat{i}$ .
B the point $P$ has a linear velocity $\frac{11}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k}\cdot$

as shown in the diagram
$V_o\left(3R\right)\omega i=0$
therefore $V_o=3R\omega i$
$V_{p,o}=\frac{-R\omega }{4}i+\frac{R\omega \sqrt{3}}{4}k$
Now,
$V_p=V_{p,o}+V_o$
$=\frac{11R\omega }{4}i+\frac{R\omega \sqrt{3}}{4}k$