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 system is in the x-z plane. The point P on the inner disc is at 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 $\frac{11}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k}$
• 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

The correct option is B The point P has a linear velocity $\frac{11}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k}$

As ring is rolling the point of contact of ring with ground (O') will be instantaneous center of rotation. Hence for pure rolling $v_0=\omega \left(3R\right)=3R\omega$ and its direction is along positive x.

The velocity at the point P
$v_P=3R\omega \hat{i}-\frac{R\omega }{2}sin{30}^o\hat{i}+\frac{R\omega }{2}cos{30}^o\hat{k}$
$v_P=\frac{11}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k}$