$A$ disc rotating about its axis with angular speed $\omega_{o}$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is $R$. What are the linear velocities of the points $A, B$ and $C$ on the disc shown in Figure? Will the disc roll in the direction indicated?

(N/A) The angular speed of the disc is $\omega_{o}$ and its radius is $R$. The linear velocity $v$ of a point at a distance $r$ from the center is given by $v = r \omega_{o}$.
For point $A$ (at distance $R$ from the center,at the top): $v_{A} = R \omega_{o}$,directed tangentially to the right.
For point $B$ (at distance $R$ from the center,at the bottom): $v_{B} = R \omega_{o}$,directed tangentially to the left.
For point $C$ (at distance $R/2$ from the center): $v_{C} = (R/2) \omega_{o}$,directed tangentially to the right.
The disc will not roll. Rolling motion requires the presence of friction to provide the necessary torque and to satisfy the condition $v_{cm} = R \omega$. Since the table is perfectly frictionless,there is no external force to cause translational motion of the center of mass,and thus the disc will simply continue to rotate about its stationary center.