Q.What is the condition for pure rolling?

Rolling is the combination of two motions happening at once: translation of the object's center of mass, and rotation about that same center of mass. For a round object of radius $R$ rolling one full turn, the center of mass advances by exactly the circumference, $2\pi R$; every other point on the rim also advances by $2\pi R$ over one full turn, but while the center of mass traces a straight line, every other point traces a looping curve called a cycloid.

Two equally valid descriptions of pure rolling:

1. Translation of the center of mass (at $v_{CM}$) combined with rotation about the center of mass (at $\omega$).
2. A momentary, pure rotation about the instantaneous point of contact with the surface, which — in pure rolling — is always at rest.

The pure-rolling condition. Every point of a rolling body has two velocity contributions: the shared translational velocity $v_{TRANS}=v_{CM}$ (the same for every point), and its own rotational velocity $v_{ROT}=r\omega$ about the center of mass (perpendicular to its own position vector from the center). At the point of contact, these two contributions point in opposite directions; in pure rolling they are exactly equal in magnitude, so the resultant velocity of the contact point is exactly zero:

At the topmost point, by contrast, the two contributions point in the same direction and add, giving a resultant speed there of $2v_{CM}$ — a useful, checkable feature of pure rolling. …