Combination of Translation and Rotation

How translation and rotation combine in rolling. If a rolling object has radius $R$, then in one complete rotation, the center of mass is displaced by exactly the object's own circumference, $2\pi R$. Every other point on the object is also displaced by the same $2\pi R$ over that one full rotation — but there is a crucial difference in the shape of the path each point takes: the center of mass alone traces a perfectly straight line, while every other point on the object instead traces a looping curve called a cycloid, tracing out first a rise and fall relative to the straight-line motion of the center as it goes through each rotation.

Two velocity contributions for every point. Every point on a rolling object can be thought of as having two separate velocity contributions superimposed: a translational velocity, $v_{TRANS}$, identical for every point (and equal to the velocity of the center of mass, $v_{CM}$, since $v_{CM}=v_{TRANS}$), and its own individual rotational velocity, $v_{ROT}=r\omega$ (where $r$ is that point's own distance from the center of mass), which is always directed perpendicular to that point's instantaneous position vector from the center of mass. The point's actual resultant velocity $v$ is the vector sum of these two contributions, and (for the specific case of the point of contact with the rolling surface) this resultant velocity always turns out to be perpendicular to the line joining that point to the point of contact.

The point of contact is momentarily at rest. In pure rolling, whichever point of the rolling object happens, at any given instant, to be touching the surface is at momentary rest — this holds for every point on the rim in turn, as each one, one after another, comes into contact with the surface, is at rest for that instant, and then moves off along its cycloid path once again. This gives two entirely equivalent ways to describe pure rolling: (i) as translation of the center of mass combined with rotation about the center of mass, or (ii) as a momentary, pure rotation about the instantaneous point of contact.

Deriving $v_{CM}=R\omega$. Since the point of contact's resultant velocity is exactly zero in pure rolling, its translational contribution ($v_{TRANS}$, forward) and its rotational contribution ($v_{ROT}$, backward, since the bottom of a forward-rolling wheel rotates backward relative to the center) must be equal in magnitude and exactly opposite in direction: $v=v_{TRANS}-v_{ROT}=0$, i.e. $v_{TRANS}=v_{ROT}$ for the point of contact. Since this must hold for every point on the rim (each one is, in turn, the momentary point of contact), and using $v_{TRANS}=v_{CM}$ and $v_{ROT}=R\omega$, this gives the defining condition of pure rolling:

$\boxed{v_{CM}=R\omega}.$ …