Obtain the necessary condition $v_{cm} = R\omega$ for a body rolling without slipping.

(N/A) For a rigid body (like a sphere,circular disc,or wheel) rolling without slipping,the point of contact with the surface must have zero instantaneous velocity relative to the surface.
Consider a circular disc of radius $R$ rolling on a horizontal surface. Let $v_{cm}$ be the velocity of the center of mass $C$ and $\omega$ be the angular velocity about the center.
The velocity of any point $P$ on the rim of the disc is the vector sum of the velocity of the center of mass and the tangential velocity due to rotation: $\vec{v}_P = \vec{v}_{cm} + \vec{v}_{rot}$.
At the point of contact $P_0$ with the ground,the velocity due to rotation $\vec{v}_{rot}$ is directed backwards with magnitude $R\omega$. The velocity of the center of mass $\vec{v}_{cm}$ is directed forwards.
For the condition of rolling without slipping,the velocity at the point of contact $P_0$ must be zero:
$\vec{v}_{P_0} = \vec{v}_{cm} + \vec{v}_{rot} = 0$
Since $\vec{v}_{cm}$ is forward and $\vec{v}_{rot}$ is backward at $P_0$:
$v_{cm} - R\omega = 0$
Therefore,the necessary condition is $v_{cm} = R\omega$.