The motion of the moving gear is a combination of the translation of its center $O_1$ and rotation about $O_1$. The velocity of the center $O_1$ is determined by the rotation of the crank arm $OO_1$, which has a length of $R+r$.

$$v_{O1} = \omega_0 (R+r)$$

For rolling without slipping, the velocity of the point of contact $C$ on the moving gear must be zero, as it is in contact with the fixed gear. The velocity of point $C$ is the sum of the center's translational velocity and the velocity due to rotation. Let $\omega$ be the angular velocity of the moving gear.

$$v_C = v_{O1} - \omega r$$

Setting $v_C = 0$ for the no-slip condition:

$$v_{O1} - \omega r = 0 \implies \omega_0 (R+r) - \omega r = 0$$

Solving for $\omega$:

$$\omega = \frac{\omega_0 (R+r)}{r}$$