Answer
$$\frac{\omega_1}{\omega_2} = \frac{R_2}{R_1} $$
Work Step by Step
The tangential speed of the turntable equals the tangential speed of the roller because there is no slipping between the surfaces.
$$v_1=v_2$$
$$\omega_1 R_1=\omega_2 R_2$$
$$\frac{\omega_1}{\omega_2} = \frac{R_2}{R_1} $$