A ring of radius r is moving without slipping on a circular track of radius R. Angular speed of the ring about its axis is  $\text{ω}$. Ignoring the fact that the ring axis is inclined. 

the angular acceleration of the ring is  $\frac{{\omega }^{2}r}{R}$

the angular acceleration of the ring is  $\frac{{\omega }^2(R-r)}{R}$

the angular velocity of centre of the ring about vertical axis passing through the centre of track is  $\frac{r\omega }{R}$

the angular velocity of centre of the ring about vertical axis passing through the centre of track is  $\frac{(R-r)\omega }{R}$