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 $ω$ . Ignoring the fact that the ring axis is inclined.

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

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

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

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

answer is A, C.

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Detailed Solution

Let angular velocity of centre of ring about vertical axis passing through center of circular path is $ω_{2}$
Then : $ω_{2} R = ω r$
$ω_{2} = \frac{ω r}{R} \vec{α} = {\vec{ω}}_{2} \times \vec{ω} \Rightarrow α = ω_{2} ω = \frac{ω^{2} r}{R}$