Question

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 $\omega$. Ignoring the fact that the ring axis is inclined.

• A. the angular acceleration of the ring is $\frac{{\omega }^{2}r}{R}$
• B. the angular acceleration of the ring is $\frac{(R-r)}{R}{\omega }^2$
• C. the angular velocity of centre of the ring about vertical axis passing through the centre of track is $\frac{r\omega }{R}$
• D. the angular velocity of centre of the ring about vertical axis passing through the centre of track is $\frac{(R-r)}{R}\omega$

Answer 1

Answer: (A), (C)

The correct options are
A the angular acceleration of the ring is $\frac{{\omega }^{2}r}{R}$
C the angular velocity of centre of the ring about vertical axis passing through the centre of track is $\frac{r\omega }{R}$

Let the direction of angular velocity vector changes by $\Delta \varphi$ in time $\Delta t$
$Then,\Delta \varphi =\frac{|\Delta \omega |}{\omega }=\frac{\Delta l}{R}=\frac{{\omega }^{\prime }R\Delta t}{R}$
Where ${\omega }^{\prime }$ is angular speed of centre of the ring.
$Also,\Delta l=r\omega \Delta t$
$\therefore {\omega }^{\prime }=\frac{r\omega }{R}$
Angular acceleration, $\propto =lo{m}_{xY}\frac{\Delta \omega }{\Delta t}={\omega }^{\prime }\omega =\frac{r{\omega }^2}{R}$