A ring of mass `m` and radius `r` has a particle of mass `m` attached to it at a point A. the ring can rotate about a smooth horizontal axis which is tangential to the ring at a point B diametrically opposite to A. The ring can rotate about a smooth horizontal axis which is tangential to the ring at a point B diametrically opposite to A. The ring is released from rest when AB is horizontal. find the angular veloicity and the angular acceleration of the body when AB has turned through an angle `(pi)/(3)`.

Explanation

Angular velocity from conservation of mechanical energy
decrease in gravitational PE
`=` increase in rotational KE
or `mgrsin60^(@)+mg(2rsin60^(@))`
`=(1)/(2)[(3mr^(2))/(2)+m(2r)^(2)]omega^(2)`
`therefore(3sqrt(3))/(2)=(11)/(4)mr^(2)omega^(2)`
`thereforeomega=sqrt((6gsqrt(3))/(11r))`
angular acceleration
`alpha=(tau)/(I)=(mgrcos60^(@)+mg(2rcos60^(@)))/([(mr^(2))/(2)+m(2r)^(2)])`
`=((3mgr)/(2))/((11)/(2)mr^(2))=(3g)/(11r)`