Question

A disc of mass `M` and radius `R` can rotate freely in a vertical plane about a horizontal axis at `O` distance `r` from the centre of the disc as shown in Fig. The disc is released from rest in the shown position. Answer the following questions based on the above information

The angular velocity of the disc in the above described case is
A. `sqrt((8gr)/(5{R^(2)+2r^(2)]))`
B. `sqrt((6gr)/(5[R^(2)+2r^(2)]))`
C. `sqrt((12gr)/(5[R^(2)+2r^(2)]))`
D. `sqrt((12gr)/(5R^(2)))`

Answer 1

Correct Answer - C
From `tau=Ialpha`

`Mgxxrcos37^@=[(MR^(2))/2+Mr^(2)]alpha`
`alpha=(8rg)/(5[R^(2)+2^(2)])`
From energy conservation
`(Iomega^(2))/2=Mgxxrsin37^@`
`[(MR^(2))/2+Mr^(2)](omega^(2))/R=Mgrxx3/5`
`omega=sqrt((12gr)/(5[R^(2)+2r^(2)]))`
For `FBD` of the disc
`R_(x)-mgsin37^@=Ma_(r)=momega^(2)r`

`Mgcos37^@-R_(y)=Ma_(t)=Mralpha`
`R_(x)=(3Mg)/5[(R^(2)+6r^(2))/(R^(2)+2r^(2))], R_(y)=(Mg)/5[(4R^(2))/(R^(2)+2r^(2))]`
`R=sqrt(R_(x)^(2)+R_(y)^(2))=(Mg)/(5[R^(2)+2r^(2)])xxsqrt(9(R^(2)+6r^(2))^(2)+(4R)^(2))`