A uniform disc of radius r is to be suspended through a small hole made in te disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for it to have minimum time period?

Explanation

Let distance between centre of disc and point of suspension is x.

`tau=-mgx sin theta`
`I=(mr^2)/(2)+mx^2`
`tau=-mgxtheta` (`:' sin theta=theta`)
`Ialpha=-mgxtheta`
`alpha=-(mgx)/(I)theta=-omega^2theta`
`T=2pisqrt((I)/(mgx))impliesT^2=4pi^2(((mr^2)/(2)+mx^2)/(mgx))`
`T^2=(4pi^2)/(g)((r^2)/(2x)+x)`
For T to be minimum, `(dT)/(dx)` or `(d(T^2))/(dx)=0`
`(d(T^2))/(dx)=-(r^2)/(2x^2)+1=0impliesx=r//sqrt2`
`T_(min)=[(4pi^2)/(g)((r^2)/(sqrt2r)+(r)/(sqrt2))]^(1/2)impliesT_(min)=2pisqrt((sqrt2r)/(g))`