A cylinder of radius R is rotating about its horizontal axis with constant `omega=sqrt((5g)/R)`. A block of mass m is kept on the inner surface of the cylinder. Block is moving in vertical circular motion without slipping. co–efficient of friction between block and surface of cylinder is `mu`. If minimum value of `mu` for complete vertical circular motion of block is `(2sqrt(6))/(3x)` then find 'x'.

Explanation

`N= mg cos theta +m omega^(2)R= mg cos theta+m((5g)/R) RrArr N =5mg +mg cos theta gt 0` so block does not leave circular motion
`f_(r)=ma_(T)` for limiting case `muN=ma_(T)rArr mu(mg(5 +cos theta))= mg sin thetarArr mu=(sin theta)/(5+cos theta)`
`(dmu)/(d theta)=((5+cos theta)(cos theta)-sin theta(-sin theta))/((5+cos theta)^(2))=0rArr cos theta=-1/5`
`mu_(max)=(sqrt(1-1/25))/(5-1/5)=(2sqrt(6))/24rArr x=8`