A cylinder of mass `m` rolls without slipping on an inclined plane of inclination `theta`. Find the liner acceleration of the sphere and friction acting on it. What should be the minimum coefficient of static friction to support pure rolling?

Explanation

Linear motion: `mgsintheta-f=ma` (`i`)
Rotational motion: `fR=Ialpha=((1)/(2)mr^(2))((a)/(r ))`
`f=(1)/(2)ma` (`ii`)
(for pure rolling `a=ralpha`)
From (`i`) and (`ii`)
`mgsintheta-(1)/(2)ma=ma`
`mgsintheta=(3)/(2)maimpliesa=(2)/(3)gsintheta`
`f=(1)/(2)ma=(1)/(3)mg sintheta`
`f_(max)=muN=mumg cos theta:` coefficient of static friction.
For pure rolling `f_(max)gef`
`mumg cos thetage(1)/(3)mg sin theta implies muge(1)/(3)tantheta`
`mu_(min)=(1)/(3)tantheta`
OR
In pure rolling
`a=(gsintheta)/(1+k^(2)//R^(2))=(gsintheta)/(1+(1)/(2))=(2)/(3)gsintheta`
`mg sintheta-f=maimpliesf=(1)/(3)mgsintheta`
`muge(tantheta)/(1+R^(2)/k^(2))`
`mu_(min)=(tantheta)/(1+2)=(1)/(3)tantheta`