By accepting a equation for the static friction of rolling body from the slope, obtain the condition for rolling on the slope :

Explanation

Here `mu_(s)=[(1)/(1+(R^())/(k^(2))]]tantheta" "....(1)`
The line (or point) of contact on the surface of the body with the inclined plane is momentarily stationary. Hence we have used the coefficient of static friction.
It is due to the friction that the body rolls without sliding. So it can be said from equation
(1) that if the condition
`thereforemu_(s)ge[(1)/(1+(R^())/(k^(2))]]tantheta" "....(2)`
The body will be able to roll on the slope without slipping only if the condition of equation (2) is maintaine.
Special cases :
(1) For thing ring, radius of gyration `k=R`
`therefore` from equation (2)
`mu_(s)ge(1)/(2)tantheta`
(2) For circular disc, `k=(R )/(sqrt(2))`
`therefore` from equation (2)
`mu_(s)ge(1)/(3)tantheta`
(3) For solid sphere `k=sqrt((2)/(5)).R`
`therefore` from equation (2)
`mu_(s)ge(2)/(7)tantheta`