In the shown figure, `M` is mass of the body, `R` its radius an `I` the moment of inertial about an axis passing through centre. Find force of friction `f` acting on the body (upwards), its linear acceleration `a` (down the plane) and type of motion if:
(a) `mu=0`
(b). `multmu_(min)`
(c). `mugtmu_(min)`
Where `mu_(min)` is the minimum value of coefficient of friction required for pure rolling

Explanation

(a). If `mu=0` then
`f=0` and `a=gsintheta=a_(1)` (say)
and the motion is only translational.
(b). If `multmu_(min)` then maximum value of friction will act. As friction is insufficient to provide acclerated pure rolling or to stop the relative motion. ltbr. `thereforef=f_(max)=mumgcostheta`
and `a=(Mgsintheta-muMgcostheta)/(m)`
`=gsintheta-mugcostheta=a_(2)` (say)
In this case, motion is rotation `+` translatio with forward slip (as `agtRalpha)`
(c). if `mugtmu_(min)`, then we have discussed in the above article that
`f=(Mgsintheta)/(1+(MR^(2))/(I)))` and `a=(gsintheta)/(1+(I)/(MR^(2)))=a_(3)` (say)
Motion i this case is rotation `+` translation with accelerated pure rolling.