A force `F` is applied at center of a uniform round object of mass `m` radius `R` and moment of inertia about its centre of mass `I_(cm)`. Find (if `(I_(cm))/(mR^(2))=k`).
`(a)` Acceleration of center of the round object if it rolls without slipping.
`(b)` Minimum coefficient of fricition required so that the round object rolls without slipping.

Explanation

Equation for translatory motion
`F-f=ma`……..`(1)`
Equation for rotatory motion
`fR=I_(cm)alpha`……..`(2)`
Condition for no slipping
`a=Ralpha`……..`(3)`
By solving the equations
`F-(I_(cm)a)/(R^(2))=ma`
`a=(F)/(m+(I_(cm))/(R^(2)))`
`=(F)/(m[1+(I_(cm))/(mR^(2))])`
`=(F)/(m(1+k))`
`f=(I_(cm)F)/(mR^(2)(1+k))=(kF)/((1+k))`
Thus `mu_(min)mg=(kF)/(1+k)`
`impliesmu_(min)=((k)/(1+k))(F)/(mg)`