A force `F` is applied on a disc at it's centre. Find acceleration of center of mass in the case of pure rolling and also find minimum coefficient of fricition required for pure rolling.

Explanation

Let the friction force acting on disc be `f_(r)`, acceleration of centre of mass is a and angular acceleration is `alpha`.
We have three unknowns
`(i)` a
`(ii) alpha`
`(iii) f_(r )`
We require `3` equations to solve them
Translatory Motion
`F-f_(r )=ma`......`(i)`
Rotatory Motion
`r * f_(r )=(mr^(2))/(2)*alpha`........`(ii)`
Condition of no slipping
`a=r alpha`..........`(iii)`
By solving `(ii)` & `(iii)`
`f_(r )=(ma)/(2)`, `F=(3ma)/(2)impliesa=(2F)/(3m)`, `f_(r )=(m)/(2)xx(2F)/(3m)`, `f_(r )=(F)/(3)`
Now, `f_(r )le mu N`
`implies(F)/(3) le mu mg`
`implies mu ge (F)/(3mg)`