A solid sphere of mass 5 kg and radius `1m` is kept over a rough surface as shown in figure. A force `F=30N` is acting at the topmost point.
(a). Check whether the pure rolling will take place or not
(b). Find direction and magnitude of friction actually acting on the sphere.
(c). Find linear acceleration `a` and angular acceleration `alpha` take `g=10m//s^(2)`

Explanation

(a). Availability of friction `N=mg=5xx10=50N`
`mu_(S)N=0.3xx50=15N`
`mu_(k)N=0.2xx50=10N`
Requirement of friction for acceleration pure rolling (or to satisfy `a=Ralpha`)
Let the friction `f` acts in forward direction.
`a=Ralpha`
`implies(F_("net"))/(m)=R((tau_("net"))/(I))`
or `(F+f)/(m)=R[(FR-fR)/((2)/(5)mR^(2))]`
solving this equation, we get
`f=(3)/(7)F`
`=(3)/(7)(30)=(90)/(7)N`
since this value of `f` is less than `mu_(s)N`. Therefore, friction is sufficient for acclerated pure rolling to take place. Hence, pure rolling is taking place.
(b). the above value of `f` is positive, hence direction of friction is forward and `(90)/(7)N` friction will
(c) The actual forces acting on the sphere area as under
`a=(F_("net"))/(m)=(30+(90)/(7))/(5)=8.57m//s^(2)`
since, `a=Ralpha`
`thereforealpha=(a)/(R)=(8.57)/(1)=8.57rad//s^(2)`