A solid shere of mass `m` and radius `R` is kept over a rough ground. A time varying force `F=2t` is acting at the topmost point as shown in figure.
(a). Find angular momentum of the sphere about the bottommost point as a function of time `t`
(b). Does this result depend on the fact whether the ground is rough or smooth?

Explanation

(a). Suppose `f` is the force of friction acting on the sphere in forward directionn as shown in figure.
taking torque about bottommost point `O`. Torque of friction is already zero. As this force already passes through `O`. torque of applied force is only there.
`tau=Fxxr_(bot)`
`=(2t)(2R)=4RT`
As, this toque is a function of time
`therefore` angular impulse `=int_(0)^(t)taudt=int_(0)^(t)(4Rt)dt=2Rt^(2)`
this angular impulse is equal to change in angular momentum. Hence, angular momentum at time `t` is `2Rt^(2)`.
(b). This result is independent of the nature of surface (smooth or rough) as the torque of friction about bottommost point is already zero. it does not contribute in angular impulse and therefore in angular momentum.