I'm confused about this "rolling without slipping" kind of situation. Or better in this case the object is rolling and slipping, just use the label "rolling without slipping" to identify the kind of problem.
Suppose to have a disk with initial velocity $v$ and angular velocity $\omega$. The motion is to the right but the angular velocity is counterclockwise.
There are no forces acting on the disk besides the kinetic friction $\mathbf{f}$.
Things are ok if I take as pivot point the center of mass.
$$\{\begin{matrix} – \mathbf{f} = m\mathbf{a_{CM}}\\ – \mathbf{r} \times \mathbf{f} =I_{cm} \mathbf{\alpha} \end{matrix}\tag{1}$$
But if I take the point $O$ on the ground, then the kinetic friction has zero torque.
$$\{\begin{matrix} – \mathbf{f} = m\mathbf{a_{CM}}\\ 0 =I_{O} \mathbf{\alpha} \end{matrix}\tag{2}$$
I assumed that the angular velocity (and so $\alpha$) is the same it I take as pivot the center of mass or the point $O$.
If this is the case than parallel axis theorem can be used and $$I_O=I_{cm}+m \mathbf{r}^2$$
But there is a contradiction since I get $\alpha=0$ from $(2)$ and $\alpha\neq0$ from$(1)$.
How can that be? Maybe $\alpha$ is not the same in the two cases?
Best Answer
The equation of motion
$$ \text{torque about stationary geometrical point O} = \text{moment of inertia w.r.t. O} \times \text{angular acceleration w.r.t. O} $$
is valid only if the motion of the body is planar rotation around an axis that passes through O. This is the case if the point O is taken to be point of contact of the body when rolling without slipping, but not when rolling with slipping. Generally valid version of torque-angular momentum theorem is
$$ \text{torque about stationary geometrical point O} = \ = \frac{d}{dt}\left(\text{angular momentum w.r.t. stationary geometrical point O}\right). $$
If the body is rolling with slipping, there is no stationary geometrical point O on the ground for which the angular momentum could be written as $I_O\omega_O$ with $I_O$ constant in time and the latter equation doesn't reduce to the former one.