I'm a graduate student in mechatronics, but I give myself headaches on this astonishingly simple 2D problem.
Imagine a ball (perfectly rigid) on an horizontal plane (perfectly rigid), initially at rest. The only forces acting on it are its weight $P$ and the normal reaction of the plane.
Now, at time $t = 0$, I apply an additional horizontal force to the ball $F$, big enough ($F > \mu \times P$) in its centre of mass. An opposite tangential friction force appears ($T = \mu \times P$). And the problem begins! Since the horizontal forces aren't equal, the ball will accelerate linearly. Moreover, the tangential force create a torque, and the ball will also rotate.
But because we are still at time $t = 0_+$, both the linear and the angular velocity are nil. So the cinematic rolling-without-slipping condition $v = R \times \omega$ is fulfilled, so in fact, there is no friction and the ball is only slipping… That's pretty paradoxical.
It's easy to know if the ball will move or stay immobile. But how can I determine if the ball will slipping and rolling, or just rolling without slipping?