I searched for my question and I found many results but all of them eventually try to solve something specific (like solving for the kinetic energy or finding when does the ball stop).
what I want is why does a ball roll at all?
Pretend I shot a ball and the ball started slipping on the ground, pretend it doesn't leave the ground and that the ground isn't incline.
Now I thought of friction force exerting a torque on the ball, if the force is $F_c$ then the torque should be:
$$\tau=u\times F_c\times r$$(There is no sin because $F_c$ is perpendicular to $r$)
The existence of $u$ is due to the fact that the force doesn't just rotate the ball but also slows it down.
What's missing in my assumption is the speed of the ball, it seems to me that the faster the ball the faster it rotates, I don't have an explanation for this because $F_c$ doesn't relate to velocity, $F_c$ is given by :
where $N$ is the normal force and $u_k$ is the kinetic friction coefficient.
So what's a better explanation to this phenomena?
Something sparked in my mind, the ball doesn't just slip, in reality it bounces, we just don't see the bounces.
But I don't know how to put this into a good formula or explanation.
what I mean by "the faster the ball" I mean its initial velocity (after shooting it) because the ball slows down.