# How friction force synchronize linear velocity and angular velocity? [closed]

I am doing a simulation about moving balls on a surface. There are frictions between different balls and the surface. In the beginning, balls have a linear velocity and an angular velocity (which are independent from each other), then because of the friction, I think the state of the balls should turn from sliding to rolling (if they haven't fully stopped yet). I want to know how to compute the change of the linear velocity and the angular velocity in a time step? Is there a formula for this?

When there is friction $$f$$ acting on a ball of mass $$M$$ and radius $$R$$, the friction delivers both linear impulse and angular impulse to the ball. In a time step $$\Delta t$$, the change in linear velocity is $$\Delta v = \frac{1}{m}f\Delta t$$ whereas the change in angular velocity is $$\Delta \omega = \frac{1}{I}fR \Delta t = \frac{5f\Delta t }{2MR}$$ assuming that you have solid balls with rotational inertia $$I=\frac{2}{5}MR^2$$.
Note that the above analysis only applies when the non-slipping condition, $$v=r\omega$$, is not satisfied.