In the applet w (omgega) is positive for clockwise spin, different than for most right handed polar coordinate systems. The images in the simulation help to describe the situation.
The simulation correctly treats situations in which r*w0 is either less than V0 or greater than V0, but the text only refers to the first situation.
The equation for omega, w = w0 - t * (r u mg)/I (u is the coefficient of friction) in the text is wrong (it needs a plus sign instead of a minus sign in the sign convention used here) as long as r*w0 < V0.
It should be pointed out that the frictional force under consideration is the
kinetic friction on the ball, and not static friction. Such distinctions
are often unclear to students.
The radius of the ball and the units of r*w should be given so that
students can more easily compare the numerical values generated by the applet
with their own calculated values.