A billiard ball set gently rolling on a billiard table slows and stops, because it is decelerated by resistance forces at the contact between the ball and table.
I expect the magnitude of the resistance force will depend on the ball radius, but what is the exponent on the radius in the equation that describes the force?
The exponent is ~3 if determined by the ball volume, ~2 if determined by the surface of the ball or contact area, and ~1 if determined by the ball perimeter. It would be ~0 if the resistance is independent of ball radius.
There may be an empirical constant of proportionality in the equation, but what I would like to know is how I could use physical reasoning to fix the exponent on the radius to an integer, and then fit this model to my experiment.
ADDENDUM: After reading some of the references suggested in the comments, I saw that my question was more naive than I realized. The subject of rolling resistance is large, complex, and with significant engineering applications. My requirements for an answer are likely more modest than what I now think might be offered here. Really, what I would like is a bit more understanding of the primary cause of rolling resistance, the underlying physics, and enough mathematics to be able to relate this to what I currently know about mechanics. Clearly experimentation is necessary to really understand this accurately, but some theoretical background would be useful for experimental design or explaining it to students.