# How does the resistance force on a rolling ball depend on the ball radius?

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.

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Hint: search for hertzian contact with friction for a ball on plane. When you also consider slipping and rolling the problem is actually rather complex. – ja72 Nov 28 '12 at 1:44
It is apparently far from being settled: en.wikipedia.org/wiki/Rolling_resistance#Depends_on_diameter – Jaime Nov 28 '12 at 3:58
ja72 and Jaime - thanks. Both comments were helpful in locating more information about this problem than I found on my own. Sometimes the first step is finding the right name or word for a thing. I've encountered the Hertzian contact fields in respect to hardness testing, and they certainly could come in handy here. Of course I should have recognized this is an instance of a wheel, a simple machine, and therefore a topic with lots of information available - Jaime's link answers my question. I will post an answer summarizing what I learn in a day or two. – Mark Rovetta Nov 28 '12 at 18:06
The assumption in the question is that the ball rolls without slipping. This may be a good approximation in many cases, but balls do slip when rolling after they are hit by the cue stick, rebound from a cushion, or are hit by another ball. – Ben Crowell Apr 29 '13 at 3:03

Here is my answer to my own question, based upon what I learned from the suggestions in comments.

The primary cause of the rolling resistance experienced by a billiard ball rolling across a billiard table is the transfer of (kinetic) energy from the ball to the table through hysteresis at the ball-table contact. A billiard table top consists of a cloth tightly stretched over a rigid base. To an approximation, the table-cloth-ball contact possesses characteristics of a linear viscoelastic material. As the ball rolls along the surface, its weight deforms the composite material (the ball and table-base deform little, the cloth much more,) the material rebounds upon unloading, but there is some dissipation of energy as heat (again mostly in the cloth.) As the ball loses kinetic energy, it slows.

If F is the rolling resistance force slowing the ball and N is the weight of the ball, a rolling resistance coefficient, C, can be defined as:

$$F=C\;N$$

The following is presented here and elsewhere as a physical formula for the rolling friction of a slow rigid wheel on a perfectly elastic surface, but I’ve so far been unable to derive it (or locate the original source.) Where z is the sinkage depth and R is the ball radius.

$$C = (\frac{z}{2 R})^\frac{1}{2}$$

The classical solution for the contact between a sphere and half-space gives the relation of sinkage depth to load (ball weight,) ball radius, and an effective modulus, E.

$$N = \frac{4}{3} E\, R^\frac{1}{2}\; z^\frac{3}{2}$$

Expressing the ball weight using radius and ball density, this analysis suggests that F may be proportional to $R^\frac{1}{3}$. So the exponent is not really close to any of the integer values I guessed at in the original question.

Some values for the physical properties of balls and tables can be found here: Physics of Pool and Billiards

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