# How to calculate friction on a spinning ball on the ground?

If there is a ball on the ground initially at rest, Assume that the plane xoy(cartesian) is the ground and that Z is the normal of the ground.

If I spin the ball so that $\omega_x=\omega_y=0$ and $\omega_z$ has some value, how to calculate the effect of friction on the spin ? .

Assume we have the mass of the ball $m$ and its rotational inertia $I$ and the torque I applied on the ball to spin it is $\tau$, also assume that air resistance is negligible .

This is the problem, Now I know that the friction always oppose the relative motion but I don't exactly know how much the force will be and how to calculate resistant torque from that.

Not sure but I feel due to the fact that there is $\omega_z$ there is added Normal force from the ground because somewhat the ball moves towards the ground but that's wrong if $\omega_z$ is the same in direction as the Normal of the ground.

Also the speed of the contact point relative to the ground is zero because the contact point is part of the rotation axis, that's what confuses me : how is there a friction when there isn't any relative speed? is there "frictional torque" as opposed to "frictional force" or I'm wrong? please help me with this problem.

## 1 Answer

You are right. In an optimal system, there isn't any friction. But in real life there is, because there aren't any balls or surfaces that are so perfect that they touch just in one point.

Also the surface of the ball is never perfectly flat, so there will always be friction between the surface of the ball and the surrounding air.

So for calculating the friction, you need to know the exact area in which the ball and the ground touches. And you have to include the friction between the moving surface of the ball and the air.

• hmmm so you say that friction comes from points hitting the ground as the ball spin ? you are right but how to know exactly which points hit and how to integrate all of those small effects ? is there some equation ? Jun 9, 2015 at 13:11
• I assumed air resistance is negligible, isn't it ? Jun 9, 2015 at 13:22
• I think you can use "boring friction" or "spinning friction" with a very small radius for the area in which the ball and the grounds touch. In the moment i can't find a good equation, but if i find one, i will let you know. Jun 9, 2015 at 13:28
• i don't know if air resistence is negligible. Cause if you have a very good iron ball on an iron surface, the air resistence wouldn't be irrelevant, cause there is nearly no friction with the ground. But if you spin a a rubber ball on dirt, you don't need to calculate the air resistance. Jun 9, 2015 at 13:32