dpapavas

11-24-2007, 12:38 PM

Hello all,

most analyses of friction on a billiard ball I've seen begin with a ball being hit without sidespin or follow/draw. Sliding friction brings it then to natural roll at which point rolling friction kicks in to decelerate it to rest.

What happens if you use sidespin though? If I'm not mistaken the initial sliding phase will still terminate in natural roll but the spin axis of the ball will then not be in the XY plane and perpendicular to the velocity vector. It will still be at right angles to the velocity vector but at some angle to the z axis which depends on the linear velocity. I haven't come to this conclusion analytically. I'm developing a billiards simulator and this is the behavior I get with only sliding friction: The spin vector starts off horizontal and rises to some angle theta relative to the z axis. It also rotates on the xy plane. Once it has reached this state it stays there and the ball rolls and spins at the same time with the velocity of the contact point with the table being zero. Again this is with only sliding friction.

If I implement rolling friction as a force acting at the center of mass of the ball with magnitude mu_r * N and direction against the linear velocity vector and also a matching torque to keep the ball rolling then this happens: The initial phase is the same as before but once the spin and velocity vector reach their "steady" positions the spin vector slowly starts to turn towards the z axis and the velocity vector diminishes. That is the rolling friction decelerates the ball but leaves the z spin component intact. At the end the ball remains spinning like a top with the contact point still having zero velocity.

This seems reasonable to me. So a third friction force or rather torque is needed. A "spinning friction" torque. Assuming a torque of magnitude mu_s * N where mu_s a suitable coefficient an N the normal force and direction opposite the spin direction the behavior is this: After the initial sliding phase the spin vector "falls" from its initial direction to the XY plane and the velocity vector diminishes. That is the extra friction torque brings the ball to a "textbook" natural roll state.

Now to my questions to the physics-savvy members of this forum are:

1) Does this behavior seem accurate to you?

2) Is the coulomb-like model for "spinning friction" correct or do you know of anything better?

3) Would you know of any experimental data regarding the coefficient of spinning friction? If not would it be easy to measure it experimentally?

I've googled for answers quite a bit (including papers on spinning tops etc.) but haven't found anything interesting so far. Any help is greatly appreciated.

PS1: It's a pity I can't show you videos of all this in action but I haven't succeeded in capturing video of my simulator. I'll keep trying though.

PS2: Do you btw know of any experimental data regarding the coefficient of restitution between ball and table? This is one of the easiest coefficients to measure and I haven't found anything on this. I need it to make jumpshots and elevated cue stick shots in general work correctly.

most analyses of friction on a billiard ball I've seen begin with a ball being hit without sidespin or follow/draw. Sliding friction brings it then to natural roll at which point rolling friction kicks in to decelerate it to rest.

What happens if you use sidespin though? If I'm not mistaken the initial sliding phase will still terminate in natural roll but the spin axis of the ball will then not be in the XY plane and perpendicular to the velocity vector. It will still be at right angles to the velocity vector but at some angle to the z axis which depends on the linear velocity. I haven't come to this conclusion analytically. I'm developing a billiards simulator and this is the behavior I get with only sliding friction: The spin vector starts off horizontal and rises to some angle theta relative to the z axis. It also rotates on the xy plane. Once it has reached this state it stays there and the ball rolls and spins at the same time with the velocity of the contact point with the table being zero. Again this is with only sliding friction.

If I implement rolling friction as a force acting at the center of mass of the ball with magnitude mu_r * N and direction against the linear velocity vector and also a matching torque to keep the ball rolling then this happens: The initial phase is the same as before but once the spin and velocity vector reach their "steady" positions the spin vector slowly starts to turn towards the z axis and the velocity vector diminishes. That is the rolling friction decelerates the ball but leaves the z spin component intact. At the end the ball remains spinning like a top with the contact point still having zero velocity.

This seems reasonable to me. So a third friction force or rather torque is needed. A "spinning friction" torque. Assuming a torque of magnitude mu_s * N where mu_s a suitable coefficient an N the normal force and direction opposite the spin direction the behavior is this: After the initial sliding phase the spin vector "falls" from its initial direction to the XY plane and the velocity vector diminishes. That is the extra friction torque brings the ball to a "textbook" natural roll state.

Now to my questions to the physics-savvy members of this forum are:

1) Does this behavior seem accurate to you?

2) Is the coulomb-like model for "spinning friction" correct or do you know of anything better?

3) Would you know of any experimental data regarding the coefficient of spinning friction? If not would it be easy to measure it experimentally?

I've googled for answers quite a bit (including papers on spinning tops etc.) but haven't found anything interesting so far. Any help is greatly appreciated.

PS1: It's a pity I can't show you videos of all this in action but I haven't succeeded in capturing video of my simulator. I'll keep trying though.

PS2: Do you btw know of any experimental data regarding the coefficient of restitution between ball and table? This is one of the easiest coefficients to measure and I haven't found anything on this. I need it to make jumpshots and elevated cue stick shots in general work correctly.