Rolling Without Slipping question

[CEJ__]

Hello:

In my readings a question has come up in my mind about a ball that is rolling. Say you have a pool ball on a billiards table and you examine it immediately after impact. It's going to be sliding along the table. But at some point, it's going to stop sliding and (ideally) start rolling without slipping. What happens at that point that causes it to start rolling?

I'd imagine that it has a lot to do with the strength of the frictional force, but the frictional force has to overpower something else. If, for example, you had a table of ice, the ball would slip for a very long time because the force of friction is so small. And, conversely, if you have an extremely rough surface the ball would start rolling rather quickly.

So, again, what happens at the time that the ball begins to roll?

Thanks,
-CEJ

 

[Gear300]

well...I'm not exactly sure how good or accurate this answer is...but for the something to rotate around an axis, you'd need a net torque through that axis. Objects rotate relatively easily, so I'd have to say the ball is rotating...just that compared to its initial translational speed, its not very noticeable. Once the translational speed is slowed to some point due to friction, the rotation becomes more noticeable.

 

[atyy]

I think there's sliding, rolling and sliding, then pure rolling.

 

[Razzor7]

I think there's sliding, rolling and sliding, then pure rolling.

Agreed, and the graph of rotational velocity to time is probably a smooth curve.

 

[DyslexicHobo]

Because you hit the ball directly in the center, you aren't applying any torque. No torque means no angular speed (angular speed = rolling). However, when the ball begins to travel without rolling, the frictional force of the table on the ball applies a torque on the ball (a force perpendicular to the surface of the sphere).

[itex]\tau = I*\alpha[/tex]
torque = moment of inertia * angular acceleration (Newton's F=ma in its rotational form)

As the ball's angular velocity increases with time, it gets to the point that it only rolls and does not slide.


At least this is how I envision it... Not 100% sure though.

 

[Danger]

Because you hit the ball directly in the center, you aren't applying any torque.

This is correct in theory, but in reality you never hit the ball in the centre (except for possibly on the break shot). A normal stroke is about 1/4 - 1/2 tip diameter above centre. If you play bottom for shape, the cue ball is rotating opposite it's direction of travel (which is sliding) until it contacts something, then it backs up.

 

[Burningmace]

If you put backspin on a ball, you are making the ball's velocity relative to the surface positive and the angular velocity relative to the surface negative.

Friction slows both the velocity and angular velocity. When the angular velocity becomes zero due to deceleration from friction, a change takes place. The friction acts only on the part of the ball that has contact with the surface, putting a force on the ball in an opposite direction to the one it is going in. However, there is no friction from the table at the top of the ball so its speed stays constant (ignoring air resistance).

Because the bottom of the ball is decelerating and the top of the ball is staying at constant speed (and the constraint exists that ball cannot rip itself apart) it must begin to rotate, so it has stopped sliding and has started rolling.

 

[rcgldr]

The math for initial all sliding, no rolling case was covered in this thread:

https://www.physicsforums.com/showthread.php?t=159337

Relation between linear and angular forces on an object (sphere used as example):

https://www.physicsforums.com/showthread.php?t=184624

 

[Phrak]

It's not clear to me that there is, or is not, an abrupt transition from sliding to rolling.

It would seem to be a smooth transition (asymptotic), as the frictional force that decreases angular velocity is proportional to the difference in velocities at the contact.

Then again, how slowly can two surfaces slide before the static coefficient of friction is dominant over the sliding coefficient, where the question is further complicated by the action of rolling?

 

[rcgldr]

It's not clear to me that there is, or is not, an abrupt transition from sliding to rolling.

In real life, it would be somewhat abrupt, depending on the amount of deformation of the sliding surfaces. Eventually the speed differences will be slow enough that the deformations will lead to a transition into a static friction state. Depending on how elastic (energy conserving) the deformation reaction is, the actual work done by friction will be a bit less than the ideal case in the math from the above links. As an extreme example, imagine a ball sliding onto a totally elastic surface with infinite friction, the surface deforms in the direction of the ball, then springs back to it's original position, and there is no work done on the ball, just a energy conserving conversion of linear energy into linear + angular energy.