# Rolling with and without slipping

1. May 2, 2006

### Benny

Hi, I don't understand the difference between rolling with and without slipping. To put my question into some kind of context, consider a ball of radius r which rolls to the right along the x (horizontal) axis. The ball is travelling at a constant velocity v and continues rolling until it reaches a curved hill, eventually stopping at some distance H above the x-axis. If there is no friction between the x-axis and the ball then H can be found by solving (1/2)mv^2 + (1/2)Iw^2 = mgH...(1).

That is assuming that there is no 'slipping.' But what does slipping actually mean in this context? I was told that if there was 'slipping' then the value of H would be less than that obtained by solving equation (1). So if there is no slipping then does that mean that there cannot be any friction? It's all quite confusing to me. In particular, I don't understand the following means.

Let G be the COG of the ball which is rolling. Let point A be the point of contact between the ball and the x-axis (as a visual aid, the line segment AG is perpendicular to the x-axis). I vaguely remember certain consequences of slipping and no slipping in this situation but I'm not really sure. If there is slipping (or no slipping), then can something be said about the relative velocity of the points A and G?

Any help would be great thanks.

2. May 2, 2006

### robphy

"Rolling without slipping" means that the point-of-contact has an instantaneous velocity of zero. There is no sliding there... hence no sliding friction.

G, travelling with constant velocity, is the green line
B (in blue) is the rotation of a point around the wheel axis--that point is initially in contact with the ground
A (in red) is the superposition of the two motions...
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Note that A has zero velocity when it makes contact with ground.
Of course, A is just representative of one point on the wheel.
Each point has it own set of curves, slightly displaced in time from A.

For this to happen, $$v_G=\omega R$$.

Last edited: May 2, 2006
3. May 3, 2006

### Benny

So if there is relative motion between two surfaces in contact then there must be a friction force?

4. May 3, 2006

### robphy

Yes...but its magnitude is proportional to the coefficent of kinetic/sliding-friction.

5. May 3, 2006

### Staff: Mentor

Just to be clear: There is certainly friction between the ball and the ground, but it will be static friction. (As robphy explains, there will be no sliding/kinetic friction.) If the surfaces were frictionless, then the ball's rotational speed would not change as it went uphill.

Just to be clearer: As the ball rolls without slipping on the horizontal surface, the static friction force is zero. But as it rolls uphill, there will be a non-zero static friction force on the ball.

6. May 3, 2006

### robphy

Just to add one thing that hasn't been explicitly mentioned...
since there is no relative motion at the point of contact when "rolling without slipping", no work is being done by the frictional forces.

7. May 5, 2006

### Benny

Thanks for the help guys.