# Rolling balls angular rotation

1. Apr 17, 2010

### SpartanG345

If a cylinder is rolling without slipping, C is the centre of zero velocity for a moment and O is the centre

Does the angular rotation about O equal to the angular rotation about C, or is there only one angular rotation when a cylinder is rolling, that is the rotation about the point of contact?

3. The attempt at a solution

This is a question that came to me, not a assignment question or anything, but anyway

I think in the ordinary frame of reference a rolling cylinder only has an angular rotation about the point of contact, not about the centre

Where in a frame of reference where you are following the cylinder, you should see the cylinder rotating about the centre and the surface is moving linearly without sliding.

I know angular rotation is always about an a line, so a single motion can have many angular velocities with respect to many axis's.

Is it possible to evaluate the angular velocity with respect to O in the normal frame of reference when the ground is stationary?

i am not really sure i guess the angular velocity for each point on the shape would vary if you measure it from O since the whole object is kind of translating... and since C has a zero velocity

2. Apr 18, 2010

### tiny-tim

Hi SpartanG345!

(btw, better to say "instant" rather than "moment", so as not to confuse with other types of moment )

Angular velocity (unlike angular momentum) is the same about any point.

Angular velocity is a "free" vector (strictly, a "free" pseudovector), so (unlike force) it has a direction, but not a specific line in that direction.

Changing to a different inertial frame will, of course, alter the velocity, but will not alter the angular velocity.

Formulas that combine I and ω use the same ω, no matter whether I (the moment of inertia) is about the centre of rotation or the centre of mass (btw, they don't generally work about any other point).
Yes, you get τ = IOω, instead of 0 = ICω - rmv, which is the same since IC = IO + mr2.

(I've used your notation, but usually we use C for centre of mass, and O for centre of rotation )

3. Apr 18, 2010