Help with parallel axis theorem?

[merry]

help with parallel axis theorem??

Hey guys,
I've attached a picture from my textbook (Intro to Classical Mechanics by David Morin) showing the beginning of the proof for the parallel axis theorem. I understand most of it except the sentence where it states that if you glue a stick to the body, the centre of mass will rotate at the same rate around the origin as the body will around the centre of mass (1st two lines in the paragraph attached).
If a stick were glued to the mass, it wouldn't rotate about the centre of mass right? yet the centre of mass will rotate about the origin, why does the book state otherwise tho?
Thanks!

 

[Doc Al]

If a stick were glued to the mass, it wouldn't rotate about the centre of mass right? yet the centre of mass will rotate about the origin, why does the book state otherwise tho?

When the book states that the body will rotate about its center of mass, they don't mean that its center of mass will be stationary, if that's what you're thinking. As the entire stick+body rotates about the origin, watch how the body changes its orientation with respect to its center of mass--it rotates about the moving center of mass. (Nothing special about the center of mass in this regard.)

 

[merry]

Wouldnt the centre of mass and all the other points around it have the same angular speed with respect to the origin though? If that would be the case then the angular speed of those points with respect to the centre of mass would be zero right?

 

[Doc Al]

Wouldnt the centre of mass and all the other points around it have the same angular speed with respect to the origin though?

Yes.

If that would be the case then the angular speed of those points with respect to the centre of mass would be zero right?

No.

Actually take a paper disk and a straw and play around (or at least imagine doing it). Say the disk had four arrows marked north, south, east, and west. As the whole thing rotates about the origin, the orientation of those four arrows changes. If all points of the disk did not rotate about the center of mass, then the arrows would always point in the same directions. But as long as the straw is glued to the disk, the arrows will change directions as they rotate.

 

[merry]

Yes.

No.

Actually take a paper disk and a straw and play around (or at least imagine doing it). Say the disk had four arrows marked north, south, east, and west. As the whole thing rotates about the origin, the orientation of those four arrows changes. If all points of the disk did not rotate about the center of mass, then the arrows would always point in the same directions. But as long as the straw is glued to the disk, the arrows will change directions as they rotate.

I don't quite understand =[
The points around the centre of mass always maintain the same constant position around it, i.e. the body doesn't rotate ABOUT the centre of mass. As the position of the centre of masss changes relative to the origin, so do the positions of the other points. I seem to be missing some vital point.

 

[Doc Al]

I don't quite understand =[
The points around the centre of mass always maintain the same constant position around it, i.e. the body doesn't rotate ABOUT the centre of mass. As the position of the centre of masss changes relative to the origin, so do the positions of the other points. I seem to be missing some vital point.

Those points maintain a constant position when viewed from a frame fixed with respect to the rotating object, not when viewed from an inertial frame.

To determine whether that disk rotates about its center, imagine that there's an arrow painted on it. As the center translates (or moves about the axis) what happens to that arrow? If its direction doesn't change, then you can say that the disk did not rotate about the center. But in this example, it obviously does change direction--thus the disk does rotate about the center. (That's not all it does, of course. The motion of the disk can be described as a movement of its center plus a rotation about its center. The net effect is that the entire object is in pure rotation about the axis.)

 

[merry]

I never thought of it that way. I guess from the cm frame, it itself is stationary and the points around it rotate about it. It makes sense now. Thanks very much! =]