Does angular velocity obey vector addition?

Question

Suppose I take a water bottle and impart two angular velocities on it simultaneously, i.e.

1. along the axis of the bottle

2. of the axis of the bottle itself about another axis perpendicular to it

what I see is not a simple rotation of the bottle along a third axis (in the direction of the diagonal of the parallelogram created by rotation "vectors" 1 and 2) but rather a complicated kind of motion. But then isn't this a contradiction to the kind of treatment we see in textbooks? Or is it any kind of a tensor? (PS: I don't have any advanced physics background but am somewhat well informed about the basic stuff)

Answer 1

http://farside.ph.utexas.edu/teaching/301/lectures/node100.html basically says you can at an instant in time or instant-to-instant, but not over an interval of time.

You're not doing your little exercise properly. Redo it using a ball with markings on it. Something like a bottle is really bad for this because the asymmetrical shape causes all sorts of illusions and prevents you from even holding the bottle to spin it around any axis other than the orthogonal ones to try and verify the rotation around the resultant axis.

Example: I am holding a squash ball up in front of my face between my left index and thumb with each finger on the north and south pole respectively. The flash mark from the molding of the ball runs around the equator. I then rotate my left wrist so north moves away from me and south moves towards me.

As I do this, I simultaneously grab the equator with two fingers in my right hand and rotate it from west to east.

I don't know about you, but for me at least, I instinctively look to the top right, expecting one that the axis of rotation passes from the bottom left through to the top right. Except that's wrong. The resulting rotation is passing from the bottom right to the top left.

Even then, it might still not be obvious due to differences in speed as you rotate both axis. It gets a lot more obvious when you observe how the equatorial mark is moving, and then use your left index and thumb to grasp the NW and SE pole and just spin it about that one axis. The equatorial mark moves the same way. So, it checks out.

Answer 2

A proper treatment requires lie algebra. The idea is that rotations do not obey a vector space structure except when we consider very small magnitude rotations.

For discrete calculations, you will find as you said but if you have a continuous motion with angular momentum around individual axes and then consider the superposition, you will see that both are same due to the situation described above.