Double gyroscope: Can a spinning pencil tumble on only one axis?
Picture an object such as item 7 on this page .. http://en.wikipedia.org/wiki/List_of_moment_of_inertia_tensors
Call that the x axis and z is in to the distance. See diagram below.
We are in deep space. The object is 10 meters long, a couple inches diameter and weighs 100kg.
OK, we make it spin on it's own length. Let's say at about 2 hz.
So it's spinning on it's own long axis, our x axis.
Now for clarity, we will refer to "All the points along the center of the rod" as ATPATCOTR.
So far it's only spinning on the x-axis. Looked at from any anywhere, it is of course obviously unchanging: Note that every ATPATCOTR is completely stationary, the coordinates x,y,z of every ATPATCOTR never change in any way.
Now: we make it also tumble on the z axis. So the left end begins to move slowly down and the right end begins to move slowly up. The whole thing tumbles end-over-end, let's say once every few seconds.
To be clear, every ATPATCOTR now descibes a perfect circle in the XY plane, with z always zero.
BUT WAIT -- here's the question.
Now, if you actually try to do this, typically it will also wobble. 53
What do I mean by wobble? Look at it from overhead, and it will be varying it's angle - note the second diagram. (ie, the angular velocity along the y axis is no longer zero - it goes back and fore a bit.)
It is no longer the case that ATPATCOTR make perfect circles (z=0), they are forming all sorts of weird spirals.
THE QUESTION -- in fact, quite simply, IS IT POSSIBLE in this universe, to have the object both spin on the x axis, and, tumble on the z axis, CLEANLY with NO rotation or angular velocity in the y axis?
In other words grab your pencil and throw it away from you, spinning it along it's length, but also make it tumble end-over-end a little through the air. In fact, can the tumbling motion be perfectly clean in one flat plane? Or is that actually impossible? (ie, you would have to use two glass sheets that forcefully retain the tumbling motion to be in one perfect plane only?
In other words, make a platonic pencil spin on it's length and lay it on a platonic frictionless table. Now make it additionally spin like a baton as seen from above. Is that in fact "impossible", i.e. it will be angry that the flat table is holding the baton-spin to one perfect plane and it will force against it?
In other words: http://www.youtube.com/watch?v=wfCgCXvw50A
it looks like (A) the board is spinning on it's own length, and, (B) slowly rotating on the z axis (using our diagram). Superficially (C) there's no rotation from overhead, rotation on the y axis. But is that simply wrong - if we observed closely there would be some rotation/wobble on the y axis also?
In other words, can you spin a pencil along it's length, and also have it tumble in one perfectly flat plane only? Or will there necessarily be "wobble", essentially some rotation or change of angle in all three axis?
In this universe can an object in fact spin in "two axis" only, or is that impossible? You can (I think!) have an object spin cleanly in one axis, but after that do you have to go straight to all-axis rotation?
Thanks! 53