Force Free Motion of a Symmetric Top: Direction of phi dot

[kennethellen]

Homework Statement

The following question is example 11.12 from Classical Dynamics of Particles and Systems by Thornton.

For a force free symmetric top with what angular velocity does the symmetry axis x_3 and omega rotate about the fixed angular momentum L?

I had no trouble finding the magnitude of the angular velocity about L, phi dot, however I cannot figure out the direction of phi dot. Since phi dot is the rotation about L I figured that the phi dot would be directed along the axis where L lies but apparently this is incorrect as it would make phi dot indistinguishable from omega. My second guess is that it would be directed along omega but I cannot think of a reason to justify this answer.

My professor told me to try and express phi dot in terms of its vector components to try and see the dependencies and relations. I did that but I still don't understand where phi dot is supposed to be directed towards.

Relevant Equations

The given figures and the solution from the book are attached. As well as my attempt at expressing phi dot in terms of its vector components.

 

[timetraveller123]

i am no expert but i will try
i think what he meant is to break omega into x3 axes and x3'axes
you get two rotations about x3 axis and about x3'.
if instead you break omega into x3 axis and the axis perpendicular to it you get two rotations one about the x3' axis and and one about the 'horizontal axis' the rotation
i am guessing on this one if after a small time the rod would have moved a little bit into the page due to x3' rotation but at the same time a little bit out of the page due to the horizontal rotation
dont hold me on this though meanwhile in the if you break into x3 and x3' you don't get this