A homemade mechanics problem: A beetle on a globe

[wrobel]

Yes and It is not a surprise. If $b = 0$, then the beetle is at rest, and thus the ball does not move. If, in the theorem cited in #29, the initial position coincides with the final position, then the axis of rotation is not defined uniquely .

 

[anuttarasammyak]

As an example, for b=R/2, this vector is
$$(\sqrt{3},0,1)^T=2* ( \frac{\sqrt{3}}{2}, 0 , \frac{1}{2} )^T$$
I would like to understand how this (x,0,z) type vector of magnitude $\frac{R}{b}$ tells us about the final angle position of the ball.

 

[wrobel]

Once again: this vector is the direction vector of the ball's rotation axis. The magnitude of this vector does not matter. The angle of rotation about this axis is given in #27.