Finding the 3 Axes of a Sakai Gyro

[Icheb]

I have a Sakai gyro that looks like this:
http://img165.imageshack.us/img165/4556/sakainn4.gif

Now I have to find the three main axes of the moment of inertia by considering symmetries.

I put the two axes into the picture which I think I know. The first one would be the upright because it is rotating around that axis.
The second one would be the one in the plane of the ring because one half of the ring is symmetrical to other half.
But where would the third axis lie? I can't seem to find another symmetry. At first I just thought it's at a 90° angle to my second axis, but one half wouldn't be symmetrical to the other half.

 

[berkeman]

Wow, that doesn't seem like an Introductory Physics question past the first axis. Would you like me to move it to the Advanced Physics homework forum?

And past that, it looks like you need to use a discrete integral to calculate the moment if inertia of that figure in the other axes. What are the integral equations that you would start with?

 

[OlderDan]

You cannot always have three axes of symmetry. In fact you cannot always have two or even one. You have picked two reasonable axes based on the symmetries you have identified. Take the third on to be perpendicular to the other two. It is probably giving you too much information, but a google search on the Sakai top brings up a rather nice paper on the subject

http://www.e20.physik.tu-muenchen.de/~cucke/ftp/lectures/SAKAIEN.PDF