Re: Can an object rotate freely about two or more axes in space?

It turns out (No pun intended!) that for a freely-rotating object without any torques applied the object will generally rotate about one axis, but a combination of rotations is possible. Consider, for instance, a coin. On this neat page the moments of inertia are given for the disc, and it can be seen that the moment around the axis normal to the plane of the disc is exactly twice that of either of the orthogonal axes (Due to symmetry the location of the orthogonal axes is arbitrary until one is chosen.). It is possible (Try it!) to cause the coin to rotate about the normal axis at the same time that it is rotating about one of the orthogonal axes, in a free flip (Okay, there are some [very small] torques being applied by gravitational acceleration and by air resistance, but in the small amount of time that it is flipping through the air the torques have almost no effect.). The easiest rotations to impart to the coin are, though, either around the axis normal to the plane, or around an (arbitrary) axis orthogonal to it (This would be the same axis as when it would be "spinning on its edge", but in a free flip it is not in contact with, for instance, a table.).

If torque(s) is(are) applied then the situation is either stable or unstable depending on the shape of the object and depending on the axis of rotation. For an object rotating about a principal axis having the smallest moment of inertia, it is stable, and may only precess. If the object is rotating about a principal axis having the largest moment of inertia it can also be stable. However, if the axis of rotation is the principal axis having intermediate (between smallest and largest) moment of inertia then the motion is unstable with applied torque(s).

There is a really good "dynamics" site on the Internet that goes into excruciating detail about the subject. It is found here, and the page that is of most interest to this question is this page. The math is fairly easy because it is mostly vectors and matrices. However, the interpretation of the results is a bit tricky, and it is important to keep in mind all the definitions of the various variables.

John Link, MadSci Physicist