how does product of inertia help in locating the plane of an additional mass in rotational stability problems.

Answer 1

To use dynamic wheel balancing as an example, if you know the products of inertia of the unbalanced wheel, the applicable terms (Pxy and Pxz if x is the rotation axis) are the M*R^2 terms needing correction. If you know Rx (the location of the corrective weight attach point relative to the center of mass, measured parallel to the rotation axis), then with a little algebra you can compute the other location axes Ry and Rz (since you know the radius of the attach point), and the mass M of the required balancing weight.