How do you add angular momentum in different dimensions?

[24forChromium]

Say a ring is spining around the z-axis, an angular impulse is then applied to it in the x-axis, what is the resultant motion qualitatively and quantitatively? How can it be calculated?

(You can make up the quantity of z-angular momentum and x-angular impulse)

 

[dipole]

I assume you mean that the impulse is delivered when the ring is perfectly in the x-z plane. Initially the ring has angular momentum \vec{L} = \omega_z\hat{z}, and the impulse introduces some angular momentum \omega_x\hat{x}. The total angular momentum will then be the vector sum of these:

\vec{L'} = (\omega_x, 0, \omega_z), and given the rotational symmetry of the ring, it will start to rotate about an axis parallel to this new vector, which now lies in the x-z plane. If the ring is spinning on a table, for example, then there is the additional complication of torque on the ring due to gravity, and it will precess about this axis.