Hi - I know that a spinning object resists being rotated in dimensions that are different from its rate of rotation. The classic example of this is a top that's spinning and resists the downward torque due to gravity.

My question: if we ignore the effect of precision, how can we characterize this torque? I know this should be an easy question, but my naive googling around has not found the answer. I swore (25 years ago) I remember doing this derivation in my physics classes, but it was some time ago. I'm trying to recreate this and making a big mess of it and I'm hoping that someone can help point me in the right direction.

One way to look at this is from a change in rotational energy, e.g., 1/2 (rotational inertia_z) (angular velocity)^2 or 1/2 I w^2.. So, if an object is spinning purely in the z dimension, its angular velocity in vector form would be w1' = 0 w_x + 0 w_y + k w_z. where 'k' is some initial speed. And then if a torque was applied to the object to turn it 90 degrees on its side, it's new angular velocity would be w2' = 0 w_x + k w_y + 0 w_z.

Then in theory, the torque applied time the angle is the work done which should equal the change in energy from the initial to the final states, right? In other words, if we want to know the torque done T,

T * radians = E1 - E2

T * radians = 1/2 I_z ((w2)^2 - (w1)^2))

T * radians = 1/2 I_z ( (k w_y - 0 w_y)^2 + (0 w_z - k w_z)^2 )^2

T = I_z |k| / (\pi/2 radians)

Does this seem right? I feel like there's some flaw in my logic or math because I haven't seen this clean of an answer anywhere.

Thank you in advance!