# Calculate total angular momentum of precessing and spinning wheel, then use result to prove gyroscopic precession formula

Consider the following scenario: https://youtu.be/8H98BgRzpOM?t=27.

How would I calculate the total angular momentum of this system? The spinning wheel is rather easy, and it's $$L_{\rm wheel}=I\omega$$. However, I am not sure how to account for the angular momentum due to precession. Is it simply $$L_{\rm press} = I\Omega$$? If so, how would I calculate $$I$$? Would I simply use inertia of a disk and the parallel axis theorem to find it?

I want to use the total angular momentum to prove the formula $$\Omega=\dfrac{mgr}{I\omega}$$ by using the fact that $$\vec{{\tau }}=\dfrac{d{\vec{L}}}{dt}$$.

Lastly, I have a conceptual question: $$\Omega$$ is said to remain constant assuming $$\omega$$ is also constant. However when the wheel is held still, $$\Omega$$ is clearly zero, until it is released, and then $$\Omega$$ becomes non zero. Why does $$\Omega$$ rise and then remain constant?

• For the process of starting gyroscopic precession, see the answer I wrote (2012) about the mechanics of gyroscopic precession Nov 17, 2020 at 20:51
• Consider to include an image of the system to make the question self-contained. Nov 17, 2020 at 22:15

I'm not sure, but I think that expression $$\Omega=\dfrac{mgr}{I\omega}$$ is in fact only valid (in appriximation) when neglecting the angular momentum associated with the precessing motion.
• Ah yes, so the fact that $\Omega$ can suddenly pick up and stop is actually not true, merely the mathematical consequence of the approximation. Nov 18, 2020 at 4:30