# Moment of Inertia of Euler's Disc

I was thinking about the Euler's disc and why its angular velocity increases over time. The conclusion I arrived on is that the gravitational potential energy of the disc gets converted to the rotational kinetic energy (I have assumed there is no translation). But when I tried to write down conservation of energy I had problems with the determining the moment of inertia term of the disc. There are two axis that the disc is spinning about; one is passing perpendicular to the disc and through its centre and the other is perpendicular to the floor. The moment of inertia about central axis is $$I=\frac{1}{2}MR^2$$ But I was not able figure out the inertia about the other axis. Also what is the net moment of inertia?

Can someone help me with this problem?

• The MMOI across a disk is $\tfrac{1}{4} M R^2$. Also I think the precession increases while the spin decreases (due to friction). – ja72 May 6 '19 at 19:55
• @ja72 that is about the x-axis or y-axis. I have clearly states that the disc is rotating about the central axis, i.e. the z-axis. – Manvendra Somvanshi May 6 '19 at 19:57
• real-world-physics-problems.com/eulers-disk.html – ja72 May 6 '19 at 19:59
• @ja72 thanks for the link. – Manvendra Somvanshi May 6 '19 at 20:02

For a disk of radius $$R$$, mass $$m$$, and MMOI of $$I$$, the total energy is $$E = m g R \sin \alpha + \tfrac{1}{2} I \Omega^2 \sin^2 \alpha$$ where $$\alpha$$ is the inclination angle w.r.t. horizontal plane, and $$\Omega$$ the precession rate.
As you can see when the inclination $$\alpha$$ becomes smaller, the precession must increase in order to maintain the overall energy approximately constant.