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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?

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  • $\begingroup$ 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). $\endgroup$ – ja72 May 6 '19 at 19:55
  • $\begingroup$ @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. $\endgroup$ – Manvendra Somvanshi May 6 '19 at 19:57
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    $\begingroup$ real-world-physics-problems.com/eulers-disk.html $\endgroup$ – ja72 May 6 '19 at 19:59
  • $\begingroup$ @ja72 thanks for the link. $\endgroup$ – Manvendra Somvanshi May 6 '19 at 20:02
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See equation (1) of this paper describing the total energy of the disk.

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.

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