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PC1672 Advanced dynamics

Research project

(This can be undertaken in the pub, on your kitchen table, or anywhere with a hard flat surface.)

Part 1

Spin a coin on a table and watch its behaviour. To understand the main features of its motion, assume that the coin rolls in a circle without slipping and that its centre of mass does not move. Also, ignore air resistance etc. Call the mass of the coin $m$ and its radius $a$. The point of contact between the coin and the table moves round the circle with angular velocity $\Omega$. The angle between the coin and the table is $\alpha$.

Treat this problem like the ``slow top'' we looked at in 4.9 Gyroscopes and introduce a precessing frame which rotates about the vertical axis at the rate $\Omega$.

Show that in this frame the coin rotates about its axis of symmetry at $\omega_3^\prime=-\Omega\cos\alpha$. Hence show that total angular velocity of the coin with respect to an inertial frame is $\Omega\sin\alpha$ about a diameter of the coin.

Use the requirement that the angular momentum of the coin should be constant in the precessing frame to show that $\Omega$ and $\alpha$ are related by

\begin{displaymath}\Omega^2={mga\over I_1\sin\alpha}\end{displaymath}

where $I_1$ is the moment of inertia of the coin about a diameter. [This explains why the coin rolls faster and faster as it sinks towards the table.]

Finally, show that the component of the coin's angular velocity about the vertical axis tends to zero as the coin sinks to the table. [You should have seen the coin turning more and more slowly while its rolling speeds up.]

Part 2

Now try to extend your description to include dissipation of energy. Possible sources of this include: air resistance, rolling resistance, slipping and friction. Decide (by experiment?) which of these are most important. How do $\Omega$ and $\alpha$ vary with time when dissipation is included? [If the dissipation is small, you could assume that the coin is always close to the steady state you found in Part 1. Also, you may find it easier work in terms of the total energy of the coin, rather than its angular momentum.]

Compare your results with those of H. K. Moffatt, Euler's disk and its finite-time singularity, Nature 404 (2000) 833. Do you find a similar singularity in $\Omega$? For more on rolling discs, see the physics examples and other pedagogic diversions for April 2000 and December 2000 by Kirk McDonald (Princeton). You might also compare what you find with the movies of simulations of a spinning disc created by Marc Shapere (University of Texas).

Textbook references

Home: PC 1672 home page | Up: 4.9 Gyroscopes | Examples | Help: Guide to using this document |

Mike Birse
5th February 2001