We discuss a project on rigid-body motion that is appropriate for students in an upper-division course in classical mechanics. We analyze the motion of *Hurricane Balls*, two spheres that are welded (or glued) together so they act as a single object that can be spun like a top. The steady-state motion consists of purely rotational motion about the center of mass, such that only one ball is in contact with the table as it rolls without slipping. We give a qualitative explanation for why one ball rises into the air, and we theoretically analyze the system using multiple approaches. We also perform a high-speed video analysis to obtain experimental data on how the orientation depends on the spin rate, and find agreement within a few percent of the theory.

## References

*tornado balls*, which appears to be identical to the original Hurricane Balls (though manufactured by a different company), is available.

We note here that the analysis is not symmetric; that is, if the left ball in Fig. 2 were to momentarily lose contact with the table, this would *not* result in the left ball rising into the air.

The fact that the steady-state motion of the double-sphere is one of steady precession *without* nutation is not completely obvious. Evidently, in dissipating energy, the frictional force (and to a lesser extent air drag) quickly damps out any nutation until the condition $\theta \u0307=0$ is reached (see Fig. 4). This observed fact allows one to approach this problem using an energy-minimization procedure, as discussed in Appendix B 2.

*Euler's Disk*(official website <http://www.eulersdisk.com>). One intriguing aspect of this toy is that while the double-sphere's rotational frequency

*decreases*as energy is lost from the system ($\theta \u2192\pi /2$), Euler's disk has the property that the rotational frequency

*increases*as energy is lost from the system ($\theta \u21920$). Thus, you can hear the rotation rate increase dramatically as both

*θ*and

**L**approach the vertical. See, for example,

Interestingly, using the condition *α* = 0 to calculate $\theta min$ leads to the expression $\theta min=2\u2009tan\u221213/7$, whereas we previously determine that $\theta min=cos\u22121(2/5)$. Students might be amused to learn that $\u2009cos\u22121(2/5)=2\u2009tan\u221213/7$ and can be challenged to prove this fact.

While higher frame rates will result in more accurate data, a 2,000 fps camera is not necessary to obtain reasonable experimental data. In order to prevent aliasing, the frame rate needs to be at least double the rotation frequency of the motion (but of course higher frame rates are always better). Thus, spin rates up to $\u223c6,000\u2009rpm\u2248100$ Hz can be analyzed using a camera with a frame rate of only 200 fps, which is within the capabilities of many cell-phone cameras. It is worth noting, however, that to make high-quality slow-motion videos of this motion, such as the movie linked to Fig. 3, requires frame rates that are about an order of magnitude higher than what is minimally necessary to obtain useable data.

*Tracker*is a free video analysis and modeling tool built on the Open Source Physics (OSP) Java framework, designed to be used in physics education. More information on Tracker can be found at <https://www.cabrillo.edu/~dbrown/tracker>.

*passive*rotation matrix is used when the coordinate axes are rotated while the vector remains fixed. An

*active*rotation matrix is used when the vector itself is rotated and the coordinate axes remain fixed. These rotation matrices are related to each other by $\theta \u2192\u2212\theta $. See, for example,

This approach is a standard feature of most textbooks in classical mechanics when dealing with a spinning top, and typically includes a discussion of the energy. See, for example, Taylor, Ref. 1, pp. 401–407.

When analyzing the motion of a spinning top with a fixed point of contact, most textbooks do *not* specifically mention that the motion of the center of mass is being neglected. But save for the trivial case of a “sleeping” top, the center of mass will be in obvious motion, even for the case of steady precession. Thus, even if explicitly stated (and valid), such an assumption can be mystifying to a student who can plainly observe such center-of-mass motion. The motion of a double-sphere, on the other hand, is such that the center of mass appears to be (very nearly) at rest.

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