The mysterious spinning cylinder—Rigid-body motion that is full of surprises

Euler's Disk even found its way onto the popular television show The Big Bang Theory (season 10, episode 16).

In addition to playing around in class, the hope is that students will show it to their friends and continue to think about what makes the $X$s (or $O$s) disappear.

Beginning students might suggest using the equation $Γnet=Iα$⁠, where I is the object's moment of inertia and $α$ is the angular acceleration vector. Unfortunately, this equation is not valid in this situation. It may be worth reminding students that the equation $Fnet=ma$ is not always valid either (e.g., when the system has a changing mass) and in such cases we must appeal to the more general form of Newton's second law: $Fnet=dp/dt$⁠, where p is the momentum of the system. Likewise, the equation $Γnet=Iα$ is not generally valid (e.g., when the angular velocity is not aligned with one of the principal axes), so we must appeal to the more general form given in Eq. (3).

Comparing Fig. 4 of Ref. 6 to Fig. 2 here shows a lot of similarities. Indeed, the analysis is similar in some ways; however, the variable aspect ratio for a cylinder adds a new layer of complexity to the problem and leads to a lot of interesting physics not found with Hurricane Balls.

The symbol on the stationary end is quite clear while the symbol on the moving end is a completely blurred out.

Another way of figuring things out is by noting that for an object that is rolling without slipping, the point of contact is instantaneously at rest. Knowing that both the center of mass C and the point of contact P are both at rest defines the instantaneous axis of rotation. All points on the instantaneous axis of rotation—including the uppermost point—are (instantaneously) at rest.

Generally speaking, it is much more difficult to calculate the inertia tensor analytically without using a set of principal axes. For example, trying to calculate the components of the inertia tensor in the (x, y, z) coordinate system in Fig. 2 would present a significant challenge.

The Lagrangian approach can be used, but it must be approached with care (see  Appendix).

In analogy with spheroids, we might instead refer to prolate and oblate cylinders instead of cylinders and disks.

Although we analyzed the situation of a cylinder rolling on its side, Eq. (15) is also valid for a cylinder rolling on its end. In this sense, we also get the steady-state solutions to Euler's Disk for “free.”

Point $D$ represents an energy maximum on the vertical line $ℓ̃=2.5$⁠. In fact, there is a curve of such maxima (forming a “ridge” in the energy landscape) that lies below and follows the boundary (dashed curve) between stable and unstable solutions, osculating this boundary at the point where $θ∞$ and θ₀ intersect.

Interestingly, while the rotation rate $Ω→∞$ as $θ→0$⁠, the face of the coin will actually become stationary. This can be understood by noting that rotations about the vertical axis and rotations about the symmetry axis cancel out as $θ→0$⁠. From Eq. (6), we see that $ψ̇→−ϕ̇$ while $ê3→êz$ (see Fig. 2). The end result is that $ω$ [see Eq. (4)] (and L) goes to zero as $θ→0$⁠.

We did our best to make sure the experimental results were obtained when the center of mass was at rest and θ was constant, but some aspect ratios were very susceptible to instabilities at specific rotation rates. Thus, while our experimental data are obtained with good precision (reasonably small error bars), it is likely that some of our spinning cylinders (and disks) were not true steady-state motions (our patience was limited).

In Fig. 7, notice that the upper (disk on its side) $ℓ̃=0.75$ curve is almost vertical near $θ=π/2$⁠. In fact, the angular speeds for some disks having aspect ratios near $ℓ̃=3/2$ are predicted to first increase slightly above $Ω̃=1$ before decreasing to $Ω̃=0$⁠.