Simultaneous three-way observation of optical, acoustic and electric properties demonstrates a theoretical prediction that a spinning prolate spheroid can spontaneously lose contact with the table in the course of its rising motion when the contact friction is weak and the spin is large enough. The durations of the first loss of contact are measured for various initial spin rates and for three aspect ratios. The measurements show good agreements with numerical simulations. It is also visually shown that a spinning hard-boiled egg can jump.
If a hard-boiled egg is spun sufficiently rapidly on a table with its axis of symmetry horizontal, then this axis will rise from the horizontal to the vertical: the centre of mass rises against the gravity. Moffatt & Shimomura (2002) analytically resolved this behaviour by finding the existence of the ‘Jellett’ constant for arbitrary bodies of revolution as an adiabatic invariant under the ‘gyroscopic approximation’ when the initial spin rate is high and the friction at the moving point of contact is weak. They obtained a first-order differential equation for the inclination of the axis of symmetry, which, for the case of a uniform prolate spheroid, did indeed describe the rise of the axis to the vertical. Based on this equation, Sasaki (2004) and Shimomura (2005) analysed the effects of the shape and the locus of the centre of mass of a hard-boiled egg to determine which end will rise. From a viewpoint of ‘dissipation-induced instabilities’ (Bloch et al. 1994), Bou-Rabee et al. (2004, 2005) studied the problem of the spinning prolate spheroid.
Following Moffatt & Shimomura (2002), Moffatt et al. (2004) discussed linear instabilities of a spinning spheroid and verified the gyroscopic approximation. They also showed that the slowly rising motion is actually accompanied by fluctuations. Shimomura et al. (2005) have recently found that these fluctuations rapidly oscillate with a slowly growing amplitude and consequently allow the normal reaction to decrease to zero in certain circumstances. Noting that these slow and fast time-scales intrinsic to the dynamics are well separated when the friction is weak, they analytically predicted and numerically confirmed self-induced jumping: a uniform spheroid, which is a simple model of a hard-boiled egg, will spontaneously lose contact with the table in the course of the rising motion after being spun sufficiently rapidly.
This prediction may be hard to believe, and it is important to strictly demonstrate self-induced jumping. On a website (http://www.kobe-u.ac.jp/info/topics/t2005_07_11_02.htm in Japanese, 2005), Nishioka et al. presented four snapshots of a spinning hard-boiled egg spun rapidly on a table by hand, which were taken using a high-speed video camera. The snapshots seemed to show a gap between the egg and the table, which existed for about 20 ms. With some consistent results of their numerical simulation based on a finite-element method, they claimed that self-induced jumping was experimentally detected. In fact, we observed such gaps in our preliminary experiment using a similar method. However, we realized that various uncontrollable oscillations will be initially introduced to the motion of a spinning egg if it is spun rapidly by hand, and they could cause a jumping motion of the egg, which, of course, means nothing to self-induced jumping: for example, a hard-boiled egg would easily jump if spun initially at a high spin rate with an upward velocity of the centre of mass. It is also impossible to know from only a few of many snapshots whether a jump is induced by itself or not, because we cannot track the course of a rising motion from the horizontal to the vertical: in order to demonstrate self-induced jumping, we should show a period prior to first jump, for which a spinning egg keeps contact with a table. Therefore, we do need to mechanically spin a body with a controllable initial rate of spin and adopt a method allowing observations all through the rising motion to definitely identify self-induced jumping.
In the present paper, we show three-way experimental evidence for self-induced jumping of a uniform spinning spheroid with simultaneous observations of optical, acoustic and electric properties. We further measure the durations of its first loss of contact in comparison with numerical simulations for various initial spin rates and for three aspect ratios. Lastly, we answer the question in the title of the present paper.
In §2, we explain the experimental method for the detection of self-induced jumping of a spinning spheroid, by which three-way evidence to demonstrate these events is presented in §3, and measured durations of the first loss of contact are compared with numerical simulations in §4. In §5, self-induced jumping of a hard-boiled egg is visually shown. Finally, in §6, we give conclusions of the present study.
2. Three-way observation
In the observations of self-induced jumping, we use a prolate spheroid made of aluminium with a uniform mass density. The dynamic friction coefficient between the spheroid and the table is 0.20. On a horizontal table of polished copper, the spheroid was mechanically spun by a rotor with its axis of symmetry initially horizontal at a controlled spin rate. The rotor has a shaft which is connected at its end to a disk with four supports for the spheroid to be fixed to it, as shown in figure 1. We release the spinning spheroid from the four supports by lifting the shaft connected to them. This method does not introduce significant disturbance to the initial spinning motion because the four supports spin at the same rate as the spheroid until the moment when we lift the shaft. We detect this moment (t=0) by an electrical contact between electrodes attached to the shaft and a support column of the apparatus, which simultaneously triggers the three measurements explained below.
The first one is a series of snapshots of a spinning spheroid, taken by a high-speed camera against the background of a tungsten lamp. The shutter speed of the camera is 1 ms, and it takes 240 frames per second. If the spheroid jumps, there should appear a gap between the lowest point of the spheroid and the table. Even when the gap width cannot be resolved for lack of enough pixels in a snapshot, its existence can be visually shown by detecting the light passing through the gap.
The second one is the time-series of a sound signal emitted from a point of contact, which is detected by a microphone. When the spheroid lands back on the table from a free flight, the impact causes a sound.
The third one is the time-series of a capacitance. Both the table and the spheroid are electrically conducting, so that they form a one-plate condenser, whose capacitance depends on their configuration. The capacitance of the table alone is 25 pF, which increases by 0.3 and 0.7 pF when the table contacts the static spheroid with its axis of symmetry horizontal and vertical, respectively.
3. Evidence for self-induced jumping
Here, we use a prolate spheroid with the major radius of 3 cm and the minor one of 2 cm, whose weight is 133 g. The initial spin rate is 1500 r.p.m.
Figure 2a depicts a series of six snapshots of the spinning spheroid. It illustrates the rising motion of the spheroid. If we observe carefully, however, a narrow gap between the spheroid and the table is recognized in the third snapshot at t=479 ms and the fifth one at t=538 ms, while the lowest point keeps contact with the table in others. We note that the gap at t=479 ms is smaller than that at t=538 ms. This suggests the first jump at t=479 ms is smaller than the second one at t=538 ms, which will be confirmed by the measurements of the capacitance below.
Figure 2b plots the time-series of a sound signal emitted from a point of contact, which is detected by a microphone. Excluding background noise, the amplitude of its oscillation suddenly expands at t=556, 616, 679 and 741 ms. This is caused by an impulse when the airborne spheroid impacts the table.
Figure 2c shows the time-series of the capacitance. As is expected, its average, obtained by filtering out contact noises of high frequency, slowly increases as the spheroid rises. However, strikingly apparent are the six periods, during which the capacitance remains relatively small for more than 15 ms: t=472–491, 529–556, 591–616, 651–679, 715–741 and 784–800 ms. In each period, the capacitance achieves a minimum value at about three-quarters of the period. This means that the jumping height attains the maximum value at that time. This is a prominent feature of self-induced jumping, derived by an analytical expression for the free flight of the spheroid after jumping (see appendix A). Hence, six jumps are electrically detected by measuring the capacitance, consistent with the data from the snapshots in figure 2a and the sound signals in figure 2b. The duration of the first loss of contact is 19 ms, from which an analytical approximation estimates the maximum gap of 0.08 mm (see appendix A).
Figure 3 shows some examples of the time-series of the capacitance at various initial spin rates. The number of jumps, which are identified by the capacitance, generally increases as the initial spin rate becomes high. It is also observed in the present case that the spheroid does not jump when the initial spin rate is below 1448 r.p.m.
In summary, figures 2 and 3 show that a spinning spheroid can really jump in the course of the rising motion. However, we may not claim without a doubt that self-induced jumping has been detected by these observations: it is undeniable that there could remain disturbances in a rapidly spinning motion of a spheroid, which would induce trivial jumping. In order to experimentally demonstrate self-induced jumping, we need to systematically measure many events at various initial spin rates and with different aspect ratios, and, furthermore, to quantitatively compare the results of measurements with numerical simulations, which is done in §4.
4. Durations of the first loss of contact
The durations of the first loss of contact are systematically measured from the separate signals of sound and capacitance. Figure 4 plots the durations of the first loss of contact T of a uniform prolate spheroid as a function of the initial spin rate in comparison with numerical simulations for the corresponding case. Here, we test three kinds of uniform prolate spheroids, whose weights M, major radii a and minor ones b are all different: figure 4a–c shows the results for an aspect ratio a/b=1.24 (M=126.6 g, a=26 mm and b=21 mm), a/b=1.50 (M=132.3 g, a=30 mm and b=20 mm) and a/b=2.00 (M=128.8 g, a=36 mm and b=18 mm), respectively. The plots indicating T=0 mean that the spheroid does not jump in the course of the rising motion.
The simulations are carried out by numerically solving a full system of six first-order ordinary differential equations using the fourth-order Runge–Kutta method with a dynamic friction coefficient of 0.2, and the time-step of 4 μs. The initial condition is given by imposing a small disturbance on each variable expressing a steady state of horizontal spinning. In the six-dimensional phase space, the direction of the initial disturbance vector is randomly chosen, while its dimensionless magnitude (or, norm) is fixed at 0.05 (see appendix A for the method of normalization).
Here let us remark about the initial disturbance. This small disturbance is necessary for a spinning spheroid to raise its axis of symmetry from the horizontal to the vertical: the body would not rise without disturbance, remaining in the initial steady state of horizontal spinning. In real experiments, this initial disturbance corresponds to inevitable and uncontrollable noise, which is very small and originates from asymmetry in the structure of spinning apparatus, the surface roughness of the body and/or the table, etc. As the body rises, this unavoidable small disturbance is amplified in time to cause self-induced jumping when the initial spin rate is high enough. This is the reason why we introduce the initial small disturbance into the present numerical simulation.
We find that all of our experiments are well explained by the present simulations with an initial disturbance whose dimensionless magnitude is 0.05. The durations of the first loss of contact T scatter depending on the distribution of the initial disturbances, though there seems an upper limit for them.
In figure 4b,c, we note that the maximum T decreases as the initial spin rate increases, which is not so clear in figure 4a. The measurements of the critical rate of initial spin, over which the spinning spheroid jumps and below which it does not, agree well with the present simulations. It is also observed that the critical rate of initial spin decreases as the aspect ratio increases, which agrees with a numerical result of Shimomura et al. (2005). Lastly, we note that the maximum T is longer for a higher aspect ratio, notably for below 2500 r.p.m.
5. Self-induced jumping of a spinning egg
Now, we should come back to the original question: can a spinning egg really jump?
We spin a hard-boiled egg on a table mechanically by an electric motor at the initial spin rate 1800 r.p.m. The initial condition is not contaminated with significant fluctuations, which could be introduced if we spin it rapidly by hand. This is assured by noting the fact that jumping is not observed until 200 ms from the initial time when the egg was mechanically set to spin. For hard-boiled eggs, the signals of sound or capacitance are difficult to detect, but a movie clip shows a gap between the egg and the table in the course of its rising motion, which strongly suggests a jump of a hard-boiled egg (see movie clip in the electronic supplementary material). Figure 5 is a sequence of the snapshots in the movie clip. We note that figure 5c–e clearly shows a gap between the egg and the table. So, we may answer the original question in the affirmative.
In conclusion, simultaneous three-way observation demonstrates that a uniform prolate spheroid spinning on a table can really jump in the course of its rising motion. The measured durations of the first loss of contact after jumping show very good agreements with numerical simulations for various initial spin rates and for three uniform prolate spheroids with a different aspect ratio. Many jumps subsequent to the first one are also observed, a topic which is not within the scope of the previous study (Shimomura et al. 2005) and their quantitative analysis is left for future works. Lastly, it is also shown by a sequence of snapshots that a spinning hard-boiled egg can spontaneously jump.
The spinning-egg problem is a so-called toy problem, but it provides a real example of general phenomena, in which fluctuations cause unexpected events. In this context, other toy problems are also expected to have the same potential to predict an unexpected phenomenon in a range of practical problems of much greater complexity and importance.
We thank A. Yoshizawa for making valuable comments on our work, and S. Kimata and M. Omote for providing PCs for the experiments. Y.S. is grateful to H. K. Moffatt and M. Branicki for stimulating discussions, and to N. Osada for giving him an opportunity to collaborate with K.A. This work is partially supported by the Keio Gijyuku Academic Development Fund.