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Rolling/Slipping Motion of Euler’s Disk H.Caps, S.Dorbolo, S.Ponte, H.Croisier, and N.Vandewalle GRASP, Department of Physics B5, University of Liège, B-4000 Liège, Belgium. (Dated: Received: February 2, 2008/ Revised version: date) 4 We present an experimental study of the motion of a circular disk spun onto a table. With the 0 help of a high speed video system, the temporal evolutions of (i) the inclination angle α, (ii) the 0 angularvelocity ω and(iii) theprecession rateΩarestudied. Theinfluenceofthemassofthedisk 2 and thefriction between the disk and the supporting surface are considered. The inclination angle n α and the angular velocity are observed to decrease according to a power law. We also show that a theprecession rate Ω diverges as the disk stops. Exponentsare measured verynear thecollapse as J well as on long range times. Collapsing times have been also measured. The results are compared 6 with previoustheoretical and experimental works. The major source of energy dissipation is found 1 tobe theslipping of thedisk on the plane. ] PACSnumbers: 05.45.-a,45.40.-f h c e I. INTRODUCTION where n , n and n are the dynamical exponents to m α Ω ω bedetermined[6]. Dependingonthedissipationprocess, t- A common straightforward experiment is the follow- different values can be found for these exponents. The a ing. A coin spun onto a table rotates with an increasing prefactors of those laws, and hence collapsing times t0, t s precession rate and a decreasing inclination angle until also depend on the dissipation mechanism. Moreover, it t. it abruptly stops. This complex motion is known as the shouldbe noticed thatthe power lawsmay be valid only a Euler’s disk problem. in a limited range of inclination angles since the major m Within the classical formalism of mechanics, the disk dissipationmechanismmaydependonthedynamicalpa- - shouldspinadinfinitum. Moffat’swork[1,2]emphasized rameters α and Ω values. d n thenecessityofconsideringenergydissipationinorderto Nowadays, no consensus has been found by the phys- o avoid such forever persisting motion. Many theoretical ical community. Other experiments are thus required in c and numerical studies [3, 4, 5, 6] have been devoted to ordertoemphasizethe majordissipationprocessleading [ the finding of the major dissipation process leading to to the brutal stop of the disk. 1 the stop of the disk. However, only a few experimental v result can be found in the scientific literature [2, 7]. 8 ForadiskofradiusR,massm,andaninertiamomen- Ω 7 tum I, the total energy reads ω 2 1 1 0 E =mgRsin(α)+ IΩ2sin2(α), (1) 2 4 0 where α is the inclinationangle with respectto the hori- / t zontal and Ω is the precession rate. A sketch of the disk a is illustrated in Figure 1. Parameters α, Ω and ω are m emphasized. α - Inordertomeetobservations,onehastoconsiderthat d n the energy E is dissipated with a rate φ. The goal is o thus to find the majormechanism responsible for energy c dissipation φ. : FIG.1: Sketchofthediskanditsdynamicalparameters: the v Considering that the energy is dissipated through a inclination angle α, the precession speed Ω and the angular i velocity-dependent process, the temporal evolutions of X velocity ω. α, Ω and ω can be described in a general way with r a α∼(t −t)nα, (2) 0 Inthe nextsection,we presentthe experimentalsetup 1 nΩ and procedures. In Section III, we report the experi- Ω∼ t −t (3) mental results concerning the temporal evolution of the (cid:18) 0 (cid:19) inclination angle, the precession frequency and the ro- and tation velocity as well as times to collapse. We discuss the results in Section IV. A summary of our findings is ω ∼(t −t)nω, (4) eventually given in Section V. 0 2 II. EXPERIMENTAL PROCEDURES δ camera γ The experiment consists in spinning a disk by hand onto a table and to observe the evolution of the three α r different dynamical parameters with the help of a high speed video system. We used three different coins: (i) a stainless disk having a diameter D = 75 ± 0.01mm, a thickness 2 e = 10±0.01mm and a mass m = 353.4±0.1g ; (ii) 2 2 an alumna disk with a diameter D = 75 ± 0.01mm, 1 a thickness e = 9.9 ± 0.01mm and a mass m = 1 1 123.9±0.1g; (iii) an alumna torus with an internal diameter D = 63.65 ± 0.01mm, an external diameter Ti D = 78 ± 0.01mm, a thickness e = 10 ± 0.02mm FIG.2: Experimentalsetupfortheinclinationanglemeasure- Te T and a mass m =124.5±0.1g. ment. A laser beam is reflected by the disk and creates an T The used surfaces on which the disks spin were 2 mm ‘ellipse’ on a screen situated on the other side. A high speed videocamera records images of the‘ellipse’. thicknessplatesofalumna,glassandroughplasticsheets. All these plates where supported by the same table in ordertochangeonlythesurfaceroughness. Startingfrom the lowestfrictioncoefficienttotheuppest, onefindsthe glass, the alumna and the rough plastic surface. Herebelow, we describe the experimental procedures we used for the data acquisitions. Note that despite the factthatthedisksarespunbyhand,ourresultsarequite reproducible. This has been also mentioned in [7]. A. Inclination angle α FIG.3: Pathofthelaserbeamduringthediskrotation. The The inclination angle measurementis based onthe re- size of theellipse decreases with time. flectivityofthedisks. Whenalaserbeamisdirectedtoa diskwithafiniteangleγ relativetothehorizontalplane, one can observe an “ellipse” formed by the laser beam Geometrical arguments give us the relation between on a screen situated on the opposite side. This ellipse is the size of the ellipse and the inclination angle α. The due to the motion of the disk, which changes the inci- mainaxislengthAoftheellipseandtheinclinationangle dentangleofthe laserbeamonthe disk. Infact, acircle α are correlated through should be observed on the screen. However, the angle betweenthe laserbeamandthe horizontalplaneinduces 2rsin(2α) a deformation of that circle into an ellipse. In order to A= (5) minimize this effect, the laser beam is placed as verti- sin(γ+δ)− cos2(γ+δ)tan2(2α) cally as possible (γ ≈80◦ in practice); its position being sin(γ+δ) limitedbythe screen. Asketchofthe setupisillustrated where δ is the angle betweenthe screenandthe horizon- in Figure 2. talplaneandr isthedistancefromthecenterofthedisk Sincetheinclinationangleαofthediskdecreaseswith to the center of the ellipse. Typically, r = 0.3 m. This time, the size of the ellipse accordingly decreases. Mea- relation is invertible for determining α from the knowl- suringthesizeoftheellipsegivesustheinclinationangle edgeofA,buthasarathercomplicatedformulation. The αofthediskasafunctionoftime. InFigure3,wepresent minor axis case is more simple. Calculations lead to the the whole pathof the laserbeamduring the lastseconds followingrelationbetweenthe verticalaxislengthB and ofthe disk motion. One canobservea spiralpatterndue the inclination angle to the decrease of α with time. The black circle in the center of the picture is an artifact due to the width of 1 1−2cos(2β)+cos 2arcsin B the trajectory on the picture. Experimentally, the high α= arccos 2r (6) speedvideocameraallowsus to distinguishthe trajecto- 2 4sin2(β(cid:0)) (cid:0) (cid:1)(cid:1)! riesuntil0.0005sbeforethediskmotionstops. However, directobservationsfromthesideofthedisksshowedthat For each measured value of A and B, we have cal- arecordingrateof500framesper secondislargeenough culated α from Eqs.5 and 6 and we have averaged the for an accurate determination of α. values. 3 B. Precession rate Ω C. Angular velocity ω In order to measure the precession rate Ω, we have We have measured the angular velocity ω of the disk proceededasfollows. Alaserbeamhasbeenhorizontally by recording the motion of two diametrically opposed placed in front of the table in such a way that the beam whitemarkssituatedonthetopfaceofthedisk. Theex- skimsthetable[seeFigure4]. Duringtherotatingmotion perimental setup is illustrated in Figure 6. The angular ofthedisk,thelaserbeamisintercepted,leadingtoafast position, and then the angular velocity ω, are calculated decrease of the light intensity transmitted on the other from the displacement of both dots between two consec- side of the table. Such a light extinction occurs twice a utive frames. The images have been recorded at a rate rotation of the disk. The transmitted light is observed of 125 frames per second. This has been found to be with a high speed video camera situated at the same sufficient for detecting the stop of the disk. Top view height as the laser but on the opposite side of the plate. recordings also allowed us to measure the motion of the The images are then transfered to a computer for image center of mass of the disk. analysis of the intensity of the light spot created by the laser beam. The images are recorded at a frame rate of 500 frames per second, allowing us a high precision for the spot extinction detection. FIG.4: Sketchoftheexperimentalsetupforthemeasurement FIG.6: Sketchoftheexperimentalsetupfordeterminingthe of the precession rate Ω. A laser beam skims the supporting angular velocity ω. Two diametrically white dots are glued surface and is intercepted by the disk twice a precession. A on the disk. Their motions are recorded with the help of a high speed video camera records the intensity of the trans- high speed video camera, from the top. mitted beam. Figure5presentsatypicalcurveforthe temporalevo- lution of the intensity I of the laser spot in arbitrary III. EXPERIMENTAL RESULTS units. Eachsharppeakcorrespondstoaninterceptionof the laser beam by the disk. The peaks are alternatively due to the front and to the back of the disk. The peaks In this section, we present the results of the experi- aredetectedby fitting aGaussiancurveoneachpointof ments performedwiththree ‘disks’onthe three different the signal. In so doing, times corresponding to succes- surfaces. This corresponds to nine different experimen- sive minima of the intensity are recorded. This method tal situations. The measurements have been performed allows a precision of 0.002 s. a largenumber oftimes for eachexperimentalcondition. Wewillfirstconsidertimestocollapset andlongtime 0 behaviour of the dynamical parameters. Then, the last 14 stage of the disk’s motion will be further analysed. 12 nits) 10 A. Time to collapse t0 b. u 8 ar I ( 6 Figures7presentstwotypicalevolutionsofthepreces- 4 sion rate Ω and the inclination angle α as a function of time t. Black lines correspond to fits using Eq.(4). 2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 At the beginning of the disk’s motion, a larger dis- t (s) persion of the measurement is observed; due to the way the disk was set into motion. The center of mass of the FIG.5: Typicalcurveofthetemporalevolution oftheinten- disk may, for exemple, own some small linear momen- sityI ofthelaserbeam. AfitusingaGaussiancurve(dashed tum. However,themotionconsideredintheintroduction curve) is also illustrated. is rapidly reached and we have observed that the early stage of motion has not significant effect on the results presented hereafter. This was also mentionned in [7]. 4 One can seen that the power law behavior previoulsy 1.2 proposedisconsistentwithobservationovertwodecades in times. This means that, as close as we can approach 1.0 the collapse, the major mechanism of energy dissipation 0.8 remains the same whatever its nature. g) de 0.6 α ( 1000 Disk 1 / Glass 0.4 Disk 2 / Glass 0.2 0.0 -1s) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ad t (s) Ω (r 100 FIG.8: Temporal evolutionoftheinclination angle αforthe stainless steel disk (disk 1) case on thealumna surface. A fit using Eq.(2) is also illustrated. 0.1 1 10 t-t (s) 0 FIG. 7: Temporal evolution of the precession rate Ω over an nα glass alumna plastic entireruninalog-logscale. Twocasesareillustrated,aswell disk 1 0.51 ± 0.01 0.60 ± 0.01 0.56 ± 0.03 as fits using Eq.(4). disk 2 0.49 ± 0.01 0.49 ± 0.05 0.59 ± 0.02 torus 0.52 ± 0.02 0.56 ± 0.03 0.66 ± 0.03 If the friction between the disk and the surface is increased, the time to collapse t decreases. For the 0 alumna disk (disk 1) on a glass surface, we have t = TABLE I: Values of the exponent nα of Eq.(2) for the three 12.389±0.007while t =3.393±0.168for the same d0isk ‘disks’ on thedifferent surfaces. 0 on the alumna surface (similar results are obtained for other experimental conditions). Since the geometry of the system is the same in both cases, the energy dissipation may not come from the air drag. Friction seems C. Precession rate Ω take the main source of dissipation. The precession rate Ω is observed to diverge after a finite time, as illustrated in Figure 9. The divergence (Eq.(3)) is illustrated (continuous curve of Figure 9). B. Inclination angle α Note that around the black curve, small oscillations of Ω are observed. They were already reported in other Figure 8 presentsa typicalcurve ofthe inclinationan- measurements [3]. gle α as a function of time t. The angle is observed to decrease and to vanish quite abruptly at the time 280 t0 = 0.571s. A fit using Eq.(2) emphasizes the power 260 law behavior. 240 The values of the exponent n are presented in Ta- 220 α bseleemI.s Wtoeinhcarveeasoebswehrveendtthheasttdateiscpiftreictthioenecroroerffibcaiernst, nµα -1d s) 128000 increases. This tends to show that the friction at thes Ω (ra 114600 rolling contact point of the disk may play an important 120 role in the dissipation process of the disk energy. One 100 should also note the small differences between the val- 80 ues obtainedfor the three ‘disks’. Particularly,the torus 60 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 behaves in nearly the same way as the two disks. This t (s) result suggests that the air layer under the disk is not a major source for the energy dissipation. Another ob- FIG. 9: Temporal evolution of the precession rate Ω for the servation is that n does not depend on the mass of the alumnadiskcaseonthealumnasurface. AfitusingEq.(3)is α spinningcoin,inagreementwiththeequationsofmotion also illustrated. derived in [6]. 5 AdjustingthedatawiththepowerlawEq.(3),wehave found different n values for different experimental con- TABLE III: Values of the exponent nω for the three ‘disks’ Ω on the different surfaces. ditions [see Table II]. It is observed that the precession rate diverges as rapidly as the static friction increases. Once again, the torus and the disks behave in similar IV. DISCUSSION ways,inthe sensethatthe valuesofn arenotso differ- Ω ent for these different coins. This also suggests that the Assuming that the main dissipation mechanism that air drag is not a majorsource for the energy dissipation. leads to the singular behavior is the air drag, Moffatt concluded that n should be 1/3, and n = 1/6. Bild- nΩ glass alumna plastic sten pointed outαa mistake in the MoffatΩt’s calculations disk 1 0.39 ± 0.03 0.34 ± 0.01 0.39 ± 0.03 and found the values n = 4/9 and n = 2/9. These α Ω disk 2 0.32 ± 0.01 0.30 ± 0.01 0.34 ± 0.01 theoreticalprevisionsareatoolowtoexplaintheexperi- torus 0.30 ± 0.03 0.31 ± 0.02 0.35 ± 0.02 mentalresults, suggestingthat airdrag does not playan important role in the energy dissipation, as also shown bytorusexperiments. Experimentsperformedinvacuum conditions [2] have also emphasized the small effects of TABLEII:ValuesoftheexponentnΩ of Eq.(3)for thethree viscous drag. ‘disks’ on thethree different surfaces. Fromthevaluesofbothexponentsn andn ,wethus α Ω see that the rolling friction seems to play an important role in the energy dissipation mechanism. Indeed, one would expect [7] n =1/3 in the case of rolling without Ω D. Angular velocity ω slipping, what is quite close to our values. Thisresultisalsosupportedbyplotsofthedissipation We have measured the angular velocity of the differ- rate φ of energy during the experiments. We have cal- ent coins and reported the values of nω in Table III. A culated the total energy E of the disks (Eq.(1)) for each typicalcurve,aswellasafitusingEq.(4)arereportedin measured values of α and Ω. Then we have computed Figure 10. the dissipated rate of energy φ by finite differencing of the energy E. 6 0.1 5 Disk 2/glass Disk 2/alumna 4 -1s) d 3 ω (ra 2 -1φ (J s) 0.01 1 0 0.36 0.38 0.4 0.42 0.44 0.46 0.001 t (s) 0.01 0.1 α (rad) FIG. 10: Temporal evolution of the angular velocity ω for the stainless steel disk case on the glass surface. A fit using FIG. 11: Disipation rate of energy φ as a function of the Eq.(4) is also plotted. inclination angle α. Two cases are illustrated. As observed for both α and Ω, an increase in the fric- In Figure 11, one can find a typical evolution of the tion between the disk and the surface causes a faster dissipationrateφasafunctionofthe inclinationangleα energy dissipation. Here, we observe that the different for the stainless disk (disk 2) on both glass and alumna coins give different values of the exponent nω, contrary surfaces. Indeed, the disk is osberved to dissipate more to previous observations. energy by time unit on the surface with the largest friction. On the same figure, we have plotted the curves nω glass alumna plastic corresponding to a dissipation rate due to friction [7] disk 1 0.38± 0.07 0.46± 0.02 0.41± 0.01 disk 2 0.44± 0.01 0.53± 0.06 0.51± 0.02 φ ∼cos(α)Ω (7) torus 0.42± 0.01 0.42± 0.01 0.65± 0.03 frict . 6 The agreement between experimental and theoretical As noticed, the center of this circular path has a ten- resultsemphasizetheroleplayedbyfrictionintheenergy dency to move the most at the early stage of the motion dissipation. thanattheend. Inthecaseofthecircularring,this‘sta- Moreover, one should note that a proportionality be- bilization’ is less important. We suppose that the large tween the precession rate Ω of the disk and its angular surfaceofthediskshelpthemtogetthisstabilization. It velocity ω is expected [1, 6]. This is not so obvious here, will be interesting to check whether this mechanism al- evenifthedataarequitedispersed[seeTablesIIandIII]. ways occurs in vacuum condition, in order to determine Moreover, for a considered surface, the angular velocity the role of the air in this phenomenon. exponent n depends on the disk, and therefore on the Another important observation is that the center ω static friction. mass of the disk exhibits small quasi-periodic excursions Theseobservationsleadustobelievethatthediskmay aroundthe circulartrajectory. These oscillationsarenot slip while it is rotating. The influence of the slipping of exactlyperiodicbutarequite frequent,especiallyduring the disk was also suggested in [2], but from a theoretical the last angular rotations of the disk. point of view. This is in contradiction with [8] where These small oscillations in the circular trajectory of the energy dissipationby slipping is ruled out. However, the center of mass of the disk are due to the slipping thesepreliminaryexperimentswereperformedduringthe of the contact point of the disk along the surface. The early stage of the disk rotation. Indeed, only one exper- resolution of our setup does not allows us to determine imental condition was measured and moreover at a low wheter there is some kind of ‘stick-slip’ of this point on frame recording rate. the surface. However, these slips are responsible for the energy dissipation leading to the final stop of the disk. 1.5 1 V. SUMMARY 0.5 We have performed new experiments for studying the y (cm) 0 Euler’s disk motion. The divergence of the precession rate as the inclination angle and the angular velocity -0.5 vanish have been observed after a finite time. All these -1 parametersfollow powerlaw behaviors. The time to col- laspe is observed to decrease with the friction. -1.5 Contrarytoprevioustheoreticalworks,theairisfound -1.5 -1 -0.5 0 0.5 1 1.5 x (cm) to be a minor source of energy dissipation. On the contrary,themajorenergydissipationprocessisfoundtobe FIG. 12: Path of the center of mass of the disk 1 on the therolling/slippingofthediskonthesupportingsurface. alumna surface. In order to emphasize the slipping or not of the disk, Acknowledgments we haverecordedthe motionof the center ofmass ofthe diskwithahighprecision(250fps,1024×1024pixelsim- HC is financially supported by the FRIA (Brussels, ages). Figure12illustratesthetrajectoryofthecenterof Belgium). SD acknowledges the FNRS (Brussels, Bel- mass within one and a half angular revolution. One can gium) for financial support. This work is also supported ◦ see that the center of mass moves along a circular path. through the ARC contract n 02/07-293. [1] H.K. Moffatt, Nature 404, (2000) 833 [6] K.T. 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