The rise and fall of spinning topsRod Cross Citation: Am. J. Phys. 81, 280 (2013); doi: 10.1119/1.4776195 View online: http://dx.doi.org/10.1119/1.4776195 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v81/i4 Published by the American Association of Physics Teachers Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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The rise and fall of spinning tops

Rod Crossa)

Department of Physics, University of Sydney, Sydney NSW 2006, Australia

(Received 25 June 2012; accepted 31 December 2012)

The motion of four different spinning tops was filmed with a high-speed video camera. Unlike

pointed tops, tops with a rounded peg precess initially about a vertical axis that lies well outside

the top, and then spiral inward until the precession axis passes through a point close to the

center-of-mass. The center-of-mass of a top with a rounded peg can rise as a result of rolling rather

than sliding friction, contrary to the explanation normally given for the rise of spinning tops. A

tippe top was also filmed and was observed to jump vertically off a horizontal surface several times

while the center-of-mass was rising, contrary to the usual assumption that the normal reaction force

on a tippe top remains approximately equal to its weight. It was found that the center-of-mass of a

tippe top rises as a result of rolling friction at low spin frequencies and as a result of sliding friction

at high spin frequencies. It was also found that, at low spin frequencies, a tippe top can precess at

two different frequencies simultaneously. VC 2013 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4776195]

I. INTRODUCTION

The most fascinating aspect of a spinning top is that it cantemporarily defy gravity by moving sideways and upwardbefore it eventually falls. If the top spins fast enough it canrise to a sleeping position where the spin axis remains verti-cal. Early experimenters and theoreticians1–4 established thatthe rising of the center-of-mass of a spinning top (or an eggor a football or a tippe top) is due to a torque arising fromsliding friction at the bottom end. Parkyn5 disagreed, givingexperimental evidence that rolling friction was responsible.Air friction and friction at the base of the top act to decreasethe angular velocity of the top and it eventually starts to fallaway from the vertical position. As it does so, the top startsto precess, the rate of precession being proportional to theheight of its center-of-mass and inversely proportional to theangular momentum of the top; at least, that is the case for theidealized top treated in elementary physics textbooks.

There is now an extensive literature on the theory of spin-ning tops,6–16 but very little data on measured precessionrates. In fact, the author was unable to find any such data.The early experimenters measured the angular velocity of atop using a stroboscope, and could measure the inclinationangle and the path of the bottom end on carbon paper orgraphite, but did not provide any measurements of preces-sion rates. The advent of relatively inexpensive high-speedvideo cameras makes such a measurement straightforward,and suitable for an experiment or project in an undergraduatelaboratory. Results obtained by the author are presentedbelow and are compared with simple theoretical estimates.The experiment described here is similar to others describedpreviously concerning the precession of a spinning disk.17,18

Results were obtained for (a) a sharply pointed top, (b) atop with a rounded peg, and (c) a tippe top. All three topsbehave in qualitatively different manners. If the bottom endof a top is sharply pointed then the bottom end moves in atight circular path of very small radius, in which case thecenter-of-mass precesses slowly about a vertical axis passingthrough the bottom end. If the top has a rounded peg at thebottom then the bottom end can roll along a relatively large-radius, approximately circular path. In that case, the upperand lower ends of the top, as well as the center-of-mass, allprecess around a common vertical axis located well outside

the top. If the top then rises to a more vertical orientation,the precession axis can pass through the center-of-mass, inwhich case the center-of-mass of the top remains fixed inspace.

A tippe top consists of a truncated sphere with a short pegon top. When the peg is spun between the fingers, the tippetop precesses rapidly about a vertical axis, while the wholetop rotates slowly about a horizontal axis until it ends upspinning upright on the peg. A similar inversion and rise ofthe center-of-mass occurs when a circular disk with a largehole on one side is spun about a vertical axis. If the hole isinitially at the top then the disk rolls along its edge until thehole is at the bottom, the spin axis remaining vertical. If thehole is initially at the bottom, it remains at the bottom.

II. SIMPLIFIED THEORETICAL DESCRIPTION OF

A TOP

The equations describing the dynamics of a spinning topare often cast in forms that are far too complicated for stu-dents to understand the underlying physics. Steady preces-sion of a spinning top (or gyroscope) can be described in asimple and more intuitive manner by reference to Fig. 1. Ele-mentary treatments are given by Crabtree,7 by Deimel,8 andby Barger and Olsson.9 If the top or gyroscope is spinningwith angular velocity x about a horizontal (x) axis and issupported at the left end as in Fig. 1(a), then it will precess atangular velocity X about the vertical (z) axis. The angularmomentum L is in the x-direction, while the change in theangular momentum is in the same direction as the gravita-tional torque s ¼ MgH, which points in the y-direction.Since s ¼ dL=dt and L ¼ Ix, it is easy to show thatX ¼ MgH=ðIx), where I is the moment of inertia of the topabout the spin axis. This is the standard result derived inundergraduate physics textbooks, and it also applies to a topinclined at an arbitrary angle from the vertical.

A spinning top is usually supported at its bottom end at apoint O on a horizontal surface and the spin axis is inclinedat an angle h from the vertical, as indicated in Fig. 1(b). Inthis case, the torque in the y-direction about an axis throughO is s ¼ MgH sin h. Any force through O, including the fric-tional force required to rotate the top about the z-axis, doesnot contribute to this torque. A complication in Fig. 1(b) is

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that the moment of inertia I3 for rotation about the spin axisis usually smaller than the moment of inertia I1 for rotationabout an axis perpendicular to the spin axis and passingthrough O. In Fig. 1(b), the top precesses at angular velocityX about the z-axis, by rotating into the plane of the pageabout the pivot point at O. Consequently, the top also rotatesabout the perpendicular axis shown in Fig. 1(b).

The component of the angular momentum of the top in adirection along the perpendicular axis is L? ¼ I1X sin h. Thecomponent of X in a direction parallel to the spin axis isX cos h. The total angular momentum in a direction parallelto the spin axis is therefore Lk ¼ I3ðxþ X cos hÞ ¼ I3x3,where x is the spin imparted to the top about the spin axisand x3 ¼ xþ X cos h. The component of Lk along the x-axis is Lx ¼ Lksinh. The total angular momentum in the x-direction is therefore I3x3sinh� I1 Xsinhcosh. By equatingthe torque in the y-direction to the rate of change of angularmomentum in the x-direction, as in Fig. 1(a), we find that

MgH ¼ I3x3X� I1X2cos h; (1)

which is quadratic in X and therefore yields two real solu-tions for X, provided that

x > ð2=I3Þ½MgHðI1 � I3Þcos h�1=2: (2)

The lower frequency solution (denoted X1) is the one usuallydescribed in elementary textbooks and is the one that is usu-ally observed experimentally, while the higher frequency solu-tion (denoted X2) is comparable to x3. The textbook result isrecovered when x� X, in which case the second term on theright side of Eq. (1) can be ignored so that X � MgH=ðI3xÞ,regardless of the angle of inclination h.

If the spin x is less than that given by Eq. (2), then the topwill not precess in a steady manner and will instead fall rap-

idly to the ground. Otherwise, a rapidly spinning top tends toprecess slowly, at the lower frequency, while simultaneouslyprecessing in small sub-loops at high frequency. The sub-loops grow in amplitude and decrease in frequency as the topslows down, arising from nutation of the top. That is, the in-clination of the top varies periodically with time, accordingto the relation9

I1

d2hdt2¼ ðMgH þ I1X

2cos h� I3x3XÞsin h: (3)

Numerical solutions of Eq. (3) can be obtained by notingthat in the absence of friction the angular momentum parallelto the spin axis remains constant in time, as does the angularmomentum in the z-direction Lz ¼ I1X sin2hþ I3x3 cosh.The latter conditions determine xðtÞ and XðtÞ at each timestep for any given values of Lk and Lz. The particular solu-tion of Eq. (3) given by Eq. (1) corresponds to steady preces-sion without nutation.

Analogous solutions are obtained if the bottom end of thetop is rounded rather than being tapered to a sharp point. Inthis case, the bottom end of the top tends to roll along thesurface supporting the top. The bottom end is not fixed inspace but follows a spiral path as indicated in Fig. 2. If thebottom end is moving, valid solutions are best obtained byconsidering the torque acting about the top’s center-of-mass.19,20 A component of that torque arises from the cen-tripetal force F ¼ MR X2, where R is the radius of the pathfollowed by the center-of-mass. The centripetal force is pro-vided by friction at the bottom end of the top, as indicated inFig. 3. The normal reaction force N is equal to the weightMg of the top provided that the top is not rising or falling.

Fig. 1. (a) Gyroscopic precession when the spin axis is horizontal and the

axle is supported at the left end. (b) Precession of a gyroscope or a top when

the spin axis is inclined at an angle h to the vertical. The top precesses by

pivoting about point O, rotating into the page about an axis in the xz-plane

that is perpendicular to the spin axis.

Fig. 2. Motion of a spinning top with a spherical bottom end. The top

initially rolls along a path that spirals inward as shown in (a). The top leans

in toward the center of the path and precesses slowly about a vertical axis

through the center of the path. As the top slows down, the radius of the

path decreases and the top can assume a sleeping position as in (b) or it

can precess as shown in (c). The circular disk has been omitted from part (c)

for clarity.

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If A is the radius at the bottom end and D is the distance QGin Fig. 3, then the torque s about the center-of-mass G is givenby s ¼ MgD sin h�MR X2ðAþ D cos hÞ. Consequently,

MgD�MRX2ðAþDcoshÞ=sinh¼ I3x3X� IcmX2 cosh;

(4)

where Icm ¼ I1 �MðDþ AÞ2 is the moment of inertia aboutthe perpendicular axis through G. If the top rolls along thehorizontal surface then

rx ¼ R0X; (5)

where r ¼ A sin h is the perpendicular distance from the spinaxis to the contact point P and R0 ¼ ðRþ D sin hÞ is the ra-dius of the spiral path traced out by the contact point, as indi-cated in Figs. 2 and 3. Equations (4) and (5) can becombined to eliminate R, in which case it is found that thereare two possible precession frequencies, as before. If R¼ 0then F¼ 0 and the precession axis passes through G, as indi-cated in Fig. 2(c). The direction of F is reversed in Fig. 3 ifthe precession axis passes through the contact point or if it islocated anywhere else on the left side of G. In the latter case,R is negative (or the sign of the term containing R in Eq. (4)needs to be reversed) because the torque due to F then actsin the same direction as the torque due to N.

III. EXPERIMENTAL METHOD

The arrangement used in the present experiment is shownin Fig. 4. A versatile top (essentially a gyrostat or a gyro-scope without its gymbals) was constructed using an 8-mmthick aluminum disk of diameter 76 mm and mass 100.0 g. A4-mm diameter, 60-mm long threaded steel rod of mass 5.8 gwas inserted through a hole in the center-of-the disk andfixed to the disk with a nut above and below the disk. Thelength of the peg at the bottom end was fixed at either 17 or27 mm in order to vary the height of the center-of-mass ofthe top. The bottom end of the rod was ground to a sharppoint. An experiment was also conducted with a large radiuspeg at the bottom end using a spherical, metal drawer knob

of diameter 15 mm screwed onto the bottom end as indicatedin Fig. 2. Results obtained by filming a small, plastic tippetop are also presented.

Each top was spun on a smooth, horizontal surface either byhand at low speed or by wrapping a length of string around thethreaded rod to increase the speed. When using string to spinthe top, the upper end of the rod was allowed to spin inside avertical cylinder to keep the top approximately vertical and thecylinder was then lifted clear. Marks drawn on the spinning topwere observed by filming at 300 fps with a Casio EX-F1 cam-era, viewing either from directly above the top or from the sidein order to observe motion of the peg on the surface.21 Bothviews provided the same information on the spin angular veloc-ity of the top x and the angular velocity of precession X. Thetilt angle h with respect to the vertical was measured for con-venience from the side view, by recording the angle to the leftand right of the vertical axis.

A subtle feature regarding the measurement of x is that xis conventionally defined in a rotating coordinate systemattached to the top and rotating about the z-axis at angular ve-locity X. The moments of inertia are also defined in this rotat-ing coordinate system so that they remain constant in time.The top precesses at angular velocity X about the z-axis, sothe apparent spin recorded by a fixed camera mounted abovethe top is xþ X.17 A camera rotating at angular velocity Xaround the z-axis to follow the top would record its spin as x.In this paper, most measurements of x were obtained by sub-tracting the measured value of X from the apparent spinrecorded from the fixed camera. Some measurements of xwere also obtained by recording the rotation angle of the toponly when the top reached a fixed point in its precession cycle,for example, when it was leaning to the left or to the right.

Depending on the initial spin and the actual surface onwhich the tops were spun, the tops were observed to spin forup to about two minutes and rotated up to about 1000 timesbefore falling. Measurements were made of (a) the time atwhich each top completed successive sets of ten spin revolu-tions, (b) the time to complete each successive precession rev-olution, and (c) the tilt angle at about 50 different times fromthe start to the end of each spin. The angular speeds could bemeasured to an accuracy of better than 1% and the tilt anglecould be measured to within 0:5�. However, the tilt angle

Fig. 3. Details of a spinning top with a rounded bottom peg of radius A.

Here, G denotes the center-of-mass of the top. The normal reaction force Nand the centripetal force F both act through the contact point P at the bottom

end of the top.

Fig. 4. Experimental top constructed from a 76-mm diameter, 8-mm thick

aluminum disk with a 60-mm long threaded rod through the center of the

disk. The bottom end of the rod was tapered to a sharp point. Alternatively,

a spherical ball could be screwed onto the bottom end.

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itself varied by up to about 5� during each precession revolu-tion, due to nutation of the top. Side view measurements ofthe tilt angle were made when the top tilted to the right andagain when the top tilted to the left to obtain an average of thetwo tilt angles. At high spin rates, the tops were observed torise slowly, while at low spin rates the tops were observed tofall slowly. Under some conditions, the tops maintained a con-stant tilt angle when set spinning on a horizontal surface.

Properties of each top, including the peg radius A, totalmass M, and the distance H from the bottom end to the cen-ter-of-mass, are listed in Table I. Also listed are the moment-of-inertia I3 about the central axis, and the moment-of-inertiaI1 about a transverse axis through the bottom end. Top 3included a 16.8 g, 15-mm diameter ball at the bottom end.Top 4 was a hollow, plastic tippe top with an 11-mm longpeg attached to a truncated, 34.8-mm diameter sphere, spunby hand on a horizontal sheet of aluminum.

IV. RESULTS WITH TOPS 1 AND 2

A typical result obtained with Top 1 is shown in Fig. 5.The top had an initial spin of x¼ 126 rad/s that decreased to44 rad/s over 31 s before the top fell onto the horizontal sur-face. During that time the precession frequency X increasedfrom 2.7 rad/s to 8.6 rad/s and the angle of inclination of thetop increased from 8� to 21�. The top did not rise to a sleep-ing position.

The results in Fig. 5 were obtained by plotting the (x, y)coordinates of the upper end of the threaded rod at 0.01-sintervals during four different precession cycles. The firstcycle, from t¼ 0.73–3.05 s, took 2.32 s to complete one revo-lution, corresponding to an average precession frequency ofX¼ 2.7 rad/s. During that time the top also rotated manytimes in small-radius sub-loops, at 12360:5 rad=s, coinci-dent with the spin frequency x of the top. The standard ex-planation of the sub-loops is that they correspond to nutationof the top; however, nutation is expected at a frequencylower than the spin frequency when x� X. For example,numerical solution of Eq. (3) gives an expected nutation fre-quency of 102 rad/s for Top 1 when x¼ 123 rad/s. A likelyexplanation of the nutation shown in Fig. 5 is that the topwas slightly asymmetrical and therefore dynamically unbal-anced, despite care being taken to avoid this problem. Theproblem remained unresolved and persisted even when thesharply pointed tip was re-sharpened several times in casethere was an asymmetry in the tip itself.

Different behavior was observed when Top 1 was spun atlow frequency, as shown in Fig. 6. In this case the top pre-cessed at a relatively large tilt angle h and was stronglymodulated by nutation at a frequency about three timeshigher than the precession frequency. The result is welldescribed by solutions of Eq. (3). Nutation has the effect ofintroducing strong modulation of both x and X during eachprecession cycle, a result that was apparent simply when

observing the top by eye. The motion was quite “jerky,” de-spite the fact that the apparent spin observed in the labora-tory reference frame remained almost constant with timeduring any given precession cycle. Solutions of Eq. (3) showthe same effect. That is, xþ X remains almost constant intime, despite the fact that x and X both vary strongly withtime and reverse sign several times during each precession

TABLE I. Parameters of the four tops.

Top A (mm) M (g) H (mm) I3 (kg�m2Þ I1 (kg�m2)

1 0.1 105 21.0 7:23� 10�5 8:48� 10�5

2 0.1 105 31.0 7:23� 10�5 1:39� 10�4

3 7.5 123 21.5 7:27� 10�5 9:11� 10�5

4 17.4 6.3 15.0 1:12� 10�6 2:56� 10�6

Fig. 5. Motion of the upper end of Top 1, viewed from above, recorded over

four different time intervals during a single spin of the top. Each time interval

corresponds to one low-frequency precession cycle. The (x, y) coordinates of

the upper end are plotted at intervals of 0.01 s, as indicated by the dots in the

outer two trajectories. As time passes the angle of inclination of the top

increases until it eventually falls at t¼ 31 s after completing 430 revolutions.

Fig. 6. Motion of the upper end of Top 1, viewed from above, when the spin

is initially small, showing the first precession cycle. The (x, y) coordinates of

the upper end are plotted at intervals of 1/150 s, as indicated by the dots. The

top fell at t¼ 7.2 s after completing 50 spin revolutions. A 300-fps video of

this motion can be viewed in the online version of the paper or downloaded

from the online supplement (enhanced online) [URL: http://dx.doi.org/

10.1119/1.4776195.1] [URL: http://dx.doi.org/10.1119/1.4776195.2]

[URL: http://dx.doi.org/10.1119/1.4776195.3] [URL: http://dx.doi.org/

10.1119/1.4776195.4] [URL: http://dx.doi.org/10.1119/1.4776195.5]

[URL: http://dx.doi.org/10.1119/1.4776195.6].21

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cycle. The values of x and X quoted in Fig. 6 are time aver-aged values over one complete precession cycle.

A comparison between the observed and predicted steadyprecession frequencies of Tops 1 and 2 is shown in Fig. 7. Solu-tions of Eq. (1) are relatively insensitive to the assumed angleof inclination h so the simplifying assumption that h ¼ 10� wasmade in Fig. 7, corresponding to a typical tilt angle. Tops 1 and2 were observed to fall when x decreased below about 15 rad/sand 40 rad/s respectively, as expected from Eq. (2).

An interesting result was obtained with Tops 1 and 2 afterthey fell onto the horizontal table. The precession rapidlyreversed direction because the tops started rolling on theouter edge of the disk about a vertical axis through the pointyend. The pointy end remained fixed on the table so the diskrolled along a circular path, with the disk in Fig. 4 resting onthe table. The rolling condition was accurately described byEq. (5), r being the radius of the disk (38 mm) and R0 beingthe horizontal distance from the sharp end of the peg tothe edge of the disk. Video taken at 30 fps with a 20-mmlong peg showed that the disk started rolling on its edgewhen X ¼ �8:8 rad=s and x ¼ þ10:0 rad=s, correspondingto an apparent spin of xþ X ¼ 1:2 rad=s measured inthe laboratory frame of reference. According to Eq. (5), x¼ �R0X=r ¼ �1:135X when the disk was rolling, as meas-ured experimentally to within 1% over a 20-s interval whilethe disk gradually rolled to a stop. The significance of thisresult is described in Sec. VII A.

V. RESULTS WITH TOP 3

Results obtained with Top 3 are shown in Fig. 8. The topwas set spinning at x¼ 56 rad/s and it continued to spin for80 s before falling. During that time, the tilt angle decreasedsteadily from 23� at t¼ 0 to 4:7� at t¼ 50 s before increasingsharply at t 60 s, marking the beginning of the fall. Thesteady rise of the top is shown in Fig. 8. During the risephase, the top spiraled slowly inwards to the center of its ini-tial 62-mm radius path until the radius decreased to about1 mm, by which time the spin x had decreased to about32 rad/s. During the whole rise phase a small amplitude,high-frequency precession of the top was observed at thesame frequency as the spin frequency x indicating that the

top was slightly asymmetrical. The low-frequency preces-sion of the top was in excellent agreement with Eq. (4), asindicated in Fig. 8. The solution of Eq. (4) shown in Fig. 8was obtained using best-fit curves to the x, R, and h data inorder to calculate X as a function of time.

The top spiraled inwards by rolling rather than sliding,with rx equal to R0X within experimental error up tot¼ 50 s. Beyond that time it was not possible to ascertainwhether the top was rolling or sliding due to the relativelylarge percentage fluctuations in both R0 and h as theyapproached zero. Beyond t¼ 50 s the top appeared visuallyto roll around a vertical precession axis passing through orclose to the center-of-mass of the top.

VI. BEHAVIOR OF THE TIPPE TOP

The most interesting behavior of a tippe top occurs whenit is spun rapidly, in which case the top quickly inverts andends up spinning on its peg. The behavior at low spin-rates isalso relevant and of interest in its own right. A tippe top thenbehaves more like a regular top but the low center-of-massgives rise to several major differences. One difference is thatthe spin axis remains nearly vertical, even though the pegitself (as well as the whole top) rotates away from its initialvertical position. At low spin-rates, the peg rotates awayfrom the vertical until it reaches a limiting tilt angle, withoutinverting, and then continues to spin at that limiting anglefor some time before righting itself.

A. Low spin behavior

At low spin-rates, the top was observed to precess at twodifferent frequencies simultaneously. This behavior was par-ticularly evident when the top was spun with its peg inclinedinitially about 10� away from the vertical. The behavior ofthe top was then qualitatively similar to Top 3 since the topspiralled inwards as it precessed slowly in an approximatelycircular path. A typical result is shown in Fig. 9 where theinitial spin about an axis through the peg was 2660:2 rad=s,measured in the laboratory reference frame. The spin

Fig. 7. Precession data obtained with Top 1 (solid dots) and Top 2 (open

squares). The solid and dashed curves are solutions of Eq. (1) for these tops,

assuming h ¼ 10�.

Fig. 8. Precession data obtained with Top 3. The solid dots (open circles)

are the experimental data for the spin x (precession frequency X) vs. time.

Also shown are best fit curves to the experimental data for h and R0 vs. time.

The curve passing through the X data is the solution given by Eq. (4). Vid-

eos of the motion, taken at 300 fps, can be viewed in the online version of

the paper or downloaded from the online supplement.21

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decreased linearly to 11 rad/s over 12 s. Only the first 2 s isshown in Fig. 9, while the tilt angle increased from about 5�

to about 20�. The peg precessed about a vertical axis throughits center-of-mass at an initial rate X1 ¼ 3060:2 rad=s and itsimultaneously precessed at X2 ¼ �2:460:2 rad=s about avertical axis located near the outside edge of the 35-mm di-ameter top. The two precession frequencies are not simplythe two solutions of Eq. (4) for this top, nor do they corre-spond to a low-frequency precession combined with a high-frequency nutation. The observed high-frequency precessioncorresponds to the low-frequency solution of Eq. (4) whenR¼ 0. In that case, Eq. (4) reduces to

MgD ¼ I3xXþ ðI3 � IcmÞX2 cos h: (6)

For a tippe top, Icm is approximately equal to I3 so the secondterm on the right-hand-side of Eq. (6) can be ignored at lowspin frequencies, giving X � MgD=ðI3xÞ. The precessionfrequency is then the standard textbook result but there aretwo unusual features for a tippe top. The first is that D is neg-ative because the center-of-mass is below the center-of-cur-vature. The second is that x is also negative because the topprecessed at a higher frequency than the observed spin of thetop in the laboratory reference frame. Taking D ¼ �2:4 mm,I3 from Table I, and x ¼ 26� 30 ¼ �4:0 rad=s givesX ¼ 29:3 rad=s, essentially as observed.

Because the tippe top precessed at the higher frequencyabout an axis through its center-of-mass, the rolling condi-tion is given from Eq. (5) by Ax ¼ DX or X=x ¼ �7:25 forthis top. The observed ratio was X=x ¼ �7:560:7, consist-ent with visual and slow motion video observations that thetippe top rolled on the horizontal surface while its center-of-mass was rising.

The observed low-frequency precession corresponds tothe solution of Eq. (4) when R is taken as about –16 mm and

h � 11�. A negative value of R is required in Eq. (4) becausethe torque on the tippe top due to the centripetal force acts inthe same direction as that due to the normal reaction force.The tilt angle h does not remain constant while the top pre-cesses at high frequency and while it slowly tilts, but thequoted value can be taken as a time average during one low-frequency precession cycle. In that case, Eq. (4) indicatesthat X ¼ �2:4 rad=s (as observed) when R ¼ �16 mm andx ¼ 26þ 2:4 ¼ 28:4 rad=s (i.e., the spin of the top in a coor-dinate system rotating at –2.4 rad/s).

B. Fast spin behavior

A preliminary experiment with the tippe top showed thatit occasionally inverted when spun at high speed, but it didso by pausing for about one second after the peg had rotatedthrough an angle of about 100�. In that position, the center-of-mass was directly above the contact point on the horizon-tal surface so the normal reaction force then passed throughthe center-of-mass. However, the apparent stability of thetop in that orientation was traced to an almost imperceptibleridge where the two halves of the plastic top were joined.The ridge was removed with a fine file, with the result thatthe one second pause was eliminated and the top invertedalmost every time it was spun rather than just occasionally.

At high spin-rates, there was no observable low-frequencyprecession of a tippe top because the top inverted well beforeit completed one low-frequency precession cycle. Instead,the top precessed about a vertical axis passing through thecenter-of-mass while the axis of symmetry (passing throughthe peg) tilted slowly away from the vertical until the topwas fully inverted.

A typical inversion result for the modified tippe top isshown in Fig. 10. The top was given an initial spin of X¼ 228 rad=s by hand about a vertical axis. The top precessedrapidly about this axis the entire time while rotating slowlyabout a horizontal axis. During the interval t¼ 0 to t¼ 1.06 s,the top rotated on its spherical shell. The peg first touchedthe surface at t¼ 1.06 s and the shell lost contact with thesurface at t¼ 1.09 s. At t¼ 1.17 s the top became airborne;the peg then landed back on the surface, bounced severaltimes before becoming airborne again, jumping to a heightof 2 mm off the aluminum surface. Jumping and bouncingcontinued up to t¼ 1.41 s, and from then on the pegremained in contact with the surface. By t¼ 1.50 s, the tophad completely inverted, having rotated by 180�. The air-borne phase was reproducible and was not caused by anyirregularities in the horizontal surface, despite the fact thatthe aluminum surface was slightly scratched. The effect wasalso observed on smoother surfaces. An aluminum surfacewas used to record data for the tippe top in order to estimatethe jump height more accurately from the measured distancebetween the tippe top and its reflected image.

The spin x was estimated from a side-on camera viewusing two different techniques. While the peg remainedapproximately vertical, any given mark on the top rotated tothe front of the tippe top in a time that could easily be inter-preted in terms of the spin of the top as measured in the labo-ratory reference frame. The spin x in the rotating referenceframe was obtained by subtracting the measured precessionfrequency shown in Fig. 10(a). During the first 0.7 s, xremained approximately constant at about 2 rad/s. By thetime the peg had rotated into an approximately horizontalposition, marks on the bottom section of the tippe top had

Fig. 9. Observed trajectory of the tippe top peg when the top was spun at

low speed. Observed from above, the top spins counter-clockwise in the lab-

oratory reference frame, precesses slowly along a 16-mm-radius circular

path in a clockwise direction, and precesses rapidly around a small radius

path in a counter-clockwise direction. The center of the peg is shown by

dots at intervals of 0.02 s. The peg rotated slowly away from a vertical posi-

tion but the top did not invert. Videos of the motion can be viewed in the

online version of the paper or downloaded from the online supplement.21

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become visible and their angular displacement was recordedeach time the peg pointed to the left or the right or awayfrom the camera, once every precession cycle. The marksrotated slowly around the axis of symmetry, giving a directmeasure of x consistent with the first technique. As indicatedin Fig. 10(b), x reversed sign at t¼ 0.82 s when h � 90

�; this

effect was first observed and explained by Pliskin10 almost60 years ago by spinning a tippe top on carbon paper. Pliskinnoted that x was relatively small, but did not measure itsmagnitude.

The magnitude of x is of interest for two reasons. It indi-cates that (a) the top was sliding rather than rolling on thehorizontal surface and (b) the top precessed in a mannerqualitatively consistent with Eq. (6). As described previ-ously, the rolling condition for the tippe top is satisfied ifX=x ¼ �7:25. Apart from the fact that X was much largerthan 7:25x when the top was spun at high speed on its shell,x was of the wrong sign for the first 0.82 s to satisfy the roll-ing condition.

Solutions of Eq. (6) are shown in Table II for conditionsrelevant to the results in Fig. 10 and with D ¼ �2:4 mm; I3

¼ 1:12� 10�6 kg �m2, and Icm ¼ 1:14� 10�6 kg �m2. Thesolutions were obtained by varying x to obtain a value of X2

consistent with the data in Fig. 10(a). The resulting value ofx is at least qualitatively consistent with the data in Fig.10(b). The low-frequency solution of Eq. (6) was notobserved. Exact agreement with steady precession solutionsof Eq. (6) is not expected since the rise of a tippe top is adynamic process involving a frictional torque in addition tothe torque due to the normal reaction force. Nevertheless, itis clear from the data in Table II that when a tippe top spinsat high frequency it precesses at a frequency that is close tothe higher of the two available precession frequencies.

VII. DISCUSSION

A. Rolling condition

The rolling experiment with Tops 1 and 2 provided a use-ful and accurate check on the validity of Eq. (5), and alsoprovided insights into the differences between the three dif-ferent “spins” x, x3, and xþ X. Barger and Olsson9 quotean incorrect version of the rolling condition for a spinningtop, effectively replacing x in Eq. (5) with x3. It is relativelycommon for authors to describe x3 as the spin of a top aboutits axis of symmetry, even though x3 cannot be measured orviewed directly in the laboratory.9,22 From a theoretical pointof view, the components of the angular velocity vector ofgreatest interest are x1 ¼ X sin h and x3 ¼ xþ X cos h, asindicated in Fig. 1. Both components can be calculated frommeasurements of X, x, and h, which are the primary quanti-ties of interest experimentally. The rolling condition for atop, such as the one shown in Fig. 3, can be determined ei-ther from the components x1 and x3 or from X and x, butnot from x3 and X as assumed by Barger and Olsson in theirEq. (6-178). Confusion can easily arise because x and x3

both point along the axis of symmetry of a top; however,they differ in magnitude.

It is instructive to derive the rolling condition for thefallen Tops 1 and 2 using the two different approaches. Thegeometry is shown in Fig. 11. Precession on its own wouldcause the contact point P (on the edge of the top) to emergeout of the page at speed v ¼ R X, where R ¼ r þ A cos h.Spin on its own would cause P to rotate into the page atspeed Ax. If the disk is rolling (without slipping) then Premains at rest and

A x ¼ ðr þ A cos hÞX: (7)

The same result can be derived in terms of x1 and x3.The angular velocity of the disk has a component x3

¼ x� X cos h along the spin axis and a component x1

¼ X sin h perpendicular to the spin axis. The x3 componentwould cause P to rotate into the page at speed Ax3. The x1

component would cause P to rotate out of the page at speedHx1, where H is the length of the peg. Since P remains atrest we have

Fig. 10. Typical result showing (a) the precession frequency X and tilt angle

h, and (b) the spin x vs. time for the tippe top. The spin reversed direction

when h � 90�. The top became airborne and bounced several times soon af-

ter the peg touched the surface. A 300-fps video of the jumping tippe top

can be viewed in the online version of the paper or downloaded from the

online supplement.21

TABLE II. Solutions of Eq. (6) for the conditions shown in Fig. 10.

h 8� 50� 87� 110� 130�

x (rad/s) 3.5 2 –0.5 –1.83 –2.7

X1 (rad/s) –32 –51 –728 121 70

X2 (rad/s) 230 225 194 178 166

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Aðx� X cos hÞ ¼ HX sin h ¼ rX; (8)

which is the same as Eq. (7).

B. Rolling vs. sliding

The rise in the center-of-mass of a spinning top or a tippetop is usually explained in terms of sliding friction acting atthe bottom end. However, this explanation does not accountfor the observed rise of Top 3 or the rise of the tippe top atlow spin rates because the tops were found to roll on the hor-izontal surface while they were rising. But rolling friction byitself does not account for the rise of the tops either. Con-sider the situation shown in Fig. 2(a) or in Fig. 3 when thetop is on the left side of the precession axis and is rollinginto the page. Rolling friction opposes rolling motion andtherefore acts at point P in a direction out of the page.Therein lies not one but two significant problems. The first isthat rolling friction exerts a torque on the spherical peg in adirection that would increase the spin of the top. In fact, thespin of Top 3 decreased with time, a result that could perhapsbe explained by an even larger torque due to air resistance.The second problem is that the torque on the top due to roll-ing friction, acting about the center-of-mass, should causethe top to fall rather than rise.

The rise of a spinning top, resulting in a decrease in thetilt angle, can be regarded as an effect due to the friction tor-que, in the same way that the torque due to the gravitationalforce results in the precession shown in Fig. 1. That is, thetop precesses or tilts in such a way that the change in theangular momentum points in the same direction as theapplied torque. It therefore appears that the friction force act-ing at P must act in a direction into the page rather than outof the page, a result that would arise if rx was larger thanR0X and if point P in Fig. 3 was therefore sliding out of thepage while the peg as a whole moved into the page. This iswhy it is usually assumed that sliding rather than rolling fric-tion must be responsible for the rise of tops (tippe tops inparticular11,14,15) and is why Crabtree7 noted in his 1909book, as did others before him, that a horizontal force that“hurries” the precession will cause a spinning top to rise.

There is an alternative explanation of the two problems,not previously considered in relation to spinning tops, con-cerned with the nature and origin of rolling friction.23,24 Theexplanation is illustrated in Fig. 12. If a spherical ball ofmass M and radius R is rolling in a straight line on a horizon-

tal surface at speed v and angular velocity x, then v ¼ Rx atall times, even if v is decreasing with time. The rolling fric-tion force F acts to decrease v and it also exerts a torque onthe ball which has the effect of increasing x. Consequently,rolling cannot be maintained without some other torque tocounter the effect of the friction torque. In practice, it isfound that a ball can indeed slow down while continuing toroll without sliding, a result that can only be explained if thenormal reaction force N acts through a point located a dis-tance S ahead of the center of the ball, as indicated in Fig.12. In this case, the ball can roll with v ¼ Rx and withdv=dt ¼ Rdx=dt, consistent with

F ¼ �Mdv=dt and FR� NS ¼ Idx=dt; (9)

where I is the moment of inertia of the ball about an axisthrough its center-of-mass. Because N ¼ Mg we find that thecoefficient of rolling friction is given by

l ¼ F

N¼ MRS

I þMR2; (10)

and that x can decrease with time despite the fact that F byitself would lead to an increase in x—the net torque on theball is in the opposite direction to that due to F alone.

The same effect can be invoked to explain the behavior ofTop 3 as it rolls along a spiral path. The horizontal frictionforce acting on the top can be estimated from the linear decel-eration of the center-of-mass along the spiral path, givingF¼ 0.0013 N at t¼ 0, which corresponds to l ¼ 0:0011. Thecoefficient of friction decreased even further as the topslowed down along the spiral path. The friction torque actingon Top 3 can be estimated from the rate at which thespin decreases, ignoring air resistance. Since I3 ¼ 7:27�10�5 kg �m2 for Top 3 and dx=dt ¼ �0:48 rad=s2 averagedover the first 50 s, the average torque on the top about thespin axis was 3:49� 10�5 N �m. To simplify the followingcalculation, we will assume that such a torque arises from anequivalent friction force FE acting in the opposite direction tothe expected direction (due to the offset in N), in which casethe torque about the spin axis in Fig. 3 is FEr ¼ FEA sin h.The data for x and h in Fig. 8 yields a time-averaged valueFE 0:03 N. Since Mg ¼ 1:21 N for Top 3, this value of FE,and the corresponding value for F, are consistent with rollingand are much too small to be consistent with sliding.

The effect of the torque due to FE acting about the center-of-mass of the top would be to decrease the tilt angle h at arate dh=dt �FEH=ðI3xÞ �0:015 rad=s. In fact hdecreased by 19� over 50 s, as shown in Fig. 8, at an averagerate dh=dt ¼ �0:007 rad=s, or about half the estimated rate.

Fig. 11. Tops 1 and 2 rolled about point O on the edge of their disks when

they fell onto the table, with x and X in opposite directions as drawn. The

rolling condition in this case is given by Eq. (7) or Eq. (8).

Fig. 12. The forces on a rolling ball include a horizontal friction force F and

the normal reaction force N acting a distance S ahead of the center of the ball.

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Given that air resistance also acts to reduce x, it is certainlypossible that FE was about half its estimated value. A moredetailed investigation of the rate of rise is warranted, but thesimple estimate outlined here shows that the relevant forceresponsible for the rising of Top 3 is that due to rolling andnot that due to sliding.

C. The tippe top

The observed inversion of the tippe top was consistentwith previous observations of its behavior, although the air-borne phase seems not to have been previously documentedfor a tippe top. It has, however, been previously observedwith spinning eggs,22,25 and similar jumping behavior hasalso been observed with a “hopping” hoop.26 The jumpheight of the tippe top, about 2 mm, was considerably higherthan the 0.08-mm jump observed with spinning eggs. Inchecking the latter result, the author found that even a bil-liard ball can jump by about 0.1 mm when rolling along astraight line on carbon paper on a relatively smooth surface,a result that was presumably due to slight surface roughnessrather than any asymmetry in the ball.

At higher spin rates than the one shown in Fig. 10 the topalso became airborne for a brief period well before the pegtouched the surface. From the rate of change of the tilt anglejust before the largest jump it was estimated that the verticalacceleration of the center-of-mass, due to its vertical dis-placement, was only about 1 m/s2 at most. Such a result can-not explain the jump. However, the center-of-mass of the topdid not rise along a vertical path. Rather, it rose along an arcarising from a rapid spin about the vertical axis combinedwith a lower-speed spin about a horizontal axis. The exactpath and the velocity of the center-of-mass could not bemeasured, but a reasonable estimate is that the velocity wasabout 0.1 m/s and the arc radius was about 1 mm. As a result,the centripetal acceleration of the center-of-mass could havebeen as high as 10 m/s2 in a vertically downward direction,in which case the normal reaction force on the peg wouldindeed have dropped to zero as the top rose and then jumpedoff the surface.

A more fundamental question, addressed previously bymany authors, is why a tippe top actually inverts. Slidingfriction is invoked by most authors to explain the inversion,although the experimental evidence for sliding has beenbased primarily on the interpretation of skid marks on graph-ite.2,3,5 In 1957, Parkyn claimed5 that “There can be no doubtthat the fundamental motion of a top is one of rolling, andthat rolling friction is necessary to explain the nature of therise.” In the present paper, direct measurements of the spinabout the symmetry axis have shown that a tippe top rolls atlow spin frequencies and slides at high spin frequencies. Thecenter-of-mass rises in both cases. The dynamics of the pro-cess needs to be studied in more detail to understand whyrolling occurs only at low spin rates, but it is clear that roll-ing can occur only if the ratio X=x is about 7 or so, depend-ing on the geometry and inertial properties of the tippe top.The equations describing steady precession indicate that thiscondition is satisfied only if X is relatively small. Behavioranalogous to that observed with the tippe top is also observedwith a spinning egg. A spinning egg rolls and precesses attwo different frequencies simultaneously when it is spun atlow speed.27 A sphere or bowling ball that is projected alonga horizontal surface, while spinning about a near verticalaxis, also rolls if it is projected at relatively low speed.28

VIII. CONCLUSION

Four different spinning tops were investigated by filmingtheir behavior with a high-speed video camera. Two of thetops had a sharply pointed peg, one had a 15-mm diameterspherical peg, and the last was a tippe top. All were found toprecess at rates consistent with those expected for steady pre-cession. A high-frequency precession was also observed forthe tops with pointed ends, coinciding with the spin fre-quency at high spin-rates and most likely due to a smallasymmetry in each top.

The tops with a sharply pointed peg precessed about a ver-tical axis passing through the bottom of the peg, while thetop with a spherical peg precessed initially about a verticalaxis located well outside the top and then spiraled inwardsuntil the precession axis passed through a point close to thecenter-of-mass. During that time, the center-of-mass rosegradually until the top was almost vertical, a result that couldbe attributed to the fact that the spherical peg rolled along aspiral path without sliding.

The well known but still fascinating rise in the center-of-mass of a tippe top is usually attributed to sliding friction atthe base of the top, but it was found that the center-of-massrose even when the top was rolling. Rolling is not normallyconsidered as a candidate to explain the rise of a spinning topbecause the friction force acts in the wrong direction. How-ever, the net torque on a rolling sphere does act in the correctdirection due to an offset in the normal reaction force when asphere rolls along a horizontal surface. Inversion of the tippetop did not occur at a steady rate. The top was observed tojump off the surface before inverting, contrary to the usualtheoretical assumption that the normal reaction force on atippe top is approximately equal to its weight.14,15 When spunat high frequency, the tippe top was found to precess at a fre-quency close to the higher of the two available precession fre-quencies. By contrast, a conventional top usually precesses atthe lower of the two available precession frequencies.

In addition to the rolling vs. sliding question, there aremany other aspects of spinning tops and other spinningobjects that could be investigated further by video techni-ques. For example, what difference does it make if the hori-zontal surface is smooth or rough or lubricated? Is energy orangular momentum conserved when a top rises? What deter-mines the rate of rise or fall of a spinning top? All of thesequestions could be investigated as student projects. There isa large variety of tops and gyros that are available for study,recently reviewed and colorfully illustrated by Featonby.29

a)Electronic mail: cross@physics.usyd.edu.au1C. M. Braams, “On the influence of friction on the motion of a top,” Phys-

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8, 591–596 (1941).3A. D. Fokker, “The tracks of top’s pegs on the floor,” Physica (Amster-

dam) 18, 497–502 (1952).4N. M. Hugenholtz, “On tops rising by friction,” Physica (Amsterdam) 18,

515–527 (1952).5D. G. Parkyn, “The Rising of Tops with Rounded Pegs,” Physica (Amster-

dam) 24, 313–330 (1958).6J. Perry, Spinning Tops and Gyroscopic Motions (Sheldon, London, 1890,

reprinted by Dover, 1957).7H. Crabtree, An Elementary Treatment of the Theory of Spinning Tops andGyroscopic Motion (Longmans Green, London, 1909, reprinted by Chel-

sea, 1967).8R. F. Deimel, Mechanics of the Gyroscope: The Dynamics of Rotation(Dover, New York, 1952).

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9V. Barger and M. Olsson, Classical Mechanics, A Modern Perspective,

2nd ed. (McGraw-Hill, New York, 1994).10W. A. Pliskin, “The tippe top (topsy-turvy top),” Am. J. Phys. 22, 28–32 (1954).11R. J. Cohen, “The tippe top revisited,” Am. J. Phys. 45, 12–17 (1977).12L. Stefanini, “Behavior of a real top,” Am. J. Phys. 47, 346–350 (1979).13K. Schonhammer, “Elementary theoretical description of the heavy sym-

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tops and other tops with round pegs,” Am. J. Phys. 68, 821–828 (2000).15H. Soodak, “A geometric theory of rapidly spinning tops, tippe tops, and

footballs,” Am. J. Phys. 70, 815–828 (2002).16M. V. Berry and P. Shukla, “Slow manifold and Hannay angle in the spin-

ning top,” Eur. J. Phys. 32, 115–127 (2011).17D. Petrie, J. L. Hunt, and C. G. Gray, “Does the Euler Disk slip during its

motion?” Am. J. Phys. 70, 1025–1028 (2002).18H. Caps, S. Dorbolo, S. Ponte, H. Croisier, and N. Vandewalle, “Rolling

and slipping motion of Euler’s disk,” Phys. Rev. E 69, 056610-1–6 (2004).19F. R. Zypman, “Moments to remember - The conditions for equating torque

and rate of change of angular momentum,” Am. J. Phys. 58, 41–43 (1990).

20R. Cross, “The fall and bounce of pencils and other elongated objects,”

Am. J. Phys. 74, 26–30 (2006).21See supplementary material at http://dx.doi.org/10.1119/1.4776195 for

some sample movies.22H. K. Moffatt and Y. Shimomura, “Spinning eggs—a paradox resolved,”

Nature, 416, 385–386 (2002).23J. Witters and D. Duymelinck, “Rolling and sliding resistive forces on

balls moving on a flat surface,” Am. J. Phys. 54, 80–83 (1986).24A. Domenech, T. Domenech, and J. Cebrian, “Introduction to the study of

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“Can a spinning egg really jump?,” Proc. R. Soc. London, Ser. A 462,

2897–2905 (2006).26M. F. Maritz and W. F. D. Theron, “Experimental verification of the

motion of a loaded hoop,” Am. J. Phys. 80, 594–598 (2012).27R. Cross, “Spinning eggs and ballerinas,” Phys. Ed. (accepted).28R. Cross, “Rolling motion of a ball spinning about a near-vertical axis,”

Phys. Teach. 50, 25–27 (2012).29D. Featonby, “Dare we teach tops?” Phys. Ed. 45, 409–420 (2010).

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