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The rise and fall of spinning tops

Article  in  American Journal of Physics · April 2013

DOI: 10.1119/1.4776195

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 The rise and fall of spinning tops

Rod Cross∗
Department of Physics, University of Sydney, Sydney NSW 2006, Australia

Abstract
The motion of four different spinning tops was filmed with a slow motion 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. It jumped vertically off a horizontal surface several times while the center of
mass was rising, contrary to the usual theoretical 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.

1
I. INTRODUCTION

The most fascinating aspect of a spinning top is that it can temporarily defy gravity by
moving sideways and upward before it eventually falls. If the top spins fast enough it can
rise to a sleeping position where the spin axis remains vertical. Early experimenters and
theoreticians1–4 established that the rising of the center of mass of a spinning top (or an egg
or a football or a tippe top) is due to a torque arising from sliding friction at the bottom
end. Parkyn5 disagreed, giving experimental evidence that rolling friction was responsible.
Air friction and friction at the base of the top act to decrease the angular velocity of the top
and it eventually starts to fall away from the vertical position. As it does so, the top starts
to precess, the rate of precession being proportional to the height of its center of mass and
inversely proportional to the angular momentum of the top. At least, that is the case for
the idealised top treated in elementary physics textbooks.
There is now an extensive literature on the theory of spinning tops6–16 , but very little data
on measured precession rates. In fact, the author was unable to find any such data. The early
experimenters measured the angular velocity of a top using a stroboscope, and could measure
the inclination angle and the path of the bottom end on carbon paper or graphite, but did
not provide any measurements of precession rates. The advent of relatively cheap high speed
video cameras makes such a measurement straightforward, and suitable for an experiment
or project in an undergraduate laboratory. Results obtained by the author are presented
below and are compared with simple theoretical estimates. The experiment described here
is similar to others described previously concerning the precession of a spinning disk.17,18
Results were obtained for (a) a sharply pointed top, (b) a top with a rounded peg and
(c) a tippe top. All three tops behave in a qualitatively different manner. If the bottom
end of a top is sharply pointed then the bottom end moves in a tight circular path of very
small radius, in which case the center of mass precesses slowly about a vertical axis passing
through the bottom end. If the top has a rounded peg at the bottom end then the bottom
end can roll along a relatively large radius, approximately circular path. In that case, the
upper and lower ends of the top, as well as the center of mass, all precess 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, in which case the centre of mass of
the top remains fixed in space.

2
 A tippe top consists of a truncated sphere with a short peg on top. When the peg is spun
between the fingers, the tippe top precesses rapidly about a vertical axis while the whole
top rotates slowly about a horizontal axis until it ends up spinning upright on the peg. A
similar inversion and rise of the center of mass occurs when a circular disk with a large hole
on one side is spun about a vertical axis. If the hole is initially at the top then the disk rolls
along its edge until the hole is at the bottom, the spin axis remaining vertical. If the hole is
initially at the bottom, it remains at the bottom.

II. SIMPLIFIED THEORETICAL DESCRIPTION OF A TOP

z
ω
Ω
H x
(a)
Mg

z L =I3ω
3
(b) ω
Ω

L = I1 Ω sin θ
θ
Perpendicular
axis in xz plane x
O
Lx = I 3 ω sin θ
3

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 θ 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.

The equations describing the dynamics of a spinning top are often cast in forms that
are far too mathematically complicated for students to understand the underlying physics.
Steady precession of a spinning top (or gyroscope) can be described in a simple and more
intuitive manner by reference to Fig. 1. Elementary treatments are given by Crabtree7 , by
Deimel8 and by Barger and Olsson9 . If the top is spinning at angular velocity ω about a
horizontal (x) axis, and is supported at the left end as in Fig. 1(a), then the top will precess

3
at angular velocity Ω about the vertical (z) axis. The angular momentum, L, is in the x
direction while the change in the angular momentum is in the same direction as the applied
torque, τ = M gH, which points in the y direction. Since τ = dL/dt and L = Iω, it is easy
to show that Ω = M gH/Iω where I is the moment of inertia of the top about the spin axis.
This is the standard result derived in undergraduate physics textbooks, and it also applies
to a top inclined at an arbitrary angle to the vertical.
A spinning top is usually supported at its bottom end at a point O on a horizontal surface
and the spin axis is inclined at an angle θ to the vertical, as indicated in Fig. 1(b). In that
case, the torque in the y direction about an axis through O is given by τ = M gH sin θ.
Any horizontal force through O, including the centripetal force required to rotate the top
about the z axis, does not contribute to this torque. A complication in Fig. 1(b) is that the
moment of inertia, I3 , for rotation about the spin axis, is usually smaller than the moment
of inertia, I1 , for rotation about an axis perpendicular to the spin axis and passing through
O. In Fig. 1(b), the top precesses at angular velocity Ω about the z axis, by rotating into
the plane of the page about the pivot point at O. Consequently, the top also rotates about
the perpendicular axis shown in Fig. 1(b).
The component of the angular momentum of the top in a direction along the perpendicular
axis is L⊥ = I1 Ω sin θ. The component of Ω in a direction parallel to the spin axis is
Ω cos θ. The total angular momentum in a direction parallel to the spin axis is therefore
Lk = I3 (ω + Ω cos θ) = I3 ω3 where ω is the spin imparted to the top about the spin axis
and ω3 = ω + Ω cos θ. The component of Lk along the x axis is Lx = Lk sin θ. The total
angular momentum in the x direction is therefore I3 ω3 sin θ − I1 Ω sin θ cos θ. By equating
the torque in the y direction to the rate of change of angular momentum in the x direction,
as in Fig. 1(a), we find that

M gH = I3 ω3 Ω − I1 Ω2 cos θ , (1)

which is quadratic in Ω and therefore yields two real solutions for Ω, provided that

ω > (2/I3 )[M gH(I1 − I3 ) cos θ]1/2 . (2)

The lower frequency solution (denoted by Ω1 ) is the one usually described in elementary
textbooks, and is the one that is usually observed experimentally, while the higher fre-
quency solution (denoted by Ω2 ) is comparable to ω3 . The textbook result is recovered when

4
ω  Ω, in which case the second term on the right side of Eq. (1) can be ignored and then
Ω ≈ M gH/(I3 ω), regardless of the angle of inclination, θ.
If the spin, ω, is less than that given by Eq. (2) then the top will not precess in a steady
manner and will instead fall rapidly to the ground. Otherwise, a rapidly spinning top tends
to precess slowly, at the lower frequency, while simultaneously precessing in small sub-loops
at high frequency. The sub-loops grow in amplitude and decrease in frequency as the top
slows down. They arise from nutation of the top. That is, the inclination of the top varies
periodically with time, according to the relation9
d2 θ
I1 2
= (M gH + I1 Ω2 cos θ − I3 ω3 Ω) sin θ . (3)
dt
Numerical solutions of Eq. (3) can be obtained by noting that in the absence of friction the
angular momentum parallel to the spin axis remains constant in time, as does the angular
momentum in the z direction, Lz = I1 Ω sin2 θ + I3 ω3 cos θ. The latter conditions determine
ω(t) and Ω(t) at each time step for any given values of Lk and Lz . The particular solution
of Eq. (3) given by Eq. (1) corresponds to steady precession without nutation.

(a) Ω

(c) Ω
(b)

Top sleeps

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.

5
 Analogous solutions are obtained if the bottom end of the top is rounded rather than
being tapered to a sharp point. In that case, the bottom end of the top tends to roll along
the surface supporting the top. The bottom end is not fixed in space but follows a spiral
path as indicated in Fig. 2. If the bottom end is moving, then valid solutions are best
obtained by considering the torque acting about the top’s center of mass.19,20 A component
of that torque arises from the centripetal force, given by F = M RΩ2 where R is the radius
of the path followed by the center of mass. The centripetal force is provided by friction at
the bottom end of the top, as indicated in Fig. 3. The normal reaction force, N , is equal to
the weight, M g, of the top provided that the top is not rising or falling.

ω Ω

G R
N

θ D

Q
Precession
A axis
r
F

FIG. 3: Details of a spinning top with a rounded bottom peg of radius A. G denotes the center of
mass of the top. The normal reaction force, N , and the centripetal force, F , both act through the
contact point P at the bottom end of the top.

If A is the radius at the bottom end and D is the distance QG in Fig. 3, then the torque, τ ,
about the center of mass, G, is given by τ = M gD sin θ −M RΩ2 (A+D cos θ). Consequently,

M gD − M RΩ2 (A + D cos θ)/ sin θ = I3 ω3 Ω − Icm Ω2 cos θ , (4)

where Icm = I1 − M (D + A)2 is the moment of inertia about the perpendicular axis through
G. If the top rolls along the horizontal surface then

rω = R0 Ω , (5)

where r = A sin θ is the perpendicular distance from the spin axis to the contact point, P,
and R0 = (R + D sin θ) is the radius of the spiral path traced out by the contact point, as

6
indicated in Figs. 2 and 3. Equations (4) and (5) can be combined to eliminate R, in which
case it is found that there are two possible precession frequencies, as before. If R = 0 then
F = 0 and the precession axis passes through G, as indicated in Fig. 2(c). The direction
of F is reversed in Fig. 3 if the precession axis passes through the contact point or if it is
located 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) since the torque due to F then acts
in the same direction as the torque due to N .

III. EXPERIMENTAL METHOD

Precession axis

Threaded
θ rod
Ω
ω

G Circular
disk
N = Mg
H

Table

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.

The arrangement used in the present experiment is shown in Fig. 4. A versatile top
(essentially a gyrostat or a gyroscope without its gymbals) was constructed using an 8 mm
thick aluminum disk of diameter 76 mm and mass 100.0 g. A 4 mm diameter, 60 mm long
threaded steel rod of mass 5.8 g was inserted through a hole in the center of the disk and
fixed to the disk with a nut above and below the disk. The length of the peg at the bottom
end was fixed at either 17 or 27 mm in order vary the height of the center of mass of the top.
The bottom end of the rod was ground to a sharp point. An experiment was also conducted
with a large radius peg at the bottom end using a spherical, metal drawer knob of diameter
15 mm screwed onto the bottom end as indicated in Fig. 2. Results obtained by filming a
small, plastic tippe top are also presented.

7
 Each top was spun on a smooth, horizontal surface either by hand at low speed or by
wrapping a length of string around the threaded rod to increase the speed. When using
string to spin the top, the upper end of the rod was allowed to spin inside a vertical cylinder
to keep the top approximately vertical and the cylinder was then lifted clear. Marks drawn
on the spinning top were observed by filming at 300 frames/s with a Casio EX-F1 camera,
viewing either from directly above the top or from the side in order to observe motion of the
peg on the surface. Both views provided the same information on the spin angular velocity
of the top, ω, and the angular velocity of precession, Ω. The tilt angle, θ, with respect to
the vertical was measured for convenience from the side view, by recording the tilt angle to
the left and right of the vertical axis.
A subtle feature regarding the measurement of ω is that ω is conventionally defined in
a rotating coordinate system attached to the top and rotating about the z axis at angular
velocity Ω. The moments of inertia are also defined in this rotating coordinate system so
that they remain constant in time. The top precesses at angular velocity Ω about the z axis,
so the apparent spin recorded by a fixed camera mounted above the top is ω +Ω.17 A camera
rotating at angular velocity Ω around the z axis to follow the top would record its spin as
ω. In this paper, most measurements of ω were obtained by subtracting the measured value
of Ω from the apparent spin recorded from the fixed camera. Some measurements of ω were
also obtained by recording the rotation angle of the top only 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 on which the tops were spun, the
tops were observed to spin for up to about two minutes and rotated up to about 1000
times before falling. Measurements were made of (a) the time at which each top completed
successive sets of ten spin revolutions, (b) the time to complete each successive precession
revolution and (c) the tilt angle at about 50 different times from the start to the end of each
spin. The angular speeds could be measured to an accuracy of better than 1%, and the tilt
angle could be measured to within 0.5 degrees. However, the tilt angle itself varied by up to
about five degrees during each precession revolution, due to nutation of the top. Side view
measurements of the tilt angle were made when the top tilted to the right and again when
the top tilted to the left to obtain an average of the two tilt angles. At high spin rates,
the tops were observed to rise slowly, while at low spin rates the tops were observed to fall
slowly. Under some conditions, the tops maintained a constant tilt angle when set spinning

8
on a horizontal surface.
Properties of each top, including the peg radius, A, total mass, M , and the distance (H)
from the bottom end to the center of mass, are listed in Table 1. Also shown are the moment
of inertia, I3 , about the central axis, and the moment of inertia, I1 about a transverse axis
through the bottom end. Top 3 included a 16.8 g, 15 mm diameter ball at the bottom end.
Top 4 was a hollow, plastic tippe top with an 11 mm long peg attached to a truncated,
34.8 mm diameter sphere, spun by hand on a horizontal sheet of aluminum.

Table 1. Parameters of the four tops.

Top A (mm) M (g) H (mm) I3 (kg.m2 ) 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

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
ω = 126 rad/s which decreased to 44 rad/s over 31 s before falling onto the horizontal surface.
During that time, the precession frequency, Ω, increased from 2.7 rad/s to 8.6 rad/s and the
angle of inclination of the top increased from 8 to 21 degrees. The top did not rise to a
sleeping position.
The results in Fig. 5 were obtained by plotting the (x, y) coordinate of the upper end
of the threaded rod, at intervals of 0.01 s, during four different precession cycles. The first
cycle, from t = 0.73 to 3.05 s, took 2.32 s to complete one revolution, corresponding to an
average precession frequency Ω = 2.7 rad/s. During that time, the top also rotated many
times in small radius sub-loops, at 123 ± 0.5 rad/s, coincident with the spin frequency, ω, of
the top. The standard explanation of the sub-loops is that they correspond to nutation of
the top. However, nutation is expected at a frequency lower than the spin frequency when
ω  Ω. For example, numerical solution of Eq. (3) gives an expected nutation frequency of
102 rad/s for Top 1 when ω = 123 rad/s. A likely explanation of the nutation shown in Fig. 5
is that the top was slightly asymmetrical and therefore dynamically unbalanced, despite care

9
being taken to avoid this problem. The problem remained unresolved and persisted even
when the sharply pointed tip was re-sharpened several times in case there was an asymmetry
in the tip itself.

20 18.0-19.48 s 23.6 - 24.75 s

y (mm) 10

0.73 -
0 3.05 s

-10

27.80 s
-20
28.66 s 5217

-20 -10 0 10 20
x (mm)

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 t = 31 s after completing 430 revolutions.

Different behaviour was observed when Top 1 was spun at low frequency, as indicated in
Fig. 6. In that case, the top precessed at a relatively large tilt angle, θ, and was strongly mod-
ulated by nutation at a frequency about three times higher than the precession frequency.
The result is well described by solutions of Eq. (3). Nutation has the effect of introducing
strong modulation of both ω and Ω during each precession cycle, a result that was apparent
simply when observing the top by eye. The motion was quite “jerky”, despite the fact that
the apparent spin observed in the laboratory reference frame remained almost constant with
time during any given precession cycle. Solutions of Eq. (3) show the same effect. That is,
ω + Ω remains almost constant in time, despite the fact that ω and Ω both vary strongly
with time and reverse sign several times during each precession cycle. The values of ω and
Ω quoted in Fig. 6 are time averaged values over one complete precession cycle.

10
 30
ω= 18 rad/s
t = 0.82 s
t=0
20

10

y (mm)
0

-10

-20

Ω= 30.7 rad/s
5212

-30
-30 -20 -10 0 10 20 30
x (mm)

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 t = 7.2 s after completing 50 spin revolutions.
Enhanced online Top1 Fig6.MOV at 300 fps

12

10

Top 2
(rad/s)

8
Ω

Top 1
4

2
20 40 60 80 100 120 140
ω (rad/s)

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 that θ = 10◦ .

11
 A comparison between the observed and predicted steady precession frequencies of tops
1 and 2 is shown in Fig. 7. Solutions of Eq. (1) are relatively insensitive to the assumed
angle of inclination, θ, so the simplifying assumption was made in Fig. 7 that θ = 10◦ ,
corresponding to a typical tilt angle. Tops 1 and 2 were observed to fall when ω decreased
below about 15 and 40 rad/s respectively, as expected from Eq. (2).
An interesting result was obtained with tops 1 and 2 after they fell onto the horizontal
table. The precession rapidly reversed direction since the tops started rolling on the outer
edge of the disk, about a vertical axis through the pointy end. The pointy end remained
fixed on the table, so the disk rolled along a circular path, with the disk in Fig. 4 resting on
the table. The rolling condition was accurately described by Eq. (5), r being the radius of
the disk (38 mm) and R0 being the horizontal distance from the sharp end of the peg to the
edge of the disk. Film taken at 30 fps with a 20 mm long peg showed that the disk started
rolling on its edge with Ω = −8.8 rad/s and with ω = +10.0 rad/s, corresponding to an
apparent spin ω + Ω = 1.2 rad/s measured in the laboratory frame of reference. According
to Eq. (5), ω = −R0 Ω/r = −1.135Ω when the disk was rolling, as measured experimentally
to within 1% over a 20 s interval while the disk gradually rolled to a stop. The significance
of this result is described in Sec. VII A.

V. RESULTS WITH TOP 3

Results obtained with Top 3 are shown in Fig. 8. The top was set spinning at ω = 56 rad/s
and it continued to spin for 80 seconds before falling. During that time, the tilt angle
decreased steadily from 23◦ at t = 0 to 4.7◦ at t = 50 s, before increasing sharply at t ∼ 60 s,
marking the beginning of the fall. The steady rise of the top is shown in Fig. 8. During
the rise phase, the top spiraled slowly inwards to the center of its initial 62 mm radius path
until the radius decreased to about 1 mm, by which time the spin, ω, had decreased to about
32 rad/s. During the whole rise phase, a small amplitude, high frequency precession of the
top was observed at the same frequency as the spin frequency, ω, indicating that the top was
slightly asymmetrical. The low frequency precession of the top was in excellent agreement
with Eq. (4), as indicated in Fig. 8. The solution of Eq. (4) shown in Fig. 8 was obtained
using best fit curves to the ω, R and θ data in order to calculate Ω as a function of time.
The top spiraled inwards by rolling rather than sliding, with rω equal to R0 Ω within

12
experimental error up to t = 50 s. Beyond that time it was not possible to ascertain whether
the top was rolling or sliding, due to the relatively large percentage fluctuations in both
R0 and θ as R0 and θ approached zero. Beyond t = 50 s, the top appeared visually to roll
around a vertical precession axis passing through or close to the center of mass of the top.

70 25
5201

(rad/s)

Tilt angle
60
ω 20
50
ω
Spin

θ
15

(degrees)
40

θ
R (mm)

30
R 10
0
0

20

Ω
5

(rad/s)
Ω
10

0 0
0 10 20 30 40 50
t (s)

FIG. 8: Precession data obtained with Top 3. The solid dots are the experimental data for the
spin, ω, vs time. The open circles show the precession frequency, Ω, vs time. Also shown are best
fit curves to the experimental data for θ and R0 vs time. The curve passing through the Ω data is
the solution given by Eq. (4). Enhanced online Top3 2.3-4.4s.MOV, Top3 44.8-46.6s.MOV, both
at 300 frames/s.

VI. BEHAVIOUR OF THE TIPPE TOP

The most interesting behaviour of a tippe top occurs when the top is spun rapidly, in
which case the top quickly inverts and ends up spinning on its peg. The behaviour at low
spin frequencies is also relevant and of interest in its own right. A tippe top then behaves
more like a regular top but the low center of mass gives rise to several major differences. One
difference is that the spin axis remains nearly vertical, even though the peg itself (as well
as the whole top) rotates away from its initial vertical position. At low spin frequencies the
peg rotates away from the vertical until it reaches a limiting tilt angle, without inverting,
and then continues to spin at that limiting angle for some time before righting itself.

13
 A. Low spin behaviour

30
5489

20

10

y (mm)
0

t=0
-10

-20
t = 2.06 s

-30
-30 -20 -10 0 10 20 30
x (mm)

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 laboratory 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 position but the top did not invert. Enhanced
online TippeTop1.mov and TippeTop2.mov

At low spin frequencies the top was observed to precess at two different frequencies
simultaneously. This behaviour was particularly evident when the top was spun with its peg
inclined initially about ten degrees away from the vertical. The behaviour of the top was
then qualitatively similar to Top 3 since the top spiralled inwards as it precessed slowly in an
approximately circular path. A typical result is shown in Fig. 9 where the initial spin about
an axis through the peg was 26 ± 0.2 rad/s measured in the laboratory reference frame. The
spin decreased linearly to 11 rad/s over 12 seconds. Only the first 2 seconds is shown in Fig. 9,
while the tilt angle increased from about 5◦ to about 20◦ . The peg precessed about a vertical
axis through its center of mass at an initial rate Ω1 = 30 ± 0.2 rad/s and it simultaneously
precessed at Ω2 = −2.4 ± 0.2 rad/s about a vertical axis located near the outside edge of
the 35 mm diameter top. The two precession frequencies are not simply the two solutions
of Eq. (4) for this top, nor do they correspond to a low frequency precession combined with

14
a high frequency nutation. The observed high frequency precession corresponds to the low
frequency solution of Eq. (4) when R = 0. In that case, Eq. (4) reduces to

M gD = I3 ωΩ + (I3 − Icm )Ω2 cos θ (6)

For a tippe top, Icm is approximately equal to I3 so the second term on the right hand side
of Eq. (6) can be ignored at low spin frequencies, giving Ω ≈ M gD/(I3 ω). The precession
frequency is then the standard text book result, but there are two unusual features for a
tippe top. The first is that D is negative since the center of mass is below the center of
curvature. The second is that ω is also negative since the top precessed at a higher frequency
than the observed spin of the top in the laboratory reference frame. Taking D = −2.4 mm,
I3 from Table 1 and ω = 26 − 30 = −4.0 rad/s gives Ω = 29.3 rad/s, essentially as observed.
Since the tippe top precessed at the higher frequency about an axis through its center of
mass, the rolling condition is given from Eq. (5) by Aω = DΩ or Ω/ω = −7.25 for this top.
The observed ratio was Ω/ω = −7.5 ± 0.7, consistent with visual and slow motion video
observations that the tippe top rolled on the horizontal surface while its center of mass was
rising.
The observed low frequency precession corresponds the solution of Eq. (4) when R is
taken as about −16 mm and θ ≈ 11◦ . A negative value of R is required in Eq. (4) since
the torque on the tippe top due to the centripetal force acts in the same direction as that
due to the normal reaction force. The tilt angle, θ, does not remain constant while the top
precesses at high frequency and while it slowly tilts, but the quoted value can be taken as a
time average during one low frequency precession cycle. In that case, Eq. (4) indicates that
Ω = −2.4 rad/s (as observed) when R = −16 mm and ω = 26 + 2.4 = 28.4 rad/s (ie the spin
of the top in a coordinate system rotating at −2.4 rad/s).

B. Fast spin behaviour

A preliminary experiment with the tippe top showed that it occasionally inverted when
spun at high speed, but it did so by pausing for about one second after the peg had rotated
through an angle of about 100◦ . In that position, the center of mass was directly above the
contact point on the horizontal surface so the normal reaction force then passed through the
center of mass. However, the apparent stability of the top in that orientation was traced
to an almost imperceptible ridge where the two halves of the plastic top were joined. The

15
ridge was removed with a fine file, with the result that the one second pause was eliminated
and the top inverted almost every time it was spun rather than just occasionally.
At high spin frequencies, there was no observable low frequency precession of a tippe top
since the top inverted well before it completed one low frequency precession cycle. Instead,
the top precessed about a vertical axis passing through the center of mass while the axis of
symmetry (passing through the peg) tilted slowly away from the vertical until the top was
fully inverted.

(a) (b)
250 5
5423
Ω Peg
vertical 0
Ω (rad/s)

200

-5
150
Peg on (rad/s)
surface Ω -10
(deg)

100
ω

-15
Top airborne
θ

and bouncing
50
Tilt angle

Tilt angle θ -20

(of peg)
5423

0 -25
0 0.5 1 1.5 2 0 0.5 1 1.5
t (s) t (s)

FIG. 10: Typical result showing (a) the precession frequency, Ω, and tilt angle, θ, and (b) the
spin, ω, vs time for the tippe top. The spin reversed direction when θ ≈ 90◦ . The top became
airborne and bounced several times soon after the peg touched the surface. Enhanced online
TippeTopJump.Mov at 300 fps.

A typical inversion result for the modified tippe top is shown in Fig. 10. The top was
given an initial spin Ω = 228 rad/s by hand about a vertical axis. The top precessed rapidly
about this axis the whole time, while rotating slowly about a horizontal axis. During the
period t = 0 to t = 1.06 s, the top rotated on its spherical shell. The peg first touched the
surface at t = 1.06 s and the shell lost contact with the surface at t = 1.09 s. At t = 1.17 s,
the top became airborne. The peg then landed back on the surface, bounced several times
and then became airborne again, jumping to a height of 2 mm off the aluminum surface.
Jumping and bouncing continued up to t = 1.41 s, and from then on the peg remained in
contact with the surface. By t = 1.50 s, the top had completely inverted, having rotated by

16
180◦ . The airborne phase was reproducible and was not caused by any irregularities in the
horizontal surface, despite the fact that the aluminum surface was slightly scratched. The
effect was also observed on smoother surfaces. An aluminum surface was used to record data
for the tippe top in order to estimate the jump height more accurately from the measured
distance between the tippe top and its reflected image.
The spin, ω, was estimated from a side-on camera view using two different techniques.
While the peg remained approximately vertical, any given mark on the top rotated to the
front of the tippe top in a time that could easily be interpreted in terms of the spin of the
top as measured in the laboratory reference frame. The spin, ω, in the rotating reference
frame was obtained by subtracting the measured precession frequency shown in Fig. 10(a).
During the first 0.7 s, ω remained approximately constant at about 2 rad/s. By the time
that the peg had rotated into an approximately horizontal position, marks on the bottom
section of the tippe top had become visible and their angular displacement was recorded
each time the peg pointed to the left or the right or away from the camera, once every
precession cycle. The marks rotated slowly around the axis of symmetry, giving a direct
measure of ω consistent with the first technique. As indicated in Fig. 10(b), ω reversed sign
at t = 0.82 s when θ ≈ 90◦ . The latter effect was first observed and explained by Pliskin10
almost 60 years ago by spinning a tippe top on carbon paper. Pliskin noted that ω was
relatively small, but did not measure its magnitude.
The magnitude of ω is of interest for two reasons. It indicates that (a) the top was
sliding rather than rolling on the horizontal surface and (b) the top precessed in a manner
qualitatively consistent with Eq. (6). As described previously, the rolling condition for the
tippe top is satisfied if Ω/ω = −7.25. Apart from the fact that Ω was much larger than
7.25 ω when the top was spun at high speed on its shell, ω was of the wrong sign for the
first 0.82 s to satisfy the rolling condition.
Solutions of Eq. (6) are shown in Table 2 for conditions relevant to the results in Fig. 10
and with D = −2.4 mm, I3 = 1.12 × 10−6 kg.m2 and Icm = 1.14 × 10−6 kg.m2 . The solutions
were obtained by varying ω to obtain a value of Ω2 consistent with the data in Fig. 10(a).
The resulting value of ω is at least qualitatively consistent with the data in Fig. 10(b). The
low frequency solution of Eq. (6) was not observed. Exact agreement with steady precession
solutions of Eq. (6) is not expected since the rise of a tippe top is a dynamic process involving
a frictional torque in addition to the torque due to the normal reaction force. Nevertheless,

17
it is clear from the data in Table 2 that when a tippe top spins at high frequency it precesses
at a frequency that is close to the higher of the two available precession frequencies.

Table 2. Solutions of Eq. (6) for the conditions shown in Fig. 10.

θ 8◦ 50◦ 87◦ 110◦ 130◦

ω (rad/s) 3.5 2 -0.5 -1.83 -2.7
Ω1 (rad/s) -32 -51 -728 121 70
Ω2 (rad/s) 230 225 194 178 166

VII. DISCUSSION

A. Rolling condition

The rolling experiment with Tops 1 and 2 provided a useful and accurate check on the
validity of Eq. (5), and also provided insights into the differences between the three different
“spins” ω, ω3 and ω + Ω. Barger and Olsson9 quote an incorrect version of the rolling
condition for a spinning top, effectively replacing ω in Eq. (5) with ω3 . It is relatively common
for authors to describe ω3 as the spin of a top about its axis of symmetry, even though ω3
cannot be measured or viewed directly in the laboratory.9,23 From a theoretical point of
view, the components of the angular velocity vector of greatest interest are ω1 = Ω sin θ
and ω3 = ω + Ω cos θ, as indicated in Fig. 1. Both components can be calculated from
measurements of Ω, ω and θ, which are the primary quantities of interest experimentally.
The rolling condition for a top, such as the one shown in Fig. 3, can be determined either
from the components ω1 and ω3 or from Ω and ω, but not from ω3 and Ω as assumed by
Barger and Olsson in their Eq. (6-178). Confusion can easily arise because ω and ω3 both
point along the axis of symmetry of a top. However, they differ in magnitude.
It is instructive to derive the rolling condition for the fallen Tops 1 and 2 using the two
different approaches. The geometry is shown in Fig. 11. Precession on its own would cause
the contact point P to emerge out of the page at speed v = R Ω where R = r + A cos θ. Spin
on its own would cause P to rotate into the page at speed A ω. If the disk is rolling then P
remains at rest and then
A ω = (r + A cos θ)Ω . (7)

The same result can be derived in terms of ω1 and ω3 . The angular velocity of the

18
disk has a component ω3 = ω − Ω cos θ along the spin axis and a component ω1 = Ω sin θ
perpendicular to the spin axis. The ω3 component would cause P to rotate into the page at
speed A ω3 . The ω1 component would cause P to rotate out of the page at speed Hω1 where
H is the length of the peg. Since P remains at rest,

A(ω − Ω cos θ) = HΩ sin θ = rΩ , (8)

which is the same as Eq (7).

Ω
Peg

r G ω
A
Thin disk of
θ H
radius A
θ
O R P

FIG. 11: Tops 1 and 2 rolled about point O on the edge of their disks when they fell onto the
table, with ω and Ω in opposite directions as drawn. The rolling condition in this case is given by
Eq. (7) or Eq. (8).

B. Rolling vs sliding

The rise in the center of mass of a spinning top or a tippe top is usually explained in
terms of sliding friction acting at the bottom end. That explanation does not account for
the observed rise of Top 3 or the rise of the tippe top at low spin frequencies since the tops
were found to roll on the horizontal surface while they were rising. However, rolling friction
by itself does not account for the rise of the tops either. Consider the situation shown in
Fig. 2(a) or in Fig. 3 where the top is on the left side of the precession axis and is rolling into
the page. Rolling friction opposes rolling motion and therefore acts at point P in a direction
out of the page. Therein lies not one but two significant problems. The first is that rolling
friction exerts a torque on the spherical peg in a direction that would increase the spin of the
top. In fact, the spin of Top 3 decreased with time, a result that could perhaps be explained
by an even larger torque due to air resistance. The second problem is that the torque on
the top due to rolling friction, acting about the center of mass, should cause the top to fall
rather than rise.

19
 The rise of a spinning top, resulting in a decrease in the tilt angle, can be regarded as an
effect due to the friction torque, in the same way that the torque due to the gravitational
force results in the precession shown in Fig. 1. That is, the top precesses or tilts in such a
way that the change in the angular momentum points in the same direction as the applied
torque. It therefore appears that the friction force acting at P must act in a direction into
the page rather than out of the page, a result that would arise if rω was larger than R0 Ω
and if point P in Fig. 3 was therefore sliding out of the page while the peg as a whole moved
into the page. That is why it is usually assumed that sliding rather than rolling friction
must be responsible for the rise of tops, and tippe tops in particular.11,14,15 And it is why
Crabtree7 noted in his book published in 1909, as did others before him, that a horizontal
force that “hurries” the precession will cause a spinning top to rise.

ω
N v
S
F

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.

There is an alternative explanation of the two problems, not previously considered in

relation to spinning tops, concerned with the nature and origin of rolling friction.21,22 The
explanation is illustrated in Fig. 12. If a spherical ball of mass M and radius R is rolling in
a straight line on a horizontal surface at speed v and angular velocity ω then v = Rω at all
times, even if v is decreasing with time. The rolling friction force, F , acts to decrease v and
it also exerts a torque on the ball which has the effect of increasing ω. Consequently, rolling
cannot be maintained without some other torque to counter the effect of the friction torque.
In practice, it is found that a ball can indeed slow down while continuing to roll without
sliding, a result that can only be explained if the normal reaction force, N acts through a
point located a distance S ahead of the center of the ball, as indicated in Fig. 12. In that
case, the ball can roll with v = Rω and with dv/dt = Rdω/dt where

F = −M dv/dt and F R − N S = Idω/dt, (9)

20
where I is the moment of inertia of the ball about an axis through its center of mass. Since
N = M g, we find that the coefficient of rolling friction is given by

F M RS
µ= = (10)
N I + M R2

and that ω can decrease with time despite the fact that F by itself would result in an increase
in ω. The net torque on the ball is in the opposite direction to that due to F alone.
The same effect can be invoked to explain the behaviour of Top 3 as it rolls along a
spiral path. The horizontal friction force acting on the top can be estimated from the linear
deceleration of the center of mass along the spiral path, giving F = 0.0013 N at t = 0,
corresponding to µ = 0.0011. The coefficient of friction decreased even further as the top
slowed down along the spiral path. The friction torque acting on Top 3 can be estimated from
the rate at which the spin decreases, ignoring air resistance. Since I3 = 7.27 × 10−5 kg.m2 for
Top 3 and dω/dt = −0.48 rad/s2 averaged over the first 50 s, the average torque on the top
about the spin axis was 3.49×10−5 Nm. To simplify the following calculation, we can assume
that such a torque arises from an equivalent friction force FE acting in the opposite direction
to the expected direction (due to the offset in N ), in which case the torque about the spin
axis in Fig. 3 is FE r = FE A sin θ. The data for ω and θ in Fig. 8 yields a time average value
FE ∼ 0.03 N. Since M g = 1.21 N for Top 3, this value of FE , and the corresponding value
for F , are consistent with rolling and 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 θ at a rate dθ/dt ∼ −FE H/(I3 ω) ∼ −0.015 rad/s. In fact θ decreased
by 19◦ over 50 s, as shown in Fig. 8, at an average rate dθ/dt = −0.007 rad/s or about
half the estimated rate. It is possible that FE was about half its estimated value, given
that air resistance also acts to reduce ω. A more detailed investigation of the rate of rise is
warranted, but the simple estimate outlined here shows that the relevant force responsible
for the rising of Top 3 is that due to rolling, not that due to sliding.

C. The tippe top

The observed inversion of the tippe top was consistent with previous observations of its
behaviour, although the airborne phase seems not to have been previously documented for
a tippe top. It has, however, been previously observed with spinning eggs23,24 , and similar
jumping behaviour has also been observed with a “hopping” hoop.25 The jump height of

21
the tippe top, about 2 mm, was considerably higher than the 0.08 mm jump observed with
spinning eggs. In checking the latter result, the author found that even a billiard ball can
jump by about 0.1 mm when rolling along a straight line on carbon paper on a relatively
smooth surface, a result that was presumably due to slight surface roughness rather than
any asymmetry in the ball.
At higher spin rates than the one shown in Fig. 10 the top also became airborne for a
brief period well before the peg touched the surface. From the rate of change of the tilt angle
just before the largest jump it was estimated that the vertical acceleration of the center of
mass, due to its vertical displacement, was only about 1 m/s2 at most. Such a result cannot
explain the jump. However, the center of mass of the top did not rise along a vertical path.
Rather, it rose along an arc arising from a rapid spin about the vertical axis combined with
a lower speed spin about a horizontal axis. The exact path and the velocity of the center
of mass could not be measured, but a reasonable estimate is that the velocity was about
0.1 m/s and the arc radius was about 1 mm. As a result, the centripetal acceleration of the
center of mass could have been as high as 10 m/s2 , in a vertically downward direction, in
which case the normal reaction force on the peg would indeed have dropped to zero as the
top rose and then jumped off the surface.
A more fundamental question, addressed previously by many authors, is why a tippe
top actually inverts. Sliding friction is invoked by most authors to explain the inversion,
although the experimental evidence for sliding has been based primarily on the interpretation
of skid marks on graphite.2,3,5 In 1957, Parkyn claimed5 that “There can be no doubt that
the fundamental motion of a top is one of rolling, and that rolling friction is necessary to
explain the nature of the rise.” In the present paper, direct measurements of the spin about
the symmetry axis have shown that a tippe rolls at low spin frequencies and slides at high
spin frequencies. The center of mass rises in both cases. The dynamics of the process needs
to be studied in more detail to understand why rolling occurs only at low spin frequencies,
but it is clear that rolling can occur only if the ratio Ω/ω is about 7 or so, depending on the
geometry and inertial properties of the tippe top. The equations describing steady precession
indicate that this condition is satisfied only if Ω is relatively small. Behaviour analogous to
that observed with the tippe top is also observed with a spinning egg. A spinning egg rolls
and precesses at two different frequencies simultaneously when it is spun at low speed.26 . A
sphere or bowling ball that is projected along a horizontal surface, while spinning about a

22
near vertical axis, also rolls if it is projected at relatively low speed.27

VIII. CONCLUSION

Four different spinning tops were investigated by filming their behaviour with a slow
motion video camera, two with a sharply pointed peg, one with a 15 mm diameter spherical
peg and a tippe top. All were found to precess at rates consistent with those expected for
steady precession. A high frequency precession was also observed for the tops with pointed
ends, coinciding with the spin frequency at high spin frequencies and most likely due to a
small asymmetry in each top.
The tops with a sharply pointed peg precessed about a vertical axis passing through the
bottom of the peg, while the top with a spherical peg precessed initially about a vertical
axis located well outside the top. The top then spiraled inwards until the precession axis
passed through a point close to the center of mass. During that time, the center of mass
rose gradually until the top was almost vertical, a result that could be attributed to the fact
that the spherical peg rolled along a spiral 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 at the base of the top, but it was found that the center of
mass rose even when the top was rolling. Rolling is not normally considered as a candidate
to explain the rise of a spinning top since the friction force acts in the wrong direction.
However, the net torque on a rolling sphere does act in the correct direction, due to an offset
in the normal reaction force when a sphere rolls along a horizontal surface. Inversion of the
tippe top did not occur at a steady rate. The top was observed to jump off the surface
before inverting, contrary to the usual theoretical assumption that the normal reaction force
on a tippe top is approximately equal to its weight.14,15 When spun at high frequency,
the tippe top was found to precess at a frequency close to the higher of the two available
precession frequencies. By contrast, a conventional top usually precesses at the lower of the
two available precession frequencies.
In addition to the rolling vs sliding question, there are many other aspects of spinning
tops and other spinning objects that could be investigated further by video techniques.
For example, what difference does it make if the horizontal surface is smooth or rough or
lubricated, is energy and angular momentum conserved when a top rises, what determines

23
the rate of rise or fall of a spinning top, and so on, all of which could be undertaken as
student projects. There is a large variety of tops and gyros that are available for study,
recently reviewed and colorfully illustrated by Featonby.28

∗ Electronic address: cross@physics.usyd.edu.au

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(1958).
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 Phys. 32, 115-127 (2011).
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