Rolling Dynamics of an Inhomogeneous Ball

on an Inclined Track
Priya Narayanan∗
Department of Mechanical Engineering
University of Maryland, Baltimore County
Baltimore, Maryland 21250, USA

Rouben Rostamian†
Department of Mathematics and Statistics
University of Maryland, Baltimore County
Baltimore, Maryland 21250, USA

Uri Tasch‡
Department of Mechanical Engineering
University of Maryland, Baltimore County
Baltimore, Maryland 21250, USA

Alan M. Lefcourt§ and Moon S. Kim¶

Food Safety Laboratory
Beltsville Agricultural Research Center
Bldg. 303 Powder Mill Rd
Beltsville, Maryland 20705, USA
(Dated: June 25, 2008)
Experiments and computer simulations show that when a rigid ball with inhomogeneous but
symmetric density distribution rolls on an inclined track, it tends to adjust its orientation to rotate
about its major axis of inertia. We present tentative arguments toward explaining that behavior
and sketch ideas that may lead to rigorous analysis and better understanding of the phenomenon.

I. INTRODUCTION contact with both rails, with zero linear and angular ve-
locities, and its principal axes of moment of inertia ori-
In this note we report on experimental results and com- ented in a random way.
puter simulations of an interesting phenomenon observed As the ball rolls down the track, it picks up speed and
in a simple rigid body motion. spins progressively faster. In the early stages of the mo-
We consider a rigid spherical ball of radius R with an tion it maintains no-slip contact with both rails. However
inhomogeneous mass distribution. We assume that its as its angular velocity increases, the inhomogeneous dis-
mass is distributed symmetrically about the ball’s center, tribution of mass results in dynamic imbalance (due to
therefore the center of mass coincides with the geometric unequal principal moments of inertia) and a sequence of
center. However, the three principal moments of inertia events ensues. Depending on the situation, one or both
I1 ≤ I2 ≤ I3 (relative to the center) are not necessarily contacts may begin to slip. It is also possible that dy-
equal. namic forces may cause the ball to lift off from one or
We construct a “track” consisting of a pair of parallel even both rails (see the illustrations in Figure 2). Sub-
slender rigid cylinders (rails) of radius r each, with their sequently, the ball may recover contact with one or both
axes set at a distance of a apart. The track is inclined and rails or it may completely fall off the track, never to re-
makes an angle of α relative to the ground; see Figure 1. turn. The latter case is more likely to happen if the
The ball is placed on the track and allowed to roll down. distance a between tracks is small relative to the ball’s
We assume that a > 2r so that the rails do not overlap radius R. The falling off case is of no immediate interest
and 2R > a − 2r so that the ball does not fall through to us; in this article we focus on those cases where the
the gap between the tracks. ball does not fall off the tracks.
The contact between the ball and the rails is assumed The behavior just described is strictly a three-
to be of the Coulomb type, i.e., dry friction. The ball will dimensional phenomenon. The imbalance is due to
slip at the contact points if the tangential contact force torques, resulting from centrifugal forces, that tend to
exceeds what the dry friction can support. The ball is twist the ball in a plane perpendicular to the direc-
allowed to lift off the tracks, losing contact with one or tion of motion. Otherwise the ball is statically balanced
both rails, if the dynamics so dictates. about its center because of the symmetric distribution
The initial conditions of the motion may be quite of mass. If we dispense with that symmetry, then a
generic and is not instrumental to the discussion that two-dimensional counterpart exist in the classic “hopping
follows. In the simplest case, we start with the ball in hoop” (cf. Littlewood12 , Tokieda22 ) where a point mass
 2

R
R

R
rail diameter = 2r
α a a
ground

Side view View from above View down the track

FIG. 1. These schematic diagrams depict three views of the ball of radius R riding on an inclined track consisting of two
parallel cylindrical rails of radius r each, set a distance a apart. We assume 2R > a − 2r, so that the ball doesn’t fall through
the gap between the tracks.

R R R
R

a a a a

Two contacts One contact One contact No contact

FIG. 2. Diagrams of the ball in views down the track. During the motion the ball may be in contact with both rails, only
one rail, or have no contact with rails at all. The modeling of the dynamics of the one-contact cases is quite challenging—
the dynamics in nonholonomic, the contact constrained is unilateral, and the stick-slip nature of Coulomb friction introduces
discontinuities in the differential equations of motion.

is attached to a weightless hoop which rolls without slip- B. An overview of the sections
ping in a vertical plane. The hoop lifts off (hops) when
the radius vector to the weight becomes horizontal. This note is organized as follows. In Section II we de-
scribe the expected qualitative behavior of the rolling ball
and identify nine modes of motion. We describe obstacles
A. Rolling auto-orientation in solving numerically the differential equations of motion
in four modes that involve nonholonomic constraints. In
Extensive laboratory experiments, observations with Section III we outline our approach to simulating the mo-
high-speed cameras, and detailed computer simulations tion using the computer software MSC Adams and show
point to an interesting phenomenon: As the ball rolls that simulations confirm the rolling auto-orientation. In
down the track, it tends to orient itself so that in the Section IV we describe the methods and results of several
long run it spins about the major axis of inertia, that is, experiments which again confirm auto-orientation. Sec-
the axis of largest moment of inertia. We refer to this tion V describes a well-known mechanical system which
behavior as rolling auto-orientation. exhibits auto-orientation but the reasoning that justifies
We have no convincing explanation for rolling auto- that behavior does not apply to our system. Finally,
orientation in terms of basic principles of mechanics. The Section VI notes a scarcely explored but promising ap-
familiar minimum energy argument of Section V, which proach to the analysis of the rolling auto-orientation phe-
works well for unconstrained motion, does not really ap- nomenon from the stochastic point of view.
ply here. The stochastic analysis approach outlined in
Section VI holds some promise but it requires substantial
development to yield useful information. The purpose of C. A note on the origins of this study
this Note is to bring this phenomenon to the applied me-
chanics community in the hope that better explanations The study reported here had its beginnings in the anal-
than ours may emerge as result. ysis of a prototype device (whose patent is pending21 )
developed for the US Department of Agriculture whose
purpose is to optically scan apples for contaminants and
blemishes before they are shipped to the market. Ap-
ples are rolled down inclined tracks in an assembly line
fashion and are scanned as they go past optical scan-
 3

ning devices. Experiments show that most varieties of is expressed as a simple ordinary differential equation
apples orient themselves after a brief amount of rolling, that may be solved quite trivially by hand. Equations
and regardless of the initial conditions, they spin about for mode 9 are Euler’s differential equations for a freely
the stem-calyx axis in the long run. Such automatic ori- rotating rigid body. The equations for the remaining
entation is essential for proper scanning because other- modes turn out to be quite complicated—the nonholo-
wise the optical devices may confuse the stem or calyx nomic nature of the one-point contact leads to a system
for external contaminants. See9,14–16 for reports of these of 12 coupled differential-algebraic equations (DAEs) in-
experiments and description of instrumentation. volving Euler angles, their derivatives, and other kine-
The perfect sphere model presented in this note re- matic variables. The recent books by Holm6,7 and mono-
moves geometric complications due to a real apple’s ir- graphs on nonholonomic systems by Bloch1 , Chirikjian
regular shape but retains the unequal principal moments and Kyatkin3 , and Neĭmark and Fufaev17 , have a wealth
of inertia. of information on nonholonomic constraints and methods
for handling them. Neĭmark and Fufaev’s monograph, in
particular, has a detailed study of the rolling of a ball on a
II. INTEGRATING THE EQUATIONS OF general curved surface with a one-point, no-slip, contact,
MOTION but the ball there is homogeneous, therefore complica-
tions from tracking the Euler’s angles do not arise.
In a typical case, after the ball picks up sufficient an- We used the computer algebra system Maple24 to or-
gular speed, it loses contact with one rail and rolls for ganize the derivations of the equations of motion and
a while with a single point contact with the other rail; perform some of the unwieldy calculations. We used
see representative schematic diagrams in Figure 2. Dur- Maple’s differential equations solver (in the numeric,
ing the motion with one-point contact, the ball has extra not symbolic, mode) to solve and visualize the results.
degrees of freedom compared with the case of two-point Switching among the nine modes would be done through
contact. Computer simulations and experimental obser- a master controlling procedure that would monitor the
vations with high-speed cameras show that the ball tends constraint reactions at every time step and switch from
to pivot about that single point of contact while contin- one mode to another, as necessary.
uing its motion down the track. In due course the center Unfortunately the computations were not always suc-
of the ball drops due to the combined action of gravity cessful. We ran into an inherent difficulty with the equa-
and other dynamical influences and the ball regains con- tions of motion of our system.
tact with both rails. This lifting and regaining of contact As noted above, the effect of presence of nonholonomic
may happen a few times in quick succession. The ball re- constraints in modes 5–8 is that the equations of mo-
orients itself during those one-point contact phases, each tion change from a system of ordinary differential equa-
time coming closer to the final orientation where it will tions (ODEs) to a set of differential algebraic equations
be spinning about the major axis of inertia. (DAEs). The theory and algorithms for numerical solvers
Since rotation about a principal axis of inertia is free of for DAEs are not fully developed. The state of the art is
dynamical imbalance, once the ball settles into rotating described in the book of Brenan, Campbell and Petzold2 .
about a principal axis, the orientation is no longer dis- (Also see Riaza18 for an up-to-date look.) DAEs are clas-
turbed; it continues a simple accelerated rolling motion sified according to their index which is, in a sense, a mea-
down the track. sure of how far a given DAE is from being and ODE. The
The equations of motions of the ball may be obtained theory of DAEs of index 1 is quite well-developed. The
from the general principles of rigid body dynamics. They article by Shampine, Reichelt, and Kierzenka19 is a sur-
are quite involved, however, because accounting for the vey of numerical techniques for solving index 1 DAEs and
unequal principal moments of inertia requires tracking their implementation as built into Matlab25 . The cur-
the orientation of the ball through its Euler angles or the rent knowledge of DAEs of index 2 is somewhat spotty.
equivalent. The imposition of the unilateral constraints There is hardly anything useful known for DAEs of higher
at contact points and dry friction introduce further com- index.
plications. Maple’s DAE solver is limited to DAEs of index 1
The rolling motion of the ball may be divided into 9 or 2. It determines that our equations are of higher index
distinct modes. In mode 1, the simplest of all cases, the and gives up. Porting the equations to Matlab did not
ball rolls with no-slip contact with both rails. In modes 2, help either, because Matlab’s DAE solvers can handle
3, and 4 the ball retains contact with both rails however only index 1. We are not aware of general algorithms or
it slips against one or the other or both rails. In modes 5 software for solving DAEs of index higher than 2. Our
and 6 the ball is in no-slip contact with only one rail. conclusion is that developing a numerical integrator for
Modes 7 and 8 are like modes 5 and 6 but the ball slips the ball’s equations of motion would be a challenging but
at the contact point. In mode 9 the ball detaches from worthwhile undertaking. It calls for innovative ideas and
both rails and is in free flight; see illustrations in Figure 2. techniques which may be useful in other contexts as well.
The differential equations of motion are quite different At this point we should point out yet another unpleas-
in different modes. The motion in mode 1, for example, ant aspect of the mathematical model of a ball on two
 4

III. MOTION SIMULATION

The approach taken by the MSC Adams motion sim-

ulator23 overcomes the problems described in the pre-
? ? vious section by allowing local deformation of objects
upon contact according to an extended Hertz contact
1
2
mg mg 1
2
mg model4,20 . To describe the idea, consider an instant
where the ball is lifted momentarily off both tracks, as in
the rightmost diagram in Figure 2. The subsequent mo-
FIG. 3. A ball resting on parallel horizontal rails. If the
contact is frictional, then the support reactions are statically tion of the ball is a free fall under the influence of gravity
indeterminate. and the integration of the differential equations of motion
is straightforward. The numerical integrator takes small
time steps and at each step updates the ball’s phase space
rails. There is a problem even in formulating the statics! variables, that is, the location and orientation of the ball,
This is best explained by considering a horizontal track, and the corresponding linear and angular momenta. At
that is, the case of α = 0 in Figure 1, and a motionless each step the integrator checks for collisions, which in
ball resting on the rails. Figure 3 depicts the situation. our case is a matter of computing the distance d of the
The weight mg of the ball falls evenly on each support, ball’s center from either track. If d > R + r, then there
thus the vertical component of each support is mg/2. is no collision and the free fall continues to the next time
The horizontal components of the reaction are statically step.
indeterminate; they can be any pair of equal and opposite If d < R + r, then the ball has penetrated the rail. The
forces. The indeterminacy may be removed by assuming Hertz model assumes that the rail reacts by repelling the
that the resultant of the reaction force at each support is ball with a force of magnitude F = kδ  , where δ = R+r−
normal to the ball’s surface. This assumption applies if d is the depth of penetration, and the constants k and 
the surfaces of the ball and the rails are polished there- encapsulate in a very primitive, but still meaningful way,
fore the contact is frictionless, otherwise the horizontal the complex events that occur during the impact. There
components of the force are not determined from consid- is no general rule for selecting their values other than
erations of statics alone. To obtain a feel for the situ- trial and error, some judicious guessing, and accumulated
ation, imagine holding up an orange with two or three experience. In the contact of smooth objects, such as
fingers, similar to the drawing in Figure 3, and squeezing a sphere and a cylinder, the choice  = 2 amounts to
it gently. You should be able to modulate the horizontal assuming that the repulsion force F is proportional to
components of reactions within a limited range, if your the volume of the overlap region. MSC Adams’ user’s
fingers are not too slippery. guide suggests  = 2.1.
The indeterminacy of the horizontal components of re- To account for possible plastic deformation and energy
action forces is a serious issue in modeling the equations loss, MSC Adams augments the repulsion force F by a
of motion of the rolling ball. As we have noted above, damping term which is a certain nonlinear function of
the motion faces nine possible modes, each described by a the depth and speed of penetration. If the objects have a
distinct set of differential equations. The motion switches sliding motion relative to each other during the collision,
from a non-slip to slip mode when the tangential compo- a further term is added to account for dry friction. The
nent of the reaction force exceed what Coulomb friction resultant of the reaction forces is used then to determine
can support. Since the tangential component is unde- the change in linear and angular momenta of the ball,
fined if the horizontal reactions are indeterminate, then the phase space variables are updated accordingly, and
the entire scheme of mode-switching breaks down. the integrator proceeds to the next time step.
One way to remove the indeterminacy of the reactions The graphs in Figure 4, produced by MSC Adams,
is to model the ball and the rails as elastic bodies and show typical results of the simulation. The graph on
account for elastic deformations when computing the re- the left corresponds to the principal moments of inertia
actions. This is similar to the situation in the analysis of selected as I1 /I3 = I2 /I3 = 0.9. The graph on the right
statically indeterminate trusses where the indeterminacy corresponds to the principal moments of inertia selected
is resolved by accounting for the elastic properties of the as I1 /I3 = I2 /I3 = 1.1. In both cases the ball starts
truss’ members. rolling from rest, in contact with both rails, and its major
The method described in the next section accounts for axis of inertia making a 45 degree angle with the vertical.
elastic deformation at contacts therefore does not suffer The slope of the track is 20 degrees. The graphs show
from the problems that we have recounted above. that in both cases the ball orients itself to roll about the
major axis of inertia.
 5

10 10
0 0
-10 -10

-30 -30
Angle (deg)

Angle (deg)
-50 -50

-70 -70

-90 -90

-110 -110
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Length (m) Length (m)

FIG. 4. Graphs produced by MSC Adams show the orientation of the ball versus length traveled down the track. The
orientation is measured by the angle between a fixed horizontal vector which is perpendicular to the tracks and a variable
vector that points along the principal inertia axis corresponding to I3 . The graph on the left corresponds to the moments of
inertia I1 /I3 = I2 /I3 = 0.9. The graph on the right corresponds to the moments of inertia I1 /I3 = I2 /I3 = 1.1. In either case,
the ball orients itself to roll about the major axis of inertia.

ertia, Ix , Iy , Iz , of each configuration were computable

from the given data.
a. Configuration 1: Two diametrically opposite
stacks of 6 coins each on the z axis. This corresponds
R = 30mm to Ix /Iz = Iy /Iz = 1.61, thus Iz is the smaller of the
three moments of inertia. The left column in Figure 6
r = 2.5mm
shows a composite image extracted from a high speed
camera movie of the ball rolling down the track. Two
large dots mark the north and south poles (the locations
40mm
of the coins). We see that after just a few revolutions,
the ball has oriented itself to spin about an axis perpen-
FIG. 5. A scale diagram of the ball resting on the exper-
imental wooden tracks. The overall length of the track is
dicular to the z-axis.
3.5 meters but typically the ball achieved its final orientation b. Configuration 2: Two diametrically opposite
within 1 meter of its starting point. stacks of 3 coins each on the x axes plus two diametri-
cally opposite stacks of 3 coins each on the y axes. This
corresponds to Ix /Iz = Iy /Iz = 0.786, thus Iz is the
IV. EXPERIMENTS larger of the three moments of inertia. The middle col-
umn in Figure 6 shows a composite image extracted from
We performed a series of experiments to confirm our a high speed camera movie. The equators (the xy plane)
conjecture on the automatic alignment of a rolling ball. is marked by a dark circle. We see that after just a few
The track in our experiment consisted of 3.5 meter long revolutions, the ball has oriented itself to spin about the
parallel wooden rails, set at an angle of 20 degrees relative z-axis.
to the ground. The diagram in Figure 5 shows a scale c. Configuration 3: Two diametrically opposite
drawing of the track’s cross-section and the ball resting stacks of 3 coins each on the x axes plus two diamet-
on the track. rically opposite stacks of 2 coins each on the y axes plus
The thin hollow plastic ball used in our experiments a pair of diametrically opposite coins on the z axes. This
had a radius of 30 millimeters and a mass of M = corresponds to Ix /Iz = 0.858, Iy /Iz = 0.924, thus Iz is
100 grams. Its two hemispheres could be split open and the larger of the three moments of inertia. The right col-
snapped back. We modified its moments of inertia by af- umn in Figure 6 shows a composite image extracted from
fixing stacks of US quarter-dollar coins to the inside of the a high speed camera movie. The equators (the xy plane)
hollow shell. Each coin has a mass of m = 5.67 grams, is marked by a dark circle. We see that after just a few
radius of 12.13 millimeters, and thickness of 1.75 mil- revolutions, the ball has oriented itself to spin about the
limeters. We created three loaded balls, according to z-axis.
the configurations below, described in terms of an xyz In each experiment, the initial conditions consisted of
Cartesian system of coordinates centered at the center zero linear and angular velocities, the ball was in contact
of the ball. The total mass of each configuration was with both rails, and it was oriented so that one of the
M + 12m = 168 grams. The principal moments of in- principal moments of inertia was parallel to the track and
another one made an approximate 45 degree angle with
 6

of the kinetic energy, subject to constraint of constant

angular momentum, is achieved when Ii the largest, thus
in its final state the object will be spinning about the
major axis of inertia.
Such argument is not directly applicable to the ball on
the track. The angular momentum is certainly not con-
stant and the interaction with the track plays an essential
role and cannot be ignored. There is no fundamental law
of mechanics that we know of that implies the observed
large time behavior.

VI. STOCHASTIC DYNAMICS

Stochastic dynamics offers a very different approach to

the analysis of the large time behavior of the rolling ball
and may hold the key to the rigorous explanation of the
rolling auto-orientation phenomenon.
Consider a ball with unequal principal moments of in-
ertia, as we have in the previous sections. Let us consider
the ball in isolation—no tracks or gravity—as it would
be the case if it were floating freely in the outer space.
FIG. 6. The image of the rolling ball in each column is con- Now imagine the surface of the ball under continuous
structed by pasting five equally-spaced (in time) frames ex- bombardment of random torques in the form of isotropic
tracted from the experiment’s movies, thus creating a “multi- white noise. Specifically, let us look at the limiting case
ple exposure” effect. The left column, corresponding to Con- where the frequency of the bombardments goes to infinity
figuration 1, has internal masses concentrated near the north and their amplitude goes to zero. How will the ball be-
and south poles. These are marked by the large dots. The have in the long run? We expect that the ball will settle
middle column, corresponding to Configuration 2, has equal
into a (rotational) Brownian motion in the space of spe-
masses placed in 90 degree intervals around the equator which
is marked by a dark circle. The right column, corresponding cial orthogonal group SO(3). There is a large amount
to Configuration 3, has unequal pairs of masses placed in three of literature on the Brownian motion of a point mass
orthogonal directions. The equator is marked by a dark cir- but very little on Brownian motion of rigid bodies. The
cle. In all cases, the ball quickly orients itself to rotate about book by McConnell13 and articles by Liao10 and Liao
an axis of maximal moment of inertia. and Wang11 are among the few that address the issue.
Bounds for the expected values of deviation from steady-
state rotation about the major or minor axes of inertia
the plane of the track. The outcomes of the experiments are derived in10,11 .
were not sensitive to the initial orientation; within a short To connect this to the auto-orientation problem, we
distance the ball oriented itself to spin about the major may view the reactions of the rails as “random bombard-
axis of inertia. ment” of forces that toss the rolling ball about in unpre-
dictable ways. If under such random influences the ball
happens to come close to rotating about the major axis
V. AN ENERGY ARGUMENT of inertia, then the bombardment ceases, because rota-
tion about a principal axis of inertia is free of dynamic
It is tempting to explain away the auto-orientation imbalance. Thereafter the ball rolls down the track with-
phenomenon through a familiar energy argument. It out further disturbance. The preference of the random
is said that abandoned orbiting satellites tend to ori- dynamics to select the major, rather than the minor, axis
ent themselves in the long run to spin about the ma- of inertia may be attributable to the wider basin of at-
jor axis of inertia. There is an elementary explanation traction (deeper energy well) corresponding to rotation
for it (see, e.g., page 371 of Greenwood5 ): away from about the major axis of inertia.
the drag of the atmosphere and other non-conservative In a sense, the situation here is akin to the atomistic
external influences, its angular momentum, H, remains versus phenomenological view of matter. From the atom-
constant. Suppose an internal mechanism dissipates ro- istic point of view, an ideal gas consists of a very large
tational kinetic energy into heat. If the magnitude of number of point masses flying about in random direc-
the angular velocity is ω and the object spins about the tions and velocities within the confines of a rigid bound-
principal moment of inertia axis Ii , the magnitude of the ary. Each particle behaves according to the Newtonian
angular momentum is H = Ii ω and the kinetic energy is laws of motion. But tracking the motion of individual
K = Ii ω 2 /2 = (Ii ω)2 /(2Ii ) = H 2 /(2Ii ). The minimum particles is rather pointless. At the phenomenological
 7

level, the gas is viewed as a continuum, characterized by possible configurations elongates in one direction, losing
its state variables: pressure, temperature and density. In its isotropy, and giving a greater probability to spinning
statistical mechanics one reconciles the two points of view about the major axis.
by relating the state variables to certain averages of the The evolution of the cloud is governed by the Fokker-
point mass dynamics. Plank partial differential equation. (See Chirikjian and
The stochastic approach outlined in this section takes Kyatkin3 and Kadanoff8 for the theory and application
the view that the rolling auto-orientation is a phe- of the Fokker-Plank equations.) Thus the analysis of the
nomenological effect in the sense that it is the result of orientation of the rolling ball reduces to solving a partial
averaging of fine (and unknowable) details of the random differential equation and determining the evolution of the
interaction of the ball with the track. probability cloud as time goes to infinity. Working out
In this connection, it is interesting to note that math- the details of such an analysis is nontrivial because de-
ematical tools developed for the study of quantum me- termining the coefficients of the Fokker-Plank equation
chanics may be brought to bear some understanding of requires a close analysis of the ball’s dynamics.
the effects of random initial conditions on the evolution
of the ball’s motion.
To illustrate, assume that the ball starts out with ran- ACKNOWLEDGMENTS
dom initial orientation and angular velocity, both with
uniform probability distributions. Thus the set of all The first author wishes to acknowledge financial sup-
possible initial conditions may be viewed as an isotropic port by UMBC and USDA, during her doctoral studies.
spherical “cloud” in the probability space. From our The second author wishes to acknowledge informa-
experiments and simulations we know that as the ball tive conversation and exchange with Dr. Gregory
rolls down the track, it prefers to take on an orientation Chirikjian of the Johns Hopkins University.
whereby it spins about its major axis of inertia. If that is
so, then as the ball moves down the track, the cloud of all

∗
priya2@umbc.edu 129, 2008.
† 10
rostamian@umbc.edu Ming Liao. Random motion of a rigid body. Journal of
‡
tasch@engr.umbc.edu Theoretical Probability, 10(1):201–211, 1997.
§ 11
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¶
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 8

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20 23
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