Gyro-ring
Graciela Flores and H. Fearn
Physics Department, California State University at Fullerton,
800 N. State College Blvd., Fullerton CA 92834.
February 6, 2016
This paper presents a simple description of the gyro-ring (sometimes known as a wobble ring or jittermax).
We emphasize the importance of dynamic friction, without which the observed effect will not occur. Five small metal rings are set spinning on a
larger metal ring, and the large ring is immediately set rotating, either by hand or by a motor driven device. It is seen that the small rings
maintain their spin at a certain height up on the large ring. The gyroscopic precession of the small rings can be understood in terms of the torque
felt as a result of one point contact and friction with the larger ring. The angular velocity of the small rings can be shown to be related to the
large ring rotation speed.
I. INTRODUCTION
The purpose of this project is to be able to explain the mechanism of the toy Gyro-Ring (see figure 1). We will be using
the basic equations of mechanics such moment of inertia, angular momentum, torque and Euler’s equations. These equations will
help to explain the motion of the five discs working against gravity with the help of dynamic friction (see figure 2). We will show
that the motion of the large metal ring determines the velocity of the small discs. The faster the metal ring rotates the faster the
discs begin to spin and wobble (precess). A device to hold and rotate the large ring was constructed. We were able to control the
rotational speed of the large ring by adding DC speed motor controller to the motor.
Figure 1. Gyro-Ring
Figure 2. Spinning ring
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z-axis
Figure 3. “Simple text book Bead Problem”
To find the equation of motion that would correlate the angular velocity of the large ring and the velocity of the discs was quite a
challenge. Our system is not so simple as the text book bead on a wire (see figure 3). In those examples the wire and beads move
not only vertically but circulate around the z-axis (the beads are considered as point like and are not taken to be spinning annuli as
in our case.) The vertical and horizontal motion of the beads allow the potential and kinetic energy to balance, since friction is
ignored in these problems. With no friction, the problems can easily be solved by applying basic Newtons equations of motion.
In our case however the big ring spins about the x axis, in the vertical plane, not the horizontal. Friction is nesserary since it
causes a torque which keeps the discs rotating at a fixed height on the large ring. (The small dics do not fall down the large ring
under gravity- the friction with the large ring prevents this). Initial spin in the large ring and discs are required. Therefore it
cannot be treated as a simple “bead problem”.
II. STATIC AND DYNAMIC FRICTION.
Friction is when the motion of a object is resisted by a bonding between the body and the surface over which it wishes to move
[1]. There are different types of friction but for the Gyro-Ring experiment will consider mainly kinetic (dynamic) friction. The
simpler to exaplin is the static friction. For example in Figure 1, the box on top a table. The force applied to the object (pull or
push) needs to be greater than the static friction, this will over come the static force thus allowing the object to move.
N
Static friction
Force applied to the object
Box
Figure 4. Static Friction
2
The kinetic or dynamic friction occurs when two objects are moving relative to each other. The kinetic friction is the
force which opposses the motion of a sliding object [2]. For example Figure 2, shows how a box will start sliding at a
certain angle. The gravitational force acting down the slope will have a magnituded of mg sin Ө. The gravitational force
must exceed the kinetic friction in order to allow the box to slide down the slope.
N
Kinetic Friction
Force of gravity
Ө
Figure 5. Dynamic Friction
Now we need to calculate the coefficient of kinetic friction between the disc and the big ring. The coefficient of friction is
found by using the following equation:
Fk = µN [3]
N
Fk = µN
mg sin Ө
Ө
mgcos Ө
mg
Figure 6. Diagram of small ring shown on the larger ring.
Photograph 1: Set up to measure friction of ring
on an inclined plane
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When the ring just starts to slide, then we could say that mg sin Ө = µN; where N = mg cos Ө.
Therefore mg sin Ө = µ(mg cos Ө)
µ
=
µ =
tan Ө  Coefficient of Friction
tan 5.76 µ = 0.10
Data coefficient of friction
Ө1
5.00
Ө2
5.5
0
Ө3
6.00
Ө4
6.50
Ө5
5.80
Average angle
5.760
where Өaverage = 5. 760
Dynamic friction is usually less than static friction but we will use µ as an upper bond.
III. APPARATUS
motor
wheels 2 & 3
wheel #1
DC speed motor controller
Ring
Discs
Photogaph 2: Photo of the gyro-ring setup, showing the driving motor and stabilizing wheels.
The apparatus is made of solid wood with dimensions of 27.5 cm length, 38.5 cm height and 1.5 cm width. The
motor is located at the top left corner of the apparatus is attached to a skate board wheel to create a good (high
friction) contact. The motor has a minimum frequency of 4.76 Hz and a maximum frequency of 7.76 Hz; in addition
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it operates with a maximum voltage of 12 V. The apparatus consists of three additional wheels to hold the big ring in
place and to keep it in a stable vertical position. The big ring (gyro-ring) has a diameter of 28 cm and a mass of 216
grams. The five small discs have a diameter of 3.02 cm and an average mass of 15.64 grams. Lastly, the motor is
attached to a DC motor speed controller. The function of the speed controller allows us to collect data of different
rotational velocities at different
angles between (100-900). The DC speed controller circuit uses two oscillators connected as a pulse width
modulator. The maximum current it can handle is 16A and a maximum of voltage of 16 V. The output frequencies is
calculated by: [5]
f =
1.4400
{(R3 + 2R4) C2}
R3 = 33K, R4 = 2k and C2 is 100 nF
Figure 7. DC motor speed controller
Figure 8. Circuit diagram of the DC speed controller
r1 = 0.55 cm
0.7 cm
r2 = 1.51 cm
Figure 9. Small ring diagrams showing internal and external dimensions
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IV. DATA
Angular velocity of the motor, ring and discs
Position
Frequency of
Frequency
frequency of
the motor
of the ring
the disc
87.9 ± 5
10-200
20-300
30-400
40-500
50-650
rev/min
0.297 Hz
Dimensions of ring, motor and discs
857.8± 5
Outside radius
Inside
rev/min
(cm)
(cm)
1.46 Hz
14.29 Hz
Big ring
14.13
13.49
99.2 ± 5
993.3± 5
Motor wheel
2.87
-------
rev/min
Green disc
1.515
0.545
1.65 Hz
16.55 Hz
Purple disc
1.51
0.545
116.1± 5
1142.5 ±
Blue disc
1.51
0.545
5rev/min
Average disc radius
1.51
0.55
rev/min
rev/min
0.334 Hz
0.394 Hz
1.94 Hz
19.04 Hz
128.2± 5
1258.2± 5
rev/min
rev/min
Mass of ring and discs
2.14 Hz
20.97 Hz
Mass of ring
216 g
131.6± 5
1343.2± 5
Green disc
15.6 g
rev/min
Purple disc
15.6 g
Yellow disc
15.7 g
Orange disc
15.8 g
Blue disc
15.5 g
Average disc mass
15.54 g
rev/min
0.435 Hz
0.445 Hz
2.19 H
To find the angular velocity of the ring
we used the following equation;
ω1 = angular velocity of the motor
22.39 Hz
radius
ω2 = angular velocity of the ring
r = the radius of the motor wheel
R = the radius of the ring
ω 2 = r ω1
RR
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Freq
Frequency of ring/Hz

of disc/Hz
0.297
14.29
0.334
16.55
0.394
19.04
0.435
20.97
frequency of the ring vs frequency of the disc
y = 48.501x
R2 = 0.9938
25
frequency of the disc
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
frequency of the ring
Graph 1. Angular frequency of big ring plotted verse wobble frequency of disc
From these results we can see that there is linear relationship between the frequency of the big ring and the
frequency of precession of the discs. The precession angular velocity (wobble) and the spin of the small discs are
related via a simple relation, available in most text books; see results section.
V. CALCULATING MOMENT OF INERTIA
Calculating the moment of inertia of the little discs is Σ I= mi ri2 [1] where m is the mass of the ring and r is the
perpendicular distance from the axis of rotation. Since we are dealing with a ring with two radii (annulus) then the
moment of inertia becomes I = m (r12 + r22)/2, where r1 is the smaller of the two radii.
z-axis
r1
r2
Figure 10. Moment of inertia of the ring
Moment of inertia of the disc;
m = 15.54 grams 0.01554 kg, r1 = 0.55 cm  0.0055 m , r2 = 1.51 cm  0.0151 m,
I = m (r12 + r22)/2,
hence I = 2.0056 x 10-6 kg m2
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VI. RESULTS
In conclusion we discover that there is linear relationship between the angular velocities of the ring and discs. The
angular velocity of the big ring determines the angular velocity and the positions of the discs. The faster the large
ring rotates, the faster the discs rotate. We see that the discs are basically rolling without slipping down the large
ring due to the friction created between the inner surfaces (the edge) of the discs and the ring. As the discs are
rolling down the arc of the ring, their potential energy is converted to kinetic energy which is why the ring spins
faster as it falls. Also the tilt angle

decreases as it falls. This situation is very similar to a toy known as a
fiddlestick [6]. The fiddle stick consists of a wooden rod held vertical by a heavy base. A plastic disc, with a central
hole, is placed onto the rod and given a spin. The disc is seen to fall more slowly, than might be expected, due to the
fact that it is rolling without slipping down the rod. The slower fall is due to friction with the vertical rod. Friction
exerts a torque on the disc since there is a single point of contact with the rod.
Photograph 3. This picture shows a fiddlestick with one disc spinning and
falling slowly under the action of gravity and friction with the rod.
Exactly the same situation occurs with our system of the disc spinning on the large ring. Here, the disc does not fall
under gravity since the rotation of the large ring acts against this motion. The single point contact friction of the
large ring with the inner edge of the disc counteracts the pull of gravity so the spinning disc remains more or less in
the same position on the ring. The torque set up by the friction plays an important role in maintaining the height and
spin of the disc.
Another dynamic that we observed from the motion of two discs is that they encounter a weak coupling.
This coupling must be due to vibrations in the large ring. The two discs appear to be in phase (precess together in the
same direction) when they are very close together and π out of phase when they are separated by a slightly larger
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section of the large ring. Similar results can be seen from the online animations by Dr. Hunt [7] of Cambridge
University in the UK.
Photograph 4. The two pictures show the in phase and out of phase motion of two discs together.
3


Figure 11. Disc showing spin
3
and wobble

(precession rate) and tilt angle
.
A strobe light was used to measure the angular frequency of the motor and the small discs. The small disc frequency
measured was actually the wobble (or precession rate)

and not the spin
3 .
The 3 subscript denotes axis 3 or the
z-axis. In order to convert from wobble to spin we used the simple wobble to spin ratio given by “the Feynman
flying plate scenario” or torque free precessional motion; this gives a ratio
wobble 
2


.
spin
3 cos 
From any standard mechanics text book [8,9] the ratio of torque free precession to spin is given as;

2

 2.03
3 cos 
where we have used a tilt angle of approximately
ratio of the moments of the disc
 =10 degrees (see Figure 11. above) and we have used that the
I3
 2 , which can be seen from the perpendicular axis theorem assuming the
I1
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disc is sufficiently thin. That is we assume
I1  I 2  I 3 . This gives us the ratio of the angular velocity of the big
ring to the spin of the small disc to be;
disc 48.969

 24.1
RING
2.03
,
where 48.969 is the slope of Graph 1. above. This is interesting because the ratio of the inner radii of the big ring to
the inner radius of the small disc is also of the same order;
Rring
rdisc

13.49cm
 24.5 ,
0.55cm
where the radii refer to inner radii of the big ring and disc respectively. In fact we shall go further and suggest
that the ratios should in fact be equivalent since the disc rolls without slipping – hence we may write that the
velocity of the point of contact, of the disc with the ring obeys the relation;
  rdiscdisc  Rringring .
A small error in the ratios may appear due to error in measurement of the precession rates with the strobe light. It is
very difficult to get an exact frequency and there is often a small margin of error of approximately
 5 Hz where
you cannot tell exactly when the disc appears stationary, it appears stationary over a small range of values.
References:
[1] http://en.wikipedia.org/wiki/Kinetic_friction#Kinetic_friction
[2] http://www.ap.stmarys.ca/demos/content/mechanics/static_friction/static_friction.html
[3] J. Walker, Fundamentals of Physics, 8th ed. (John Wiley & Sons, Inc. 2008)
[4] http://www.electronickits.com/kit/complete/motor/ck1400.pdf
[5] J. Walker, Fundamentals of Physics, 8th ed. (John Wiley & Sons, Inc. 2008)
[6] J. Walker, The Flying Circus of Physics, 2nd ed. (John Wiley & Sons, Inc. 2007)
[7] Dr. Hugh Hunt University of Cambridge Department of Engineering http://www2.eng.cam.ac.uk/~hemh/
“jittermax”.
[8] H. Goldstein, Classical Mechanics, 2nd Edition, (Addison Wesley, 1980)
[9] J. B. Marion & S. T. Thornton, Classical Dynamics of Particles and Systems, 4th edition (Harcourt College Pubs,
1995).
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