Gyro-ringGraciela Flores and H. Fearn

Physics Department, California State University at Fullerton,

800 N. State College Blvd., Fullerton CA 92834.

May 20, 2023

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.

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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

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

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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.

Static friction Force applied to the object

Figure 4. Static Friction

Box

N

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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 = tan Ө Coefficient of Friction

µ = tan 5.76 µ = 0.10

where Өaverage = 5. 760

Dynamic friction is usually less than static friction but we will use µ as an upper bond.Data coefficient of friction

Ө1 5.00

Ө2 5.50

Ө3 6.00

Ө4 6.50

Ө5 5.80

Average angle 5.760

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III. APPARATUS

motor wheel #1 Ring

wheels 2 & 3 DC speed motor controller 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

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}

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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

r2 = 1.51 cm

Figure 9. Small ring diagrams showing internal and external dimensions

IV. DATA

0.7 cm

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To find the angular velocity of the ring

we used the following equation;

ω1 = angular velocity of the motor

ω2 = angular velocity of the ring

r = the radius of the motor wheel

R = the radius of the ring

ω2 = r ω1

RR

Angular velocity of the motor, ring and discs

Position Frequency of

the motor

Frequency

of the ring

frequency of

the disc

10-200

87.9 ± 5

rev/min

1.46 Hz

0.297 Hz

857.8± 5

rev/min

14.29 Hz

20-300

99.2 ± 5

rev/min

1.65 Hz

0.334 Hz

993.3± 5

rev/min

16.55 Hz

30-400

116.1± 5

rev/min

1.94 Hz

0.394 Hz

1142.5 ±

5rev/min

19.04 Hz

40-500

128.2± 5

rev/min

2.14 Hz

0.435 Hz

1258.2± 5

rev/min

20.97 Hz

50-650

131.6± 5

rev/min

2.19 H

0.445 Hz

1343.2± 5

rev/min

22.39 Hz

Dimensions of ring, motor and discs

Outside radius

(cm)

Inside radius

(cm)

Big ring 14.13 13.49

Motor wheel 2.87 -------

Green disc 1.515 0.545

Purple disc 1.51 0.545

Blue disc 1.51 0.545

Average disc radius 1.51 0.55

Mass of ring and discs

Mass of ring 216 g

Green disc 15.6 g

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

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Frequency of ring/Hz Freq 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 = 48.501xR2 = 0.9938

0

5

10

15

20

25

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

frequ

ency

of t

he d

isc

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= m i 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 + r2

2)/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

section of the large ring. Similar results can be seen from the online animations by Dr. Hunt [7] of Cambridge

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University in the UK.

Photograph 4. The two pictures show the in phase and out of phase motion of two discs together.

Figure 11. Disc showing spin 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 . 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

.

From any standard mechanics text book [8,9] the ratio of torque free precession to spin is given as;

where we have used a tilt angle of approximately =10 degrees (see Figure 11. above) and we have used that the

ratio of the moments of the disc , which can be seen from the perpendicular axis theorem assuming the

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disc is sufficiently thin. That is we assume . This gives us the ratio of the angular velocity of the big

ring to the spin of the small disc to be;

,

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;

,

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;

.

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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