DEMONSTRAT I ON

13-2258 STUDENT GYROSCOPES

li

RASED ON A D SIGN B PROF SSOR HAROLD OF ~^EW h E x I c Q I 6 TATE N I v E R s I T Y

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13-2209 A R E DEMONSTRATION GYROSCOPE INSTRUCTIONS AND TECHNIQUES

N o t e : This I n s t r u c t i o n M a n u a l i s c o p y r i g h t . P e r m i s s i o n t o r e p r o d u c e i l l u s t r a t i o n s or t o q u o t e d e s c r i p t i o n s o r o t h e r material i n t e x t b o o k s , l a b o r a t o r y m a n u a l s or o t h e r p u b l i c a t i o n s w i l l be g i v e n on r e q u e s t of a u t h o r s , s u b j e c t t o t h e u s u a l a c k n o w l e d g e m e n t .

CONTENTS

I Operation of the Air Gyroscope and Its Accessories

The Support System The Ball The Base The Axial Rod The Loading Weights Colored Disks The off-axis weight holder Leveling the gyro to remove air torques Removing gravitation torques

I1 emo on st rations with the 13-2209 Air Gyroscope

2.1 Introduction 2.2 Precession due to various forces 2.3 Precession of the instantaneous axis 2.4 Shifting the principal axis 2.5 Experiments to demonstrate nutation 2.6 Rigidity of a force-free gyro 2.7 Measuring the constants of the gyroscope

I OPERATION OF THE A I R GYROSCOPE AND I T S ACCESSORIES 1.1 The Suppor t System

1.1.. I A i r Supply

The a i r supp ly should be c l e a n and f r e e o f o i l o r m o i s t u r e .

However, con tamina t ion i s l e s s dangerous t h a n it i s annoying and

t h e a i r l i n e , gyro s e a t , e t c . a r e e a s i l y c l e a n e d . Very l i t t l e

p r e s s u r e i s r e q u i r e d . I n f a c t , most o f t h e gyros can be sup-

p o r t e d w i t h mouth p r e s s u r e a f t e r an i n i t i a l l i f t i n g of t h e b a l l

by hand. T h i s d o e s n ' t work f o r a v e r y long time! Use a 3/8 i n c h

I D tube t o connect t h e gyro t o t h e a i r supp ly . I f no l o c a l a i r

i s a v a i l a b l e , use t h e B a l i n g 14-913 Small Compressed A i r Source.

1 .1 .2 . The Gyro S e a t

The s e a t i s made by c a s t i n g epoxy o v e r a p e r f e c t b a l l . S i n c e

t h e r e i s some s h r i n k a g e , t h i s f i r s t c a s t i n g o p e r a t i o n i s fo l lowed

by t h e p l a c i n g of a few drops of epoxy i n t h e bot tom o f t h e s e a t

fo l lowed by a g e n t l e s e a t i n g o f t h e b a l l i n t o t h e s m a l l puddle

of epoxy u n t i l t h e s e a t i s e n t i r e l y f i l l e d . T h i s t echn ique w i l l

r e j u v e n a t e a damage s e a t i f c e r t a i n p r e c a u t i o n s a r e fo l lowed.

1 .1 .3 Procedure f o r Repa i r ing Gyroscope S e a t

a . Spray t h e b a l l w i t h a T e f l o n mold r e l e a s e s p r a y o r

some o t h e r mold r e l e a s e s u i t a b l e f o r use w i t h epoxy

r e s i n s .

.b. Mix abou t h a l f a teaspoon of epoxy. Devcon Two-Ton

brand c l e a r epoxy w i l l work w e l l a s w i l l a number o f

o t h e r s i m i l a r f o r m u l a t i o n s .

Place a smal l p lug of wood o r cork i n t h e a i r h o l e

i n t h e bottom of t h e s e a t .

T rans fe r t h e epoxy t o t h e bottom of t h e s e a t and l e t

it s i t f o r a few minutes t o g e t r i d of a s many bubbles

a s p o s s i b l e .

Gently lower t h e coated b a l l i n t o t h e s e a t and p r e s s

down u n t i l r e s i n appears i n a bead around t h e r i m of

t h e s e a t . I t i s important t h a t some r e s i n appear on

a l l s i d e s , b u t t h e r e may be more on one s i d e than t h e

o t h e r .

Allow a f u l l day f o r t h e epoxy t o s e t thoroughly.

The b a l l i s then removed by cover ing it wi th a s h e e t of

paper and s t r i k i n g with a l e a d hammer. The hammer

blow should be a s near t h e s e a t a s p o s s i b l e s o t h a t

it has an upward component.

D r i l l o u t t h e a i r ho le wi th a No. 60 d r i l l . B e su re

t o chamfer t h e edge of the ho le . I f t h e edge i s l e f t

sha rp , t h e b a l l w i l l be hard t o f l o a t i n i t i a l l y .

Clean t h e bead of r e s i n o f f t h e edge of t h e s e a t . The

edge i s chamfered a t about 45O i n t h i s a r e a and it i s

e a s i e s t t o c l ean down t o t h e ba re metal wi th a pocket

k n i f e o r s c r ape r followed with f i n e sandpaper. Take

c a r e n o t t o round t h e edge s i n c e it i s d i f f i c u l t t o do

t h i s uniformly a l l around.

F i n a l l y c l ean t h e r e l e a s e o f f t h e b a l l and t h e job i s

done.

Note t h e Baling technique f o r manufacturing a p e r f e c t gyro-

scope s e a t w i l l work w e l l i n your own l abo ra to ry i f you dec ide t o

make a i r b e a r i n g s o f o t h e r s i z e s . Very l a r g e b a l l s such a s

bowling b a l l s have been suppor ted i n t h i s way, b u t one should be

warned ahead of t i m e t h a t t h e s e b a l l s a r e n o t h e l d t o n e a r l y a s

c l o s e t o l e r a n c e s a s bea r ing b a l l s and t h a t t h e i r composit ion isnot

uniform s o they w i l l be heavy on one s i d e . This f a u l t can be

a v i r t u e f o r c e r t a i n exper iments and should n o t o v e r l y d i scourage

f u r t h e r exper imenta t ion w i th l a r g e b a l l s . See s e c t i o n 3 . 3 f o r

o t h e r t ypes o f b a l l s t h a t a r e a v a i l a b l e from Ea l ing .

1 . 2 THE BALL

The Gyro b a l l i s made from a p r e c i s i o n ground type 5100 f r e e

machining chrome s teel c a s t i n g . To make t h e counter-bor ing

o p e r a t i o n p o s s i b l e , t h e b a l l is - n o t ca se hardened. Thus, it i s

somewhat s o f t e r than t h e h a r d e s t s t e e l b a l l s and can be damaged.

Extensive tests have shown t h a t l i t t l e o r no damage r e s u l t s from

t h e d u s t and d i r t t h a t accumulate i n a d i r t y urban environment.

Such d i r t may, however, p rov ide f r i c t i o n i f n o t removed.

Note t h a t t h e b a l l , n o t be ing c a s e hardened, cannot be

dropped on a s t o n e f l o o r o r s u b j e c t t o o t h e r forms of s e r i o u s

abuse. The b a l l has been n i c k e l p l a t e d t o p r even t r u s t .

SAFETY PRECAUTION

Carry t h e b a l l c r a d l e d i n both hands. Do n o t c a r r y it i n

i t s s e a t s i n c e t h e weight o f base p l u s b a l l makes f o r awkward

handl ing .

Clean t h e b a l l wi th any convenient s o l v e n t . F i n g e r p r i n t s

w i l l n o t a f f e c t t h e o p e r a t i o n o f t h e gyro , b u t t h i c k e r d e p o s i t s

should be wiped o f f .

- S M A L L SAFETY BUMPER

Figure 1.1 The positi-in of ?he large *eight when th" safety b:i%i~ers lust come into contact.

1.3 The Iiase - The base i s a n aluminum c a s t i n g w i t h s e v e r a l u s e f u l f e a t u r e s .

First : , i t has a threaded i n s e r t every 60' around t h e periphery,

of these accommodate the 3/8-16 leveling screws while t h e

o t h e r t h r e e are a v a i l a b l e f o r mounting t h r e a d e d rods or standard

cype appara tus rods w i t h che app rop r i a t e 3/8-16 t h r e a d s . S u c h rods

a r e useful f o r m o i ~ n t i n q photocell gates, timing, m a r k e r s , etc.

The o u t s i d e edge of the base h a s a series of 3 0 @ bosses for u s e . .

when making s t roboscopic photographs. A f u r t h e r .feature of the:

l-^-ve- i s - the l oca t ion o n a n e leva ted pedestal so t h a t the accessory

disks always c lear t h e painted surface. T h i s feat-dre- also pro-

tects the a x i a l rod when t he la rge? l o a d i n g w e i q h t is u s e d . Figure

1.1 shows h o w protect i v e 0-ring:- on t h e hrsb and +-.he 1.-irgc l oad inq

m'- ., weight make; simultaneous contact. A,iui,c bumpers stop b o t h the.

la rge loading w e i g h t and the ball hub when t he w e i g h t is al.l.owed

fco f a l l f r e e and hi?: t h e base a s f r e q u e n t l y hapqe-is r- - - when one f i r s t

t r i e s to set up some of t h e i n t e r e s t i n g n u t a . t i o n a 1 motions.

1 . 4 The Axia l Rod

1 . 4 . 1 I n s t a l l a t i o n o r Replacement

The a x i a l rod i s shipped s e p a r a t e l y from t h e b a l l and must

be i n s t a l l e d by t h e u s e r . F igure 1.1 shows t h e rod p rope r ly

i n s t a l l e d . The rod screws over a s t u d i n s i d e t h e b a l l hub and

has been machined s o a s t o l i n e up p rope r ly when i n s t a l l e d . I f

you wish t o t a k e t h e rod on and o f f , merely screw it i n t i g h t l y .

Otherwise, use a drop of t h e L o c t i t e s e a l a n t provided t o per -

manently ho ld t h e rod i n p l a c e . Should t h e rod be bad ly damaged,

a Replacement Rod may be o rdered from Bal ing. (See S e c t i o n 1.1,

Replacement P a r t s ) . To remove a damaged rod a f t e r L o c t i t e has

been a p p l i e d , p l a c e t h e rod i n a v i c e and screw t h e b a l l o f f .

S ince a l a r g e amount o f t o rque is r e q u i r e d , t h i s removal w i l l

p robably damage t h e rod where it i s he ld i n t h e v i c e . Do n o t

remove a rod t h a t has been i n s t a l l e d w i th L o c t i t e u n l e s s it i s

going t o be r ep l aced .

1.4.2 U s e of t h e Axia l Rod

The prime f u n c t i o n o f t h e a x i a l rod i s t o mark one p r i n c i p a l

a x i s o f i n e r t i a f c o f t h e gyroscope. The rod a l s o s e r v e s a s a con-

v e n i e n t p l a c e t o mount. l oad ing weigh ts and t h e d i s k s used f o r P.

ana lyz ing complex motions o f t h e gyro. The a x i a l r o t 6 is marked Y

o f f w i th grooves one cen t ime te r a p a r t w i th a double groove every

f i f t h cen t ime te r . Measurement i s w i th r e s p e c t t o t h e c e n t e r of

t h e b a l l and t h e f i r s t double groove marks a p o i n t 10cm from

t h e c e n t e r .

Note t h e r ed anodized f i n i s h on t h e rod which can be seen

from a d i s t a n c e . The anodizing process may g i v e t h e rod a s l i g h t

roughness. I f t h i s cond i t i on seems exces s ive , apply a t h i n

c o a t of Simonize o r c l e a r shoe p o l i s h which w i l l seal t h e s u r f a c e .

SAFETY PRECAUTION

Do n o t l i f t t h e gyro b a l l by t h e a x i a l rod. The rod i s

made from t h e b e s t l i gh twe igh t aluminum a l l o y a v a i l a b l e , b u t it

i s n o t s t r o n g enough t o suppor t t h e weight of t h e b a l l un l e s s

t h e rod i s nea r ly v e r t i c a l . I t i s b e s t t o develop t h e h a b i t of

always l i f t i n g t h e Gyro by c r a d l i n g t h e b a l l i n both hands.

Note t h a t s e v e r a l experiments r e q u i r e t h a t t h e rod be s t r u c k

with t h e f o r e f i n g e r t o s h i f t t h e l o c a t i o n of t h e angula r momentum

vec to r away from t h e p r i n c i p a l a x i s . The rod w i l l n o t be damaged

by any impulse d e l i v e r e d by one f i n g e r t h a t a ) does no t h u r t

t h e person doing t h e experiment; and b) does n o t cause excurs ions

of t h e rod g r e a t e r than what i s r equ i r ed f o r studying t h e p a t t e r n s

on t h e accessory d i s k s provided.

, - 1.4.3 Axial Rod Bearing

The smal l bea r ing a t t h e end of t h e a x i a l rod is important

f o r changing t h e rod o r i e n t a t i o n wi thout slowing down t h e gyro.

U s e c a r e when p l ac ing t h e va r ious a c c e s s o r i e s over t h e end of

t h e rod. I f t h e bea r ing becomes damaged, it may be rep laced by

o rde r ing bear ing No, S418CHH-5 from Miniature P rec i s ion Bearings

i n Keene, New Hampshire. The bea r ing i s he ld i n p l ace wi th

L o c t i t e .

F i g u r e 1 . 2

, /

I

I I---

\ YELLOW

'. DISK FOR DEMONSTRATING

FRONT PRECESSION OF THE IN- BACK STANTANEOUS AXIS,

/-- fl

\

/*'

BLACK \

'/ ,"

DISK TO HEL.P EXPLAIN THE SHIFT OF PRINCIPAL AXIS EXPERIMENT, BACK FRONT

*... --. + ""..

BLACK s G

DISK FOR EXPERIMENTS OF THE USERS OWN DE-

FRONT BACK

1.5 Loading Weights

Each gyro i s equipped wi th two smal l blackened s t e e l 10 gm

weights , one 5 gm aluminum weight , one 1 inch d i a . 10 gm s t r o b e

d i s k , and a l a r g e 3 inch d i a . load ing weight. The Large Loading

Weight covers approximately 2 cm of t h e a x i a l rod whi le t h e 5 and

10 gm weights cover 1 cm. This is a u s e f u l f e a t u r e when s h i f t i n g

t h e c e n t e r of mass of a given weight t o a known l o c a t i o n on t h e

a x i a l rod. Each of t h e s e weights i s i n s t a l l e d by pushing it

g e n t l y over t h e smal l bea r ing on t h e end of t h e a x i a l rod and

s l i d i n g it g e n t l y i n t o p o s i t i o n . A l l of them a r e h e l d i n p l a c e

by p a i r s of rubber 0-Rings. Do n o t worry i f t h e weights a r e a

l i t t l e hard t o move a t f i r s t . They w i l l loosen a b i t w i th t i m e ,

bu t n o t s o much a s t o become sloppy. Should an 0-Ring become

damaged, it may be r ep l aced wi th a s t anda rd 0-Ring with a 1/16

inch nominal c r o s s s e c t i o n and a nominal i n s i d e diameter of

1 / 4 inch.

1.6 Colored Disks

F igure 1 . 2 . shows both s i d e s of t h e co lored d i s k s provided

with every gyro. These d i s k s a r e provided wi th 1 / 4 inch I D rubber

grommets f o r a t t a c h i n g them t o t h e a c i a l rod o r t o t h e o f f - a x i s

- - rod on t h e o f f - a x i s weight ho lder . The p a t t e r n s and c o l o r s a r e

designed t o a i d i n s p e c i f i c demonstra t ions and t h e reason f o r

each p a t t e r n i s d i scussed f u l l y i n t h e experiment s e c t i o n . Re-

placement grommets can be ob ta ined from any e l e c t r o n i c s u p p l i e r .

The Off-Axis Weight Holder

The rod on t h e o f f - a x i s weight ho lder is mounted a t an angle

of 15O from t h e a x i s of t h e opening which passes over t h e a x i a l

--~- , . ' t . .

NCM:' . .

NEW AMS SCREW

. . , . ' . .

Figure 1.3 The off-axis weight instatled so that this short axis projects tht i ->~gh the center ot the ball.

.^AIR SUPPLY .CONNECTION

a \

b -. 2; t

I / I : 960

f' - S O C K E T S FOR i "-.. - /'A I / INSERTING RODS BOSSES WITH 3 / 8 - 16

THREAD

rod. If the base of the holder is centered over the 9 cm mark

on the axial rod, the axis of the short rod will also pass

approximately through the center of the ball. The holder is

installed by clamping it in place with the alien wrench pro-

vided. This wrench can also be used to rotate the gyro about

the shifted principal axis. To shift the principal axis 7 . 5 O

requires about 15 gm to be added to the small rod. Figure 1.3

shows the proper installation of the off-axis weight holder

and the placement of the special alien wrench for use in spin-

ning the gyro about the shifted principal axis.

1.8 Leveling the Gyro Remove Air Torques

Certain precise experiments require the elimination of the

small torques which result from assymetries in the gyro seat or

from the seat not being level. These torques may be eliminated

by using the leveling screws. Study Figure 1.4 and note the

three leveling screws A, B, and C. Raise the entire base about

a half inch off the table with these leveling screws and then

turn the air supply on. Now hold the axial rod horizontal in

position (1) perpendicular to line AB using only the small ball

- bearing as a holding point.

The ball will move down on the low side with increasing

angular velocity. Thus, if the ball moves down at point (a),

raise this point with leveling screw A. Screws A and B are

adjusted until the ball remains stationary, and then the rod is

shifted 90' to position (2). Screw C is then raised or lowered

to bring the ball to a stationary position. For great precision,

position (1) should then be rechecked.

1.9 Removing Gravitational Torque

An important feature of the Air Gyro is the fact that the

side of the ball on which the rod is mounted has been bored

out so that the rod side is lighter even after the rod has been

installed. This lightness means that precession can be demon-

strated either with the rod side heavy or light. For some

demonstrations, you will want to start with all gravitational

torque balanced out. A special 1 inch diameter 10 gm weight

has been provided for this purpose. It has sufficient mass so

that the ball will just balance when this weight is placed near

the hub joining the rod to the ball. On typical gyros this

weight balances the gravitational torque when placedbetween

the 7 and 8 cm marks. This leaves most of the rod free for

adding other accessories. Figure 1.1 shows this weight properly

located at the balancing point. The balancing point is easy

to find if you have removed all air torques. Otherwise you

may rotate the ball very slowly and then observe which way the

rod precesses. This is a surprisingly sensitive test for the

balancing out of gravitational torques.

DEMONSTRATIONS WITH THE 13-220 A1 R GYROSCOPE

2.1 Introduction

Mechanics is a venerable branch of physics an,d the gyroscope

has been well described by a number of authors during the last

half century. It, therefore, seems pointless to give a theoreti-

cal discussion of the subject when it has already been done ex-

tremely well on all levels. Rather, an attempt will be made to

describe accurately a number of demonstrations and experiments

which can be done on the 13-220 Four Inch Ball Demonstration

Air Gyroscope and to discuss which points of physics they are

designed to illustrate. It is left to the teacher to select

a mathematical model which is a satisfactory explanation of

the phenomena and is of a proper degree of sophistication for

his own students.

Since, however, it is difficult to discuss nutation, principal

axes of inertia, and similar subjects without any mathematical

notation, the symbolism of olds stein' has been adapted, and

where formula have seemed desirable, they have been copied out

with no attempt at derivation. Reference to this text will clear

up any mystery as to where these relations have come from.

2.2. Precession Due to Various Forces

2.2.1 Gravitational Torque

To show that the axial rod is on the light side of the ball,

hold the rod horizontally and release it. The slow oscillatory

old stein, Classical Mechanics; Addison-Wesley, Cambridge,

1951, Chapter 5.

---- -- - -- - - - -

motion that results clearly shows that this side is light. In

similar manner, add a 10 gm weight to the axial rod and place it

at various distances from the center demonstrating the conditions

with the rod side heavy and with the gravitational torque~cancelled.

2 . 2 . 2 Response to Gravitational Torque

a. Technique for Showing Precession - For demonstration you will want relatively fast precession rates. This

will require angular speeds of only a few rps which are

easily imparted with the fingers. Note that there are no

forces to damp out nutation; therefore, it is not enough

to simply release the axis~you must move the end of the

axial rod in a horizontal plane at the expected pre-

cession rate and then release it. A few moments practice

and it is no problem to avoid nutations which mightanfuse

when first introducing the gyro. Hold the axis by the

small end bearing when easing it into its precessional

"orbit".

b. Direction of Precession - Show that with the axis light,

the gyro precesses one way and with the axis heavy, this

motion is reversed. Next repeat but change the rotation

direction. This is a convenient time to discuss whatever

notation you wish to use to describe the various angular

velocities, etc.

c. Quantitative Comparison of Precession Rates - The re-

sponse of precession rate to a change in the other variables

is easily demonstrated by altering these variables by exact

f a c t o r s of two. Precess ion r a t e may be timed wi th r e s p e c t

t o an appara tus rod screwed i n t o one of t h e 3/8-16 screw

holes i n t h e per iphery of t h e base , while r o t a t i o n r a t e

may be checked a g a i n s t a s t roboscope. To do t h i s , p l ace

t h e 1-inch diameter s t r o b e d i s k about 8 cm from t h e c e n t e r

of t h e b a l l . With a l i t t l e adjustment , t h i s d i s k w i l l

se rve two func t ions . F i r s t , it w i l l enable you t o t ime

t h e r o t a t i o n r a t e ; and second, it w i l l ba lance o u t a l l

g r a v i t a t i o n a l to rques s o t h a t it i s easy t o change t h e to rque by

a known amount wi th t h e heavy loading weight o r one of t h e

small weights . This loading weight can be placed a t 1 0 ,

15 , o r 20 cm o r a t o t h e r convenient d i s t a n c e s without d i s -

t u rb ing t h e s t r o b e d i s k which i s c l o s e r t o t h e c e n t e r of t h e

gyro b a l l . I n a d d i t i o n t o a l t e r i n g t h e r o t a t i o n r a t e o r t h e

moment arm by a f a c t o r of 2 , one canchange t h e loading mass

with t h e smal l loading weights o r a l t e r t h e component of

t h e g r a v i t a t i o n a l fo rce by r a i s i n g t h e rod t o some convenient

angle such a s 60Â above t h e h o r i z o n t a l .

2.2.3. Response t o a More General Torque

Show t h a t t h e end of t h e a x i a l rod i s capped wi th a b a l l bear-

ing . Then add t h e s t r o b e weight and balance o u t t h e g r a v i t a t i o n a l

to rque . Bring t h e b a l l up t o a speed of s e v e r a l r p s us ing your

f i n g e r s and being s u r e t h a t t h e c l a s s can s e e which way t h e b a l l

is r o t a t e d . Then show how a meter s t i c k pushed gen t ly a g a i n s t

t h e o u t e r r a c e of t h e smal l bear ing w i l l cause t h e a x i s t o move

along t h e s t i c k . To s tuden t s viewing t h i s motion a t a d i s t a n c e ,

it l o o k s l i k e t h e a x i s i s g rabb ing o n t o t h e s t i c k l i k e a t i r e

moving down a road . While t h i s i s an e x c e l l e n t mnemonic f o r

p r e d i c t i n g which way t h e a x i s w i l l move (it works f o r e i t h e r

d i r e c t i o n o f r o t a t i o n ) , it is i m p o r t a n t t o remind t h e s t u d e n t s

t h a t t h e s m a l l b e a r i n g makes a c t u a l f r i c t i o n a l c o n t a c t i m p o s s i b l e .

Another e f f e c t i v e demons t ra t ion i s t o s p r e a d t h e f i n g e r s o f

one hand a p a r t and t h e n t o touch t h e s m a l l b e a r i n g g e n t l y . The

a x i a l rod w i l l f o l l o w a l o n g t h e complex c o n t o u r of t h e o u t -

s t r e t c h e d hand.

2 . 3 P r e c e s s i o n of t h e I n s t a n t a n e o u s Axis

Due t o symmetry, t h e gyro h a s one e a s i l y i d e n t i f i e d p r i n c i -

p a l a x i s . The o t h e r two a x e s are d e g e n e r a t e b u t , of c o u r s e ,

are p e r p e n d i c u l a r t o each o t h e r and t o t h e a x i a l rod . I f t h e

gyro i s spun abou t t h e p r i n c i p a l a x i s r o d w i t h an a n g u l a r

v e l o c i t y t ~ i and t h e rod i s t h e n s t r u c k l i g h t l y , t h e gyro w i l l

p roceed t o r o t a t e abou t some new a x i s . The a x i a l r o d w i l l appear

t o wobble and, i n f a c t , w i l l g e n e r a t e a cone.

Gyroscope t h e o r y p r e d i c t s t h a t under t h e s e c i r c u m s t a n c e s ,

t h e new i n s t a n t a n e o u s a x i s w i l l p r e c e s s abou t t h e p r i n c i p a l a x i s

and t h a t i n t h e c a s e where t h e o t h e r two p r i n c i p a l moments o f . -

i n e r t i a 11 and 12 a r e e q u a l , t h e p a t h fo l lowed w i l l be a c i r c l e .

The a c t u a l e q u a t i o n f o r t h e p r e c e s s i o n r a t e Q i s

Note t h a t t h e long a x i a l rod on the A i r Gyroscope makes 13 q u i t e

a b i t s m a l l e r t h a n I i o r 12 s o t h a t t h e p r e c e s s i o n r a t e R i s r a p i d ,

negavive, and proportional to u. All these relations can be

demonstrated with the aid of a few accessories.

We seek to locate the instantaneous axis and to prove the

theorem. For this purpose, the disk with four colored sectors

is used. Place this disk about halfway down the axial rod and

face the rod toward the observer. Note how the colors blur if

the gyro is spun fast.

Strike the rotating axial rod so as to cause a wobble cone

about 20Â wide. Note how the center of the cone changes color

rapidly, while in the area outside the cone, the color is blur-

red. The color appears unblurred at the point on the disk which

is instantaneously at rest. This point is the center of the

wobble cone which is not too surprising. Many students, however,

will find the color change a bit unexpected even after they have

predicted it from theory.

A final consideration is to note the order of the colors on

the disk. Looking at the axial rod and disk, if the axis is rotat-

ing clockwise, the instantaneous axis moves about the rod counter- -+-

JJ, ",f 4~ $'t/.rc d,,ccr7> / J J , x: A * ,*Af ,

clockwise in the ball frame of reference. r S, + -\ -> -> - p

.. A> )"'-,'- ' 1 I 1 -

X

'4 If you want a q&& convincing demonstration that/ the pyth of the

/

instantaneous axis is a circle, take the disk and reverse it so the

yellow circle faces the class. It will take a little more care to

get the wobble cone the right size; but when you do, one point on the

circle will remain at rest at all times and the center of the wobble

cone will be a yellow dot.

2.4 S h i f t i n g t h e P r i n c i p a l Axis

2 . 4 . 1 S e t t i n g up t h e Off-Axis Weight

The most i n t e r e s t i n g motion o f t h e gyro f o r most obse rve r s

occurs when a mass i s added on one s i d e away from t h e a x i a l rod.

Th is experiment i s set up by clamping t h e o f f - a x i s weight ho lde r

t o t h e a x i a l rod about 9cm from t h e c e n t e r of t h e b a l l . (See

Sec t i on 1 . 7 ) . The s h o r t 2 i nch rod on t h e ho lde r makes an ang l e

of 15O wi th t h e a x i a l rod . This ang l e i s j u s t r i g h t s o t h a t t h e

a x i s o f t h e s h o r t rod a l s o ex tends through t h e c e n t e r o f t h e b a l l

when t h e ho lde r i s over t h e 9cm mark.

To perform t h e s i m p l e s t form of t h e exper iment , s l i d e t h e

5 and 1 0 gram weigh ts on to t h e s h o r t rod and r o t a t e t h e a x i a l

rod w i th t h e f i n g e r s of one hand wh i l e ho ld ing t h e sma l l b a l l

b e a r i n g w i th t h e o t h e r hand. Now s t eady t h e sma l l bea r ing and

r e l e a s e t h e a x i a l rod. Not ice how t h e a x i a l r o d immediately moves

o u t t o a wobble cone of about 30° The a x i a l rod t h e n "dewobbles"

back t o i t s o r i g i n a l p o s i t i o n and t h i s a c t i v i t y r e p e a t s i t s e l f

i n d e f i n i t e l y .

2 . 4 . 2 Accessory Disk f o r U s e w i th t h e Off-Axis Weight

The o f f - a x i s weight has an i n t e r e s t i n g h i s t o r y . Like s o

many i n v e n t i o n s , it came i n t o be ing as an a c c i d e n t . A t e c h n i c i a n

dropped a p e r f e c t gyro and t h e b e n t rod was observed t o do i n t e r -

e s t i n g t h i n g s . L a t e r , an accessory d i s k was developed t o h e l p

prove t h a t t h e observed wobble and dewobble i s e a s i l y unders tood

i n t e r m s of a p r i n c i p a l a x i s s h i f t . The d i s k w i th t h r e e rubber

grommets i s s p e c i a l l y designed t o h e l p ana lyze what i s happening.

The two o u t e r grommets a r e p laced ove r t h e long and s h o r t rods .

For emphasizing t h e na tu re of t h e problem t o s t u d e n t s i n t h e

back row, use t h e s i d e wi th t h e two l a r g e pa in t ed d o t s . Even

from a d i s t a n c e , it w i l l then be apparen t t h a t f i r s t t h e yellow

d o t and then t h e red i s t h e ins tan taneous a x i s . I f t h i s exper i -

ment i s done a s h o r t t i m e a f t e r t h e ins tan taneous a x i s experiment,

very a s t u t e observers w i l l s e e t h e s i m i l a r i t y a t t h i s p o i n t . Most

people , however w i l l need t o s e e what happens when t h e d i sk i s

reversed.

When t h e o t h e r s i d e i s used, it w i l l immediately become ap-

pa ren t t h a t t h e ins tan taneous a x i s i s moving around a c i r c l e and

t h a t t h e two rods a r e p o i n t s l y i n g on t h i s c i r c l e . Fu r the r , t h e

colored sec:tions of t h e c i r c l e w i l l appear i n t h e same o rde r a s

they d i d i n t h e ins tan taneous a x i s experiment. I t now becomes

inc reas ing ly c l e a r t h a t t h e two experiments a r e e s s e n t i a l l y t h e

same. One only needs t o assume t h a t t h e added mass s h i f t e d t h e

p r i n c i p a l a x i s t o t h e c e n t e r of t h e c i r c l e .

2 . 4 . 3 The l o c a t i o n of t h e New P r i n c i p a l Axis i s Ver i f i ed

The f i n a l test is whether o r n o t t h e c e n t e r of t h e c i r c l e i s

t h e new p r i n c i p a l a x i s . This may be checked ou t by i n s e r t i n g t h e

long hexagonal rod i n t o t h e socke t provided and r o t a t i n g t h e gyro

A * about t h i s new a x i s . (See f i g u r e 1 . 7 ) . With a l i t t l e c a r e , t h e

gyro can be span wi th t h e ho le i n t h e c e n t e r of t h e c i r c l e a t

r e s t . There should be p r a c t i c a l l y no wobble when t h i s i s done.

T h i s completes t h e demonstration showing t h a t t h e o f f - a x i s weight

experiment can be f u l l y understood i n t e r m s of a s h i f t of p r i n c i -

p a l a x i s .

2.5 Experiments t o Demonstrate N u t a t i o n

2 .5 .1 N o t a t i o n Used f o r N u t a t i o n Experiments

The problem w i t h n u t a t i o n is t h a t it i s e a s y t o show what

it is w i t h t h e A i r Gyroscope b u t h a r d t o do demons t ra t ions t h a t

a r e n o t o v e r l y d i f f i c u l t t o i n t e r p r e t . C e r t a i n l y n u t a t i o n ex-

pe r iments a r e most meaning'ful i f done i n c o n j u n c t i o n w i t h a

c o u r s e which d e r i v e s o r a n a l y s e s t h e e q u a t i o n s o f motion o f a

gyroscope. Thus it becomes a n e c e s s i t ~ y t o d e f i n e c e r t a i n v a r i -

a b l e s . The n o t a t i o n is borrowed from G o l d s t e i n ' s C l a s s i c a l

Mechanics which sugges ted some of t h e exper iments .

M = Loading mass

L = moment arm of l o a d i n g mass

= r a t e o f gyro s p i n abou t t h e a x i a l r o d a x i s

9 = a n g l e a x i a l r o d makes w i t h t h e v e r t i c a l

4> = a n g l e a x i a l r o d makes w i t h an a r b i t r a r y r e f e r e n c e i n t h e h o r i z o n t a l p l a n e

I = moment o f i n e r t i a abou t t h e a x i a l rod

I1 = 12= moment o f i n e r t i a a b o u t an a x i s p e r p e n d i c u l a r t o t h e a x i a l r o d

2.5.2 Var ious I n i t i a l Cond i t ions i n N u t a t i o n Experiments

The way t h a t p r e c e s s i o n i s mentioned i n many i n t r o d u c t o r y t e x t s

i m p l i e s t h a t i f t h e weighted a x i a l rod o f t h e gyro i s r e l e a s e d , it

w i l l immediately p r e c e s s . The a c t u a l f a c t is t h a t it always f a l l s

s t r a i g h t down i n i t i a l l y and t h i s f a l l and t h e o t h e r i n i t i a l con-

d i t i o n s t h e n de te rmine what i t s n u t a t i o n p a t t e r n w i l l be. N a t u r a l l y ,

o t h e r p a t t e r n s can be i n t r o d u c e d by changing t h e i n i t i a l c o n d i t i o n s .

I n p a r t i c u l a r , t h e average p r e c e s s i o n r a t e $ can be g i v e n t o t h e

a x i a l rod i n i t i a l l y and t h e r e w i l l be no n u t a t i o n . Pure nu-

t a t i o n - f r e e p r e c e s s i o n c e r t a i n l y i s a v e r y spec-lhl c a s e when

viewed t h i s way. While p l a y i n g w i t h v a r i o u s i n i t i a l c o n d i t i o n s , ,

one can d i s c u s s t h e energy changes t h a t t a k e p l a c e when t h e

a x i a l rod i s r e l e a s e d . I t t h e n becomes a p p a r e n t t h a t t h e con-

f i g u r a t i o n w i t h t h e a x i s p r e c e s s i n g i n a h o r i z o n t a l p l a n e i n -

Yolves more t o t a l energy t h a n t h e c o n f i g u r a t i o n w i t h t h e same

v a l u e o f a) when t h e r o d i s r e l e a s e d from t h e h o r i z o n t a l , and

t h a t t h e energy needed t o p r e c e s s t h e gyro must b e "accumulatedM

by l e t t i n g t h e l o a d i n g we igh t f a l l a b i t .

2 .5 .3 The F a s t Top Experiment

Most c h i l d r e n who have p l a y e d w i t h t o p s know t h a t t h e t o p i s

s t a b l e a t h igh speeds and t h e n goes w i l d q u i t e suddenly as t h e

speed d rops . The c r i t i c a l a n g u l a r v e l o c i t y f o r t r a n s i t i o n from

" f a s t t o p " 'I Slow t o p " a c t i v i t y i s

T h i s r o t a t i o n r a t e can b e de termined by s p i n n i n g t h e gyro f a i r l y

r a p i d l y w i t h t h e f i n g e r s and s e t t i n g t h e a x i a l rod v e r t i c a l . The

p o s i t i o n of t h e r o d i s s t a b l e . I f t h e r o t a t i o n r a t e is t h e n

g r a d u a l l y d e c r e a s e d i n s m a l l inc rements by b r a k i n g t h e r o d wi th

t h e thumb and f o r e f i n g e r , t h e r e w i l l suddenly come a r a t e a t

which t h e r o d b e g i n s t o f a l l i n a slow s p i r a l andtoen t o r e t u r n

t o o r a lmost t o t h e v e r t i c a l . T h i s r o t a t i o n r a t e shou ld be

approximate ly t h e a)' p r e d i c t e d i n t h e formula .

2 . 5 . 4 Measuring t h e Ex ten t o f Nuta t ion

Consider a f a s t t o p which i s r e l e a s e d w i th no i n i t i a l angu la r

v e l o c i t y 6 of 6 . How l a r g e w i l l t h e n u t a t i o n p a t t e r n be? Th i s

problem i s an e x c e l l e n t example o f t h e advantages o f choosing

a convenient v a r i a b l e bo th from t h e mathematical p o i n t of view

and from t h e p o i n t of view of demonstra t ion.* The new v a r i a b l e

x i s de f ined by

x = (Cos 0 - Cos 6 )

Where 0 i s t h e i n i t i a l va lue o f 0 . Golds t e in shows t h a t t h e

maximum va lue o f x i s g iven by

When t h e c o n s t a n t s a r e i gno red , t h i s equa t i on sugges t s t h a t f o r 2

a g iven 6 , X I 'V I/" . Since x i s merely t h e p r o j e c t i o n o f t h e

a x i a l rod on t h e v e r t i c a l a x i s , w e can perform an i n t e r e s t i n g

experiment. A p o i n t source o f l i g h t i s p laced some d i s t a n c e

from t h e gyroscope and a p r o j e c t i o n s c r e e n i s used t o determine

XI. F i r s t a marker l i n e i s p l aced on o r j u s t i n f r o n t of t h e

s c r e e n t o i d e n t i f y x . Then a s u i s v a r i e d , t h e maximum f a l l 2

i s noted and recorded. Na tu ra l l y t h e s i n 0 t e r m can a l s o be

i n v e s t i g a t e d w i th t h i s technique.

*See e s p e c i a l l y p. 170 Golds te in

2.5.5 Measuring t h e Nuta t ion Rate a s a Funct ion o f Sp in Rate

The n u t a t i o n frequency f i s r e l a t e d d i r e c t l y t o t h e o r i g i n a l

s p i n r a t e u by t h e formula

Th i s r e l a t i o n i s e a s i l y v e r i f i e d exper imenta l ly and, i n f a c t , i s

probably a u s e f u l way t o measure t h e r a t i o 13/11.

R i g i d i t y o f a Force F ree Gyro

The fo l lowing experiment has never been s u c c e s s f u l l y performed

by t h e au tho r b u t i s s u f f i c i e n t l y appea l i ng t h a t i t i s mentioned

i n hopes t h a t someone w i l l do it and r e p o r t back t o ~ a l i n g what

s o r t s o f d i f f i c u l t i e s a r e r e a l l y invo lved .

2 .6 .1 Proposed Rota t ion of t h e Ea r th Experiment

Completely remove a l l a i r t o rques from t h e gyro . P e r f e c t l y

ba lance o u t a l l g r a v i t a t i o n a l t o rques . Now s p i n t h e b a l l ve ry

r a p i d l y by d r i v i n g it wi th a s o f t f e l t wheel h e l d i n a h igh speed

g r i n d e r o r s i m i l a r dev ice . F i n a l l y , p l a c e t h e a x i a l r od e x a c t l y

v e r t i c a l . The rod should now move 15O p e r hour i n c lassrooms a t

t h e equa to r and less elsewhere .

~ - 2.6.2 ber ration Due t o G r a v i t a t i o n a l Torque

The f i r s t q u e s t i o n i s whether g r a v i t a t i o n a l t o rques can be

balanced o u t w e l l enough t o observe t h e d e s i r e d e f f e c t , and a

second and r e l a t e d q u e s t i o n i s whether t h e r e i s a s e n s i t i v e enough

test t o d e t e c t t h i s c o n d i t i o n when i t e x i s t s . F i r s t , t h e pre-

s e s s i o n r a t e can be t aken i n t o account by r e c a l l i n g t h a t t h e

average r a t e w i th t h e rod h o r i z o n t a l i s

- Suppose t h a t w e r e q u i r e t h a t 4 be less approximate ly 1

r e v o l u t i o n p e r day. W e then have

Next w e assume t h a t t h e b a l l can be r o t a t e d a t 1800 rpm o r 30Hz.

Thus

The moment of i n e r t i a o f t h e b a l l i s approximate ly g iven by assuming

a uniform b a l l of mass M = 4.2 x l o 3 gm and r a d i u s r = 5.08 cm.

Then

1 3 = 2 -mr2 - 2 5 - 5 (4.2 x l o 3 ) ( 5 . 0 8 ) ~ = 4.33 x 10' gm-cm2

S u b s t i t u t i n g back i n t h e equa t i on f o r 5 g i v e s

2.6.3 Detec t ion and El imina t ion o f G r a v i t a t i o n a l Torque

Assume f o r a moment t h a t t h e a x i a l rod s i d e of t h e b a l l i s

s l i g h t l y l i g h t and t h a t one can make t hepe r iod of t h e b a l l a c t -

i n g a s a pendulum 50 sec. How much g r a v i t a t i o n a l t o rque does

t h i s r e p r e s e n t ? The formula f o r t h e p e r i o d i s

ML = .697 gm-cm

S i n c e one can a d j u s t ML t o abou t 0.5 gm-cm w i t h t h e ID gm t r i m

w e i g h t , t h e e n t i r e exper iment looks f e a s i b l e , b u t d i f f i c u l t .

2.6.4 F u r t h e r C o n s i d e r a t i o n s

When used a t h i g h r o t a t i o n a l rates, t h e g y r o i s slowed down

by t h e v i s c o s i t y o f t h e a i r . The speed f a l l s by a b o u t 50% e v e r y

13 minutes . The d r a g , o f c o u r s e , a c t s on t h e gyro away from t h e

a x i s and i s , i t s e l f , a p o s s i b l e cause of p r e c e s s i o n . There i s

one p o s i t i o n , however, i n which t h i s e f f e c t i s minimized. When

t h e a x i a l r o d i s v e r t i c a l , t h e d r a g f o r c e s are d i s t r i b u t e d sym- - 3

m e t r i c a l l y . F u r t h e r , t h e g r a v i t a t i o n a l t o r q u e s a r e a l s o z e r o

. - s i n c e t h i s t o r q u e T = MLg s i n 0 goes t o z e r o f o r 0 = 0. Thus 9

it l o o k s f e a s i b l e t o do t h i s s p e c i a l form o f t h e exper iment and

i n f a c t a n e a r l y model o f t h e gyroscope w i t h bad a i r t o r q u e s h a s

a c t u a l l y g i v e n approx imate ly t h e r i g h t answers. Hopeful ly o t h e r s

w i l l p e r f e c t t h e t e c h n i q u e f u r t h e r .

2 . 7 Measurina t h e Constants of t h e Gyroscope

2 .7 .1 The P r i n c i p a l Moments of I n e r t i a

One way t o measure t h e moments of i n e r t i a of t h e b a l l i s t o

use one form o r another of t o r s i o n pendulum. The gene ra l equat ion

f o r t h e per iod T of a t o r s i o n pendulum where torque T i s pro-

p o r t i o n a l t o angle is

where *r i s t h e torque. This equa t ion can be used t o f i n d 1 1 by

f i r s t balancing o u t a l l g r a v i t a t i o n a l to rques and then s h i f t i n g

t h e balance weight inward a known amount andneasuring t h e p r i o d

of t h e b a l l considered a s a pendulum.

The measurement of 13 i s more cha l lenging s i n c e it involves e x t r a

appara tus . There a r e two methods f o r ob ta in ing it. F i r s t a t h read

can be wound around t h e hub on t h e b a l l and a f o r c e can be app l i ed

t o t h e th read by a weight and pu l l ey arrangement. The c a l c u l a t i o n

of 13 from t h e angular a c c e l e r a t i o n is s t r a igh t fo rward . For g r e a t -

e r e legance, one could use a l a r g e c lock sp r ing and b u i l d a t o r s i o n - -

s p r i n g pendulum. For t h i s ca se , I i s computed from t h e per iod T

- which is given by

Wherek i s t h e t o r s i o n sp r ing cons t an t which i s found from an inde-

pendent measurement us ing t h e equa t ion T = k8.

2 . 7 . 2 How t h e Ba l l i s Manufactured

I t i s p o s s i b l e t o c a l c u l a t e t h e moments of i n e r t i a of t h e gyro-

scope from t h e phys i ca l c h a r a c t e r i s t i c s of t h e var ious components.

The fol lowing t a b l e of numbers should be taken a s only approximate

s i n c e t h e gyro b a l l s a r e machined one a t a t i m e and t h e r e i s un-

doubtedly some v a r i a t i o n from one b a l l t o t h e nex t . You can e a s i l y

check t h e f igures given f o r t h e a c c e s s o r i e s b u t once t h e b a l l it-

s e l f i s assembled, it becomes unaccess ib le t o f u r t h e r i n v e s t i g a t i o n .

The b a l l i s made from 430F "Free Machining" s t a i n l e s s s t e e l

and i s ground t o e x a c t l y 4 inches i n diameter . One s i d e of t i e

b a l l i s then bored o u t and a hub i s l i g h t l y p r e s s - f i t i n t o p l ace

wi th L o c t i t e . The bored-out reg ion i s a c y l i n d e r (7/8) inch d i a .

and ex tends from approximately t h e c e n t e r of t h e b a l l o u t t o a

s t e p 1 / 4 inch from t h e su r f ace (1-3/4 inch from t h e c e n t e r ) . Here

t h e diameter of t h e ho le i s increased 1/16 inch t o 15/16 inch t o

provide a shoulder f o r t h e hub t o rest a g a i n s t .

2.7.3 Important Weights

B a l l be fo re bor ing:

B a l l a f t e r bor ing:

Hub with o-ring:

S h a f t wi th smal l bear ing:

. - 2.7.4 Weights of Accessor ies

- - Large weight wi th o-r ing: 146 g m

S t e e l t r i m weights l e s s any decora t ion : 10.0 gm

Aluminum t r i m weight less ' any decora t ion :

S t robe d i s k : 10.0 gm

Disk wi th one rubber grommet: 1 4 CPU

Disk wi th t h r e e rubber grommets: 15 gin

Decorative p l a s t i c caps i f suppl ied .25 gm