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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
Note: This instruction manual is copyright. Permission to reproduce illustrations or to quote descriptions or
,'--. other material in textbooks, laboratory manuals or other publications will be given on request of authors, subject to the usual acknowledgement,
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