The lecturer enteredfrom the back of the theatre,placed a gyroscope on a sloping wire from the backrow of seats down to the lecture bench and it sliddown ahead of him into the waiting hands of MrOwen .

'Twas brillig, and the slithy tovesDid gyre and gimble in the wabe :

All mimsy were the borogoves,And the mome raths outgrabe .

`Beware the Jabberwock, my son!The jaws that bite, the claws that catch!

Beware the Jubjub bird and shunThe frumious Bandersnatch!'

The Introduction was read by a latter-day `Alice'(actually Louise, see Plate 4.1) .

Plate 4 .1 A latter-day `Alice' (Louise) reads`The Jabberwock' poem .

40

CHAPTER 4The Jabberwock

The Jabberwock was a monster with manyheads . As such it resembles, in some way, themanner in which we divide our science intoPhysics, Chemistry, Biology, etc., and thenPhysics into Heat, Light, Sound, Magnetism andElectricity . Often one can spot the various headsas being Laws of Physics, and some of them lookinto mirrors, see their reflections and think thatthe total number of their kind is bigger than itreally is . Thus they attempt to co-exist with theirown shadows and reflections. One of the bestexamples I can give you is the collection of Lawsof Electromagnetic Induction . When I was atschool, I was taught Fleming's left- and right-hand rules, and taught to remember what thefingers and thumbs represented by emphasisingthe initial letters of the electrical quantities thus :

thuMb

-MotionForeFinger - FieldseCond finger -Current (see Fig . 4 .1)

Then we had to remember which hand to usefor motor and which for generator . After that

Fig . 4 .1 Fleming's left-hand rule .

we were taught Lenz's law, the Gripping rule,Corkscrew rule and Ampere's swimming rule .What a business! They were all, apparently,separate, independent heads . But those were thebad old days-I hope . Electromagnetism is agood deal easier than that . From what we haveconsidered already about left and right hands,and about motors on one side of the LookingGlass being generators on the other, I think youcan see that the mirror really is Lenz's law itself,for it changes hands for you as you go throughthe mirror and changes the motor to a generatorat the same time .The idea of science as a monster is certainly

not a new one, but a neat way of expressing thesame sentiment is due to Martin Gardner whowrote : `Laboratory bangs can range all the wayfrom an exploding test tube to the explosion ofa hydrogen bomb . But the really Big Bangs arethe bangs that occur inside the heads of theoreti-cal physicists when they try to put together thepieces handed to them by the experimentalphysicists .'One of the things you may not have been

taught is that there is apparently no simple con-nection between gravitational mass and inertialmass. Yet if we compare two masses, either byweighing them on the same spring balance, orby subjecting each to a known force and measur-ing their relative accelerations, we always get thesame answer . Of those two heads, one must bein some way, a `reflection' of the other . Whenyou think about it, there are many interpreta-tions of the word `reflect' . In novels, forexample : `He reflected on what mighthappen . . .' `The same attitude is reflected in . . .'etc. `When I use a word,' says Humpty Dumpty,`it means just what I choose it to mean, neithermore nor less.' Lewis Carroll created lovely`combination words' in the nonsense rhyme .'Slithy' is a mixture of `lithe' and `slimy' and isbeautifully onomatopoeic . 'Mimsy' is likewise amixture of `flimsy' and `miserable' . There is anold Lancashire word 'witchit', which is notmerely a dialect word for `very wet' . It impliessomething even more emphatic than `soaked tothe skin' . It probably originated as a mixture of`wet and wretched' .Within a generation, technology has put a pre-

THE JABBERWOCK

cise meaning on a combination word . 'Stiction'is a deliberate mixture of `sticky' and `friction' .Whilst not attempting to quantify it, it signifiesthat the frictional force necessary to start onesurface sliding over another is greater than thatrequired to keep it going, once started . In the1966 Christmas lectures I did an experiment withfive brass blocks on an inclined plane, which Ishould now like to repeat .' The blocks measure2„ x2„ x z „ 1 „ xI„ x4 „

a„ xz„ xg„ 4 „ x4 „ x 6„

s

and a" x 8" x s" and were all placed with a largesurface on the plane . The assumption of a con-tant value of frictional coefficient u,j or of `angle

of friction' A, tells us that as the plane is tiltedto greater and greater angles, all the blocks willbegin to slide at the same time . In practice, theygo off in order, biggest first . But to emphasisethe beautiful descriptiveness of the word 'stic-tion', let us have `jam today' and smear eachblock with strawberry jam, and try again! Thetwo smallest blocks remain stationary when theplane reaches an angle of 90°, where p= oc . Butafter a minute or two the blocks will be seen tobegin to move. The viscosity of strawberry jamis not infinite. We are in trouble with the µ=ooonly because we created the infinity ourselves bydefining ii as equal to tan A .

Circularity is a powerful concept, the idea ofa closed loop even more so . In circular motionthere is magic, just as there is in electro-magnet-ism. But it only manifests itself when it is, like(shall we say for the moment, rather than a 're-flection' of) its `neighbouring head', truly three-dimensional .

Here is a circular coil of wire carrying alternat-ing current. Here is a similar coil connected toan a .c . voltmeter. We can induce current into theone from the other by a means totally unintelli-gible to us, but to which we give the name 'elec-tromagnetic induction' . But if I place one coilwith its axis at right-angles to that of the other,there is no induced voltage . It is as if the two cir-cuits lived in different worlds and each neverknew of the existence of the other . The artistM .C. Escher made a splendid drawing to illus-trate this by using a staircase that could be

' The Engineer in Wonderland (English Universities Press,1967) .

41

ENGINEER THROUGH THE LOOKING-GLASS

imagined to lead either up or down. Of the menusing the staircase, he said :

Here we have three forces of gravity working perpendi-cular to one another . Three earth-planes cut across eachother at right angles, and human beings are living oneach of them . It is impossible for the inhabitants of dif-ferent worlds to walk or sit or stand on the same floor,because they have differing concepts of what is horizon-tal and what is vertical . Yet they may well share theuse of the same staircase . On the top staircase illu-strated here, two people are moving side by side andin the same direction, and yet one of them is goingdownstairs and the other upstairs. Contact betweenthem is out of the question, because they live in differentworlds and therefore can have no knowledge of eachother's existence .'

Although the artist has only used a trick ofperspective to influence the mind of the observerit is a lively influence indeed . What is the mean-ing of perspective in a four-dimensional space?

Gyroscopes are essentially circular things . Ascientific gyro is a wheel mounted in gimbal ringsas shown in Plate 4 .2. The two rings in thisexample consist of a complete inner ring whosepivot axis is always horizontal, and a half circu-lar outer ring mounted on a vertical axis . In thissituation the rotor axis is perpendicular to eachof the others and all three axes pass through thecentre of mass of the wheel . With this apparatusyou shall see two different worlds, separated foryou visually as they never are in electromagnet-ism, because with a gyroscope you can touch andfeel rather than merely observe the result, as youdo with electricity. There one gets the feeling thata voltmeter in an electromagnetic arrangementis just another mysterious instrument that we donot understand . Apparently we are using it totranslate the `magnetic language' into the`electric language' by the same shady means asthose by which we first made the current in a coilproduce an entirely fictitious `magnetic field' . Tosee is to believe, they say, but to believe is notnecessarily to see .

Let me try to twist the gyroscope about itsvertical axis . It refuses to go in the direction inwhich it is pushed . Instead it rotates about its

'The Graphic Works of M. C. Escher(Macdonald and Co .,London, 2nd edn, 1967) .

42

Plate 4.2 A scientific gyroscope consists of a wheelmounted in gimbal rings .

horizontal, inner ring axis . This motion is called`precession' . So I was unable to give it energy,for it refused to be moved . The ring that movedhad no load torque to combat (except bearingfriction, and there is very little of that) so itsenergy output was zero for a different reason -lots of precessional velocity, this time, but notorque or twisting force . It is at this point that wemay return momentarily to the electromagneticworld and contemplate for a moment the possi-bility of a coil of wire that had no resistance (theequivalent of no bearing friction in our gyro) .We should find that when a changing currentflowed through it, we could measure a voltageacross it, yet it would not get hot. It has current,voltage, but no power .

This should be a more staggering result, forthose of us who have been taught Ohm's law,than if we were to find that a gyroscope clearlydisplayed mass, velocity and no momentum . But

we were enlightened about the electromagnetic`impossibility' by the idea of inductance, and inalternating current calculations, where thecurrent is forever changing, we know that theresult of such changes is to produce a voltagethat can conveniently be regarded as quite separ-ate from the Ohm's law (IR) drop, for it doesnot rise and fall in phase with the IR drop .Mathematically it is then again convenient toput the letter j in front of such voltages, wherej= ~- (-1),' and this in turn leads to the ideathat you could transfer the j to the quantity Lwe have called `inductance', and hence obtaina still further useful idea of `impedance' (Z) :

Z=(R+jLw)

where w is the angular frequency of the cyclicalrate of change of currents and voltages . A newform of Ohm's law now emerges as

E= IZ

and life becomes simple again .I use this well-known example to show that

there is no suggestion either that Ohm's law asoriginally written is `wrong', nor that it is `true'that real quantities should be naturallyexpressed as imaginary numbers. The only ques-tions that ever need be asked are :

1 Does the new notation and set of rules giveanswers that are supported by experiment?

2 Is the method convenient? (In this case, thealternative would have been the solution ofthe differential equation L(di/dt)+Ri=Esin wt . The skilled physicist knows thatthe solution E=I(R+jL(o) neglects theinitial transients that occur when a circuitis switched on or off, but the engineer knowsfrom 100 years' experience just when thesetransients matter and when, therefore,he must solve the differential equationand when they can be ignored . The lattersituation largely predominates in practice .)

Mathematicians often use `i' for ti (-1), but electricalengineers reserve i for the instantaneous value of a current .`I' they retain for direct current or the R .M .S . value of alter-nating current .

THE JABBERWOCK

There are several phenomena that we canobserve with a gyroscope that suggest that itmight conveniently be treated as analogous to anelectromagnetic device . First, in which directionwill it precess if I hang a mass on one side ofthe inner ring (Plate 4.3)? Viewed from above,

Plate 4 .3 The gyro `precesses' about a vertical axiswhen a weight is hung from the inner ring .

will it rotate clockwise or anti-clockwise? Asimple rule is illustrated in Fig . 4 .2. Imagine theforce, in this case the weight, transferred so thatit pushes directly on the rotor itself, as at F .

Fig . 4 .2 Rule relating torque,spin and precession of a gyro .

43

ENGINEER THROUGH THE LOOKING-GLASS

Allow this force to be transferred to position F',having been, as it were, `dragged around' 90° inthe direction of the rotor movement . If now therotor be regarded as stationary, F' will producerotation in the `common sense' direction, in thiscase, clockwise from above .

This method brings out the 90° shift that oneobserves with the phase of the current in an in-ductance as compared with the current in a re-sistance. Alternatively we could agree to regarda spinning wheel as having an angular momen-tum vector M (as shown in Fig . 4.3a) pointingin such a direction as would make the rotationappear right handed (as in Maxwell's corkscrewrule). If we then regard twisting force (torque,T) in a similar manner, likewise angular velocity

44

y+

z

Fig . 4.3 The left-hand rule for electric motorsused on a gyro .

of precession, Q, we may hold up the fingers andthumb of the left hand, as for an electric motor,and declare the quantities to correspond as fol-lows :

thuMb

-Motion, as before, but inthis case it isprecessional motion, Q

ForeFinger -Force, in this case twistingforce, T

*Middlefinger -Momentum (angular)

In such a concept, might we not be temptedto look for 'wattless current', now universallyreferred to as `reactive volt-amps' (the result ofinductance) in a gyroscope? Let us do somemore experiments . If I treble the mass that I hangfrom the inner ring I treble the precession rate .If I treble the rotor speed, I divide the precessionrate by 3, for the same weight hung . It appearsthat the rule connecting M, T and Q is simplyT= MQ. If the moment of inertia of the gyrorotor on its own axis of spin is I, and its rotationspeed is w then M=Iw and T=(Iw)52 . Noticehowever that if the torque, T, is on axis x, (say),and we denote it T, then Q is rotation in axisy . Let us call it Q,, therefore, so that

T =(Iw)52,.

(1)

Earlier we saw that a torque on the vertical (y)axis produced precession about the x-axis, so

T. = (IW)0,.

But examination of Fig . 4.3b shows that we musttake account of backward vectors (anti-clock-wise) as negative and therefore strictly

T,,= -(Iw)Q .

. . . (2)

Equations (1) and (2) are the two worlds ofEscher, co-existing and co-related but from cer-tain viewpoints `unaware' of each other .

The analogy with the electric motor is streng-thened by this concept (see Fig . 4 .4) . A magneticfield B,, induces current I only in the loop (1),whose axis is horizontal, and the current is there-fore to be designated I,. . A field Br as shown onlyinduces current I,, in loop (2) and, taking magni-tudes into account, if A be the area of each coiland R its resistance,

andI,.= ~

(RoJ)B.,.

I,=- 2(Rw)Bv,

which compare nicely with equations (1) and (2) .

--loop I

Fig . 4 .4 Gyro analogy with a .c . machine .

The whole of modern electrical machine theory('Generalised machine theory') is based on thisidea of the two independent axes, co-existing, co-related but nevertheless identifiably separate . Wedeal with complicated matters when we deal withrates of change of current, matters that requirenot only the Special Theory of Relativity, but theGeneral Theory (the world of relative accelera-tions) to justify them to a theoretical physicist .Might there not exist a similar complexity alsoin the gyroscope, if rates of change of accelera-tion are involved? The usual interpretations ofNewton's Laws of motion are very sound whenapplied to situations in which the accelerationdoes not change. Work on rates of change of ac-celeration (American scientists have called itsurge') is very sparse .Let us subject our gyro/electric motor analogy

to sterner tests . Let us exert torques on two axessimultaneously and see if the precessions appearto occur in appropriate senses . Attaching theweight to the inner ring again as in Plate 4.3, Ipush on the outer ring so as to try to increasethe precession rate. The velocity appears un-affected by the push, but the weight rises! Thetwo worlds can co-exist .Now to do the experiment designed to look

THE JABBERWOCK

for momentumless motion . I hang a large mass(1 kg) on the inner ring, giving the gyro a preces-sion rate of over 5 radians/second . As the 1 kgweight goes by I lift it as fast as I can, and thegyro ring system appears to come to rest in zerotime! If we were to photograph it with a high-speed camera we would be able to examine justwhat happened, in slow motion . We should un-doubtedly be impressed by the small amount ofmovement that occurs, the net result of whichwould be a slight upward movement of the pointof suspension of the weight . This small move-ment occurred in the reverse direction when theweight was first added and here lies the secretof the momentumless motion .The gyro rotor has a very large angular

momentum about its own axis . Suppose that axisis precisely horizontal as the weight is droppedon to the inner ring . A deflection of only 2°may suffice to give the gyro wheel a com-ponent of vertical angular momentum of(Iw) sin 2° ;:t 0-03(Iaw), just the amount to counter-act the precession angular momentum about thevertical . But this last is a strange momentum in-deed. It is in one sense `unavailable' momentuminasmuch as the removal of the cause reduces itto zero in the same order of time as switchingoff an inductive circuit takes to reduce thecurrent to zero. We can, if we like, declare theprecession angular momentum to be only`apparent' momentum and write it mathematic-ally asj(I'Q) where I' is the momentum of all theparts about the vertical axis .

Perhaps the most amazing thing about thislast experiment is that the moment of inertia ofthe gimbal rings about the vertical axis is almostexactly equal to that of the rotor itself about thesame axis, so the gimbal rings must have had areal momentum immediately before the weightwas lifted . In such circumstances (where the'dead' mass has a moment of inertia comparablewith that of the `live' rotor) we can only concludethat the rotor exhibited some negative, realmomentum in the sense that we are building thisanalogy-and why not? Negative resistance iscommonly accepted in modern electronics but itis usually a term applied to a system that`appears' to exhibit negative R . We must neverlet ourselves become obsessed with 'realities' in

45

ENGINEER THROUGH THE LOOKING-GLASS

engineering -we must concentrate on concepts .That a perfect gyro released from a horizontal

axis never displays angular momentum is not indoubt. Authorities on the subject will confirmthis . What I am pointing out to you now is thatit is a strange situation to say the least, for sinceall points on the wheel move in a plane atright-angles to its axis, and all precessionalmovement takes place in planes at right-anglesto this, the velocity of any point on the gyrowheel relative to the table on which theapparatus stands can never be zero, at any time .Yet the whole displays no momentum .

What is often not stated is that if a weight beadded when the gyro axis is not horizontal, butdisplaced at an angle 0 from it, a similar tiny de-flection in 0 results from adding, or subtractinga weight as before . In this case the gyro startswith a real momentum and the extra momentum(plus or minus) due to adding or subtracting theweight on the inner ring is partly `apparent',partly `negative real' as before, returnable to`source' on detaching the weight, just as the mag-netic energy stored in an inductance attempts toreturn to source if d .c. is switched off, or doesso periodically and continuously in the case ofa . c .

Let me also say very firmly here that I knowno property of a gyroscope that conflicts in anyway with the conservation of energy . I must saythis and underline it before some journalist saysthat I claim to produce energy out of nothingand therefore perpetual motion. Perpetualmotion is in the same state today as it was in thefifteenth century when Leonardo da Vincidenounced it so properly. He knew of themachines shown in Plates 4 .4 and 4.5 and otherslike them, equally incapable of generatingenergy out of nothing. If you really want to seeperpetual motion, look into the sky on a cloud-less night and marvel at the size and movementwithin the Universe . We have no reason tobelieve that if we fire a missile from an orbitingspace station into space there is anything to stopit except a chance interference with another motion of a kind and let that be an end of it . Inbody, which is probably as unlikely an occur- space-who knows? But so long as we remainrence as the molecules in a glass jar simulta- earth-bound, we shall have friction and this, asneously organising their movements such that Osborne Reynolds declared 100 years ago, is justthe jar jumps off the shelf! There is perpetual

as well, or this would be no fit place to live, for

46

Plate 4 .4 An early `perpetual motion' machineillustrated by Leonardo da Vinci .

Plate 4 .5 An alternative `perpetual motion' machinethat doesn't work either!

the air would be filled with flying objects! Andtalking of perpetual motion . . .

At this point, the lecturer spun a small topon a plastic dish (see Plate 4.6) . The top con-

Plate 4 .6 A tiny spinning top thatwill run for 5 12 days .

tinued to spin for the remainder of the lecture!Its mechanism is quite complex . Under the dishis a tiny mica chip containing transistors in aprinted circuit fed from a battery . Under the verycentre of the dish is an electromagnet whosecurrent is controlled from the printed circuit .The coil acts both as detector of the presence ofiron near the centre of the dish, and as force-pro-ducer, when fed with current. The bottom partof the spinning top is a steel disc . A part of the

Fig . 4 .5 The `perpetual motion' spinning top .

disc top is raised - the shape of an exclamationmark (see Fig . 4 .5) . When the top is pulledagainst and along it, the shape of the spindleis such that friction between spindle and raisedportion imparts more spin to replace the energy

THE JABBERWOCK

lost in friction, much as one does in a toy `whipand top' exercise (see enlarged view, Fig . 4 .6) .With a new battery that top will spin con-tinuously for 5z days .

Fig . 4 .6 The secret of the top shownin Fig . 4.5 (enlarged) .

Toy gyroscopes are fun-but they have muchmore to offer us than fun ! The toy manufacturercannot afford to put his wheel in gimbal rings .He has to sell it for 50p and make a profit . Buthe discovered by experiment long ago that asimple ring to hold the pivots would allow theuser to perform many almost unbelievable ex-periments, such as balancing the whole on atightrope (Plate 4 .7) or on the edge of a wineglass, or on a model Eiffel Tower (Plate 4.8) .

Plate 4 .7 A toy gyroscope on a tightrope .

47

ENGINEER THROUGH THE LOOKING-GLASS

Plate 4 .8 A toy gyro on a model of the Eiffel Tower .

(The resemblance to the real Eiffel Tower has de-creased in these days of affluence!) The preces-sion rate is still as predicted, in this case thetorque being provided by gravity on the wheelmass and a vertical reaction from the tower .

Look carefully as the wheel precesses and seeif you get the same unusual physical experienceas I do. Things just don't behave like that . Thegyro weighs over 40 times the tower weight, yetthe wheel precesses around the tower, not thelight tower around the wheel, as elementarymechanics would suggest or require that itshould. Let us assume that friction betweentower and bench is responsible for this effect andrepeat the experiment with a tower base set inice and free to slide on melting ice, when the co-efficient of friction is about 0 . 02, and let us usean old-fashioned wheel made of lead (over 400times the mass of the tower) . Still the towerhardly moves, yet still we might be deceived, forif true it should be possible to start the gyro fromone end of a diameter and catch it at the other,thus displacing mass through space . Once dis-placed it would be a relatively simple matter topush it back again by ordinary mechanisedmeans and thus progress the experimenter in anychosen direction -which seems unlikely!

Let us repeat the experiment with a muchlarger wheel (18-lb mass on a 6-lb shaft) on astand, as shown in Plate 4.9. If I release the wheelfrom a position where its axis is horizontal it per-forms curious dipping motions called 'nuta-tions', as illustrated in Fig . 4.7. These are the

48

Plate 4.9 An enlarged version of the gyro on tower .The wheel weighs 24 lb .

equivalent of transients in an electric circuit thatoccur on switch-on . The nutations soon die out .Now it costs but little effort to stop the precess-ing wheel, large though it is, for its momentumis only apparent, and to lift the shaft through afraction of an inch stops it dead .

If I now set it precessing again and applytorque to the shaft to try to increase the preces-sion rate, you will see the spring beneath itscentral pivot block (see Plate 4 .9) expandmomentarily. This is a dangerous experimentnot recommended for anyone but an `old handat the game', for during the transient the standof the gyro suffers toppling torques, and they are

Fig . 4 .7 `Nutation' of a gyro-on-tower .

not always in the direction that you mightexpect, i .e . centrifugal forces trying to pull thetower along the wheel shaft .The way that Ohm's Law is generally taught

is that in an electrical conductor, an e .m.f. causesa current. This is not the way Ohm himselfexpressed it, rather he defined a constant (R) asbeing the ratio E/I. The fact that we elected tosupply all houses and factories at a constant volt-age ensured that most of us would always tendto think of e.m.f. as cause and current as effect,for all houses are wired in parallel . The alterna-tive of a series-connected national network wasopen to us, but we wisely rejected it purely onthe grounds of cost. But locally one may builda constant-current system, as shown in Fig . 4 .8,by having a very high voltage feeding a load rthrough a very high resistor R . If r is always lessthan 1 per cent of R, but variable, we can say

Fig . 4 .8 Electric current can be the cause of avoltage.

to a close approximation that the current is con-stant (in this case at 2 amps) and that the voltageacross the load is simply 2r, whatever value r hasbelow 10,000 ohms . We thus tend to regard I ascause and E as effect .

So it is with gyros . A torque may be treatedas cause, if cause it is . However, one may alwaysrotate a gyro at a demanded precession speed,irrespective of what then happens : the resultingtorque on the perpendicular axis can be seen aseffect, rather than cause .

Let us leave the big wheel for a few minutesin favour of a less dangerous, but still quite largeoffset gyro. Plate 4.10 shows the apparatus hav-

THE JABBERWOCK

Plate 4 .10 A 5" diameter brass wheel that can beforce-precessed .

ing a 5" diameter wheel, 1" thick and mountedon a base . I can `communicate' with thegyro by pushing in a handle in the base tocause bevel gears to engage . As before, if I tryto increase precession speed, the wheel rises ; ifI try to retard it, it falls . Now let us stop the wheeland push the gyro stand until it overhangs thebench-as shown in Fig . 4 .9-and is on the pointof toppling. Now spin up the wheel with com-pressed air and release it. It does not fall at all,for the precession produces a torque that trans-fers the weight of the wheel to the top of thetower, as shown in Fig . 4.10 .

Let us rebalance the system when the wheelis not spinning, and its axis is parallel to the edgeof the bench, as shown in plan view in Fig . 4 .11 .Now if the wheel transfers its weight to the towerwhen spinning and precessing freely, we shouldstill be exactly balanced . Any centrifugal forceshould therefore topple it, in theory . The preces-sion rate for this experiment is of the order of0.5 revolution per second . The wheel weighsabout 6 lb f. Centrifugal force for a radius of 8"(the shaft length, tower centre to centre of wheel)should be of the order of 1-0lbf. We can counter-weight this l - 01bf. a t

5" (the height of the tower)

by a counterweight of 0 .5 lbf at 10" from the edge .It now appears possible to move the counter-weight nearer to the edge of the table withoutthe system toppling, which suggests that perhapsapart of the centrifugal force appears to be miss-ing . From one point of view this might beexpected from a rotating device with no angular

49

ENGINEER THROUGH THE LOOKING-GLASS

50

Fig . 4 .9 Balancing the offset gyro .

Fig . 4.10 The precessing gyro does not overbalance .

Fig . 4.11 Plan view of stationary wheel inbalance position .

momentum, for do we not calculate centrifugal(or centripetal, depending on where you arestanding when you observe it, not on how youwere taught!) force by the change in angularmomentum? Yet is this not also an excitingthought, for it is not every day that one discoversthe absence of something where it ought to havebeen!

Edward de Bono encourages us to do `lateralthinking' and his books contain delightfulexamples . Well, here is a beauty he might liketo add to the collection . Gyroscopes do notexhibit a new .force . They show a lack of a forcewhere we would have expected one! That is whyit was so hard to see . `I see nobody on the road,'said Alice . `I only wish I had such eyes,' the Kingremarked in a fretful tone. `To be able to seeNobody! and at that distance too!' Lewis Car-roll must be laughing at us all . If there is a lackof force, the rest is just engineering .What surprises me is that the gyro did not

avail itself of the opportunity to tilt as shown inFig. 4.12, thus lowering the centre of gravity ofthe whole system without changing its axis ofspin in this outermost position . For it has twoeffective pivots A and B in the same plane, andA will prevent any torque from being transferredto B. Perhaps it is because the gyro would haveto move away from the bench to do so . (Thisthought occurred to me only three hours beforethe lecture began .) So let us turn the experimentupside down, as shown in Fig . 4.13, for now atilt involves the wheel in moving nearer thetable . But the result is the same, stability all thetime .

Fig . 4.12 Why does the gyro not do this?

Fig. 4.13 An upside-down gyro does not answerthe question posed by Fig . 4 .12 .

Fig . 4 .14 The start of the double joint experiment .

bearing ring hanging freely from the secondpivot . The wheel is then spun and raised untilthe whole gyro axle is horizontal and from thisposition it is released . If the spinning wheel suc-ceeds in transferring its weight to the secondpivot by precession, the bearing ring and pieceof spindle up to the second joint are 'dead-weight', and must surely cause the gyro to pre-cess about the vertical tower pivot whilst adopt-ing an attitude as shown in Fig . 4.15 .

Fig . 4 .15 The expected result of the double jointexperiment .

What did happen was that the gyro end raiseditself to the position shown in Plate 4 .11, whilstprecessing about the vertical tower axis in theexpected direction. There appeared to be noordinary explanation for this but it reaffirmedmy belief, which I first expressed in a Friday even-ing Discourse at the Royal Institution on8 November 1974, that a gyro exhibited pheno-mena that were not to be found in any othermechanical object, and could well be worthy ofa study at a level not hitherto attempted . (Thatmy audience at this lecture was able to share in

Plate 4 .11 The double-joint experiment .

this thrilling experiment, only performed for thefirst time three hours earlier, and now only forthe second time, has continued to bring me greatjoy whenever I recall the demonstration .)

Gyro experiments are inclined perhaps tomake some of us think about atomic physicswhere orbiting electrons are said to spin also .Difficulties always begin when we cannot seewhat goes on. Let me show you what I mean byreplacing the second pivot on the gyro we havejust used by a solid connector, and removing thecounterbalance weight, to leave the gyro in thecondition shown in Fig . 4.16 .

Now the gyro can be spun and the whole crossarm assembly will rotate about the vertical axis .But suppose we choose its orbit so that its totalangular momentum about the vertical is muchsmaller than would be calculable from its dimen-sions and the formula, momentum=Iw, andsuppose we now enclose the gyro in a light box,as shown in Fig . 4 .17 . We can measure the speedabout the vertical axis, measure the angularmomentum (by, for example, allowing the

THE JABBERWOCK

5 1

The idea of this second joint at the bench edgeintrigued me to the extent that I told Mr Coatesthat we needed a second joint proper in theradius arm. Whereupon with no hesitation at all,he took a hacksaw to a very precious gyro andsawed its shaft in half, inserting a pivot at thecut. We only had time to try it out once beforethis lecture, but what an experiment it turned outto be!

First the system is balanced by means of theadjustable weight about the tower pivot, asshown in Fig . 4 .14, with the stationary gyro and

Fig . 4.16 The offset gyro as seen .

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Fig . 4.17 The gyro wheel is enclosed in a box .

revolving shaft to run up a small inclined planemounted on wheels, as shown in Fig . 4.18 andmeasuring the velocity of the trolley when thearm has come to rest relative to it) and deducethat there is very little mass inside the box .

We can then detach the box still containingthe spinning wheel, weigh it on a spring balanceand find the mass to be far greater than the firstexperiment had indicated .

Fig . 4.18 Demonstration of a precessing gyro thathas no momentum about the axis of precession .

It is a disturbing thought that an angularmomentum may not always be that which wepredict on the basis of weighing a mass andobserving its subsequent rotation, unless we canbe sure that nothing inside the mass, that we can-not see, is rotating also .

One of the things that has long worried me isthat rotary things appear to belong to a different`world' from things that move in a straight line .Yet there are analogous rules in which angularvelocity, Q, corresponds to linear velocity, V,and moment of inertia, I, corresponds to mass,M. So we can calculate momentum m as

m =IQ (rotary)orm = MV (linear)

dS1Twisting force (torque) = I

( dror

Linear Force

= M (dV)Energy= ZIQ 2

orEnergy = 21MV2

This seems very `tidy', but for one thing . Weknow that there is a very big difference betweenforce and energy (or work done), likewisebetween torque and energy . In either case theformer can exist without motion and thereforewithout energy . Only when force and motion co-exist is there energy .

Yet when we examine the relationship betweenrotary and linear motion we find that the dimen-sions of linear energy-for example, kineticenergy (zMV2)-and of rotary force (torque) arethe same, and each is equal to [force x distance]or [ML 2 T - 2 ] in the mass, length and timenotation .

If we examine the rotary and linear interpreta-tions of Newton's laws of motion dimensionallywe find that on the one hand

dVF= M-t has dimensions [MLT-2 ]

whereas torque

=IdQ

has dimensions [ML 2T -2 ]

The second equation appears to have beenmultiplied throughout by length to get the firstand there would appear to be no reason why, inmechanical systems, that `length' need be con-stant with the passage of time . There is a `magic'in spin that I have seen elsewhere only once inmy professional capacity - and I learned that itwas called 'electromagnetism'!

I am now going to repeat the `gyro on EiffelTower experiment' on the grand scale . I amgoing to ask a human guinea-pig called Dennis,who is aged nine, to submit himself to somerough treatment . I am going to have him standon a circular rotatable platform and besecurely fastened to a vertical pole that will re-volve with it (Plate 4 .12). The platform shaft islinked by sprockets and chain to another shaftcarrying a handle . If I turn this handle I canmake Dennis rotate in either direction . But I cando the same by giving him a spinning, offset gyroto hold, and I propose to give him the big one!But first I would like a much older volunteer totry to hold the 24-lb wheel and axle at arm'slength, allowing the shaft to hang vertically -hecannot do this, nor to be honest, can I . But whenit is spun up, using a drill (Plate 4 .13), not onlyyou and I, but young Dennis also, can hold itout horizontally, provided it is allowed to pre-cess (see Plate 4.14) .

When he is revolving, I will turn the handleso as to try to speed him up and you see anapparently Herculean act, no less, as Dennisraises (without knowing how he does so) the24-Ib mass into the air (Plate 4 .15). Notice theangle of his shoulder and elbow joint and therelaxed expression on his face . This is not the

Plate 4 .12 Dennis is securely tied to a poleon the turntable .

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Plate 4.13 Bill and Barry spin the big wheelup to 2400 r .p.m .

Plate 4.14 Dennis rotates as he holds the big wheel .

Plate 4.15 As torque is applied the wheel liftswithout pressing down on Dennis .Is this the face of a 9-year-old boy under stress?

face of a 9-year-old boy holding a 24-lb weightin that position. This act was involuntary onDennis' part. He just found the gyro rising in hishands (Plate 4 .16) . Stopping him by catching thegyro, as we know, requires little effort, and free-ing him again brings relief, for I know that hewas much concerned lest he drop the gyro . I

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Plate 4.16 Eyes full of wonder, Dennis knows thatthe wheel rises without conscious action by himself.

want to shake this brave young man by the hand .This is no experiment to try yourselves . It is moredangerous than holding a sizeable sky rocketwhile it is burning. The big wheel is more danger-ous than connecting apparatus to the householdelectricity supply. If the wheel were to bedropped and to run amok, I can tell you nowthat its energy is sufficient to throw it 200 feetinto the air - and this theatre is less than sixty feethigh !

You see, rotating mass is a very compact formof storing energy, more compact than most ofus realise. In the league table of energy/weightratio it comes a good third to those much soughtand used creatures, nuclear fuel and chemicalfuel . Electromagnetism, hydraulics, pneumatics,fuel cells, solar batteries and so on, are muchlower down the table, with electrostatics bottomof the league, and car batteries not leading themby very much (which is why we have not got ourelectric cars yet) . But we did have flywheel-driven buses -in Switzerland, where the route isall up hills and down . The one thing a petrolengine will not do is to pour petrol back into thetank as you go downhill . But a flywheel willaccept the energy back, so there is enough energyin a flywheel to drive a bus for half a day . A nicefigure to remember is that a high tensile steel fly-wheel spun up to its bursting speed (oh, I forgotto tell you, sometimes they burst, too!) is equi-valent in energy to the same volume of water asthe volume of the flywheel raised to 100,000 feet .When my colleague Professor Eastham and Iaccidentally burst a flywheel only 1 foot indiameter at Imperial College some two years ago,

54

pieces of it split the 3-foot-square cast-iron lidof a circuit breaker 7 feet up the wall and hurleda half of it some 15 feet across the room . OnlyProvidence herself dictated that we were out ofthe room at the time .

As one final danger signal here is a modestgyro (Plate 4 .17). It is a ball race 3" in diameter .I propose to spin it up with an airjet and let itgo free on a rubber mat where the frictional forceis considerable. (The result was that the gyroleapt some 10 feet in the air and was caught byBill Coates in a butterfly net .) In rehearsal whenthere was no fear of hurting members of theaudience, we ran the wheel faster and it leaptsome 40 feet, just short of the ceiling of thislecture theatre . So stick to small gyros for anyexperiments you might like to make . By allmeans hang a toy gyro on a string and study itsmotion. There are many good projects that beginthis way .

Plate 4 .17 A 3"-diameter ball race to be usedas a gyro .

Perhaps the only proper way to understand agyroscope is to make its parts more flexible (aswe did with the iron filings and mini magnets) .Plate 4 .18a shows a `wheel' in which the heavyrim is broken up into small pieces and each piecemounted on its own spring steel, wire spoke . Itis driven by a small electric motor and batterymounted on the inner ring . (The outer ring is theframe fixed to the bench .) When it has been runup to speed I give the inner ring a twist and youcan see it precess. As the plane of the wheel

a

b

Plate 4 .18 (a) A gyro in which the wheel is made upof masses on individual springy spokes . (b) Thespoked gyro tilts due to enforced precession .

comes into line with your eye, you can see thatits plane is tilted due to attempted precessionabout a horizontal axis, as in Plate 4 .18b. Whatis curious is that I do not need to keep applyingtorque about the vertical axis in order to keepthe inner ring rotating . It appears to be capableof 'free-wheeling', as if nothing inside it wasspinning at all .

It is a special case of a `rate' gyro in which thebench and earth on which it stands are still tobe seen as the `outer' gimbal ring . The frame Ispun was indeed the inner ring . It is easier tounderstand first the action of a more con-ventional rate gyro as shown in Fig . 4 .19. If theturntable is rotated, that in itself constitutes pre-cession of the outer ring . Such precession, as wehave seen by analogy with a constant current

Fig . 4 .19 A gyro to measure rate-of-turn .

electric circuit, is the cause of a torque thataccelerates the inner ring until the restrainingspring tension matches the demanded torque .The inner ring's deflection therefore acts as ameasure of the rate of turn of the outer ring . Thissystem is used as an aircraft instrument, and inspace vehicles, to measure, simply, `rate of turn' .

But beyond a few applications such as these,the gyro compass and the ship stabiliser, the sub-ject of gyros, although older than that of induc-tion motors, whose action they `reflect' so well,is still relatively `new' . In this subject the seas areremarkably uncharted and full of exciting ex-ploratory journeys . So far I have only had timeto make replicas of a few of the real machinesI would like to try (Plate 4 .19). They form a seriesof gyros that began with the gimballed gyro with

THE JABBERWOCK

Plate 4 .19 A series of curiously offset gyros .

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ENGINEER THROUGH THE LOOKING-GLASS

all three axes coincident and mutually at right-angles (Plate 4 .2) . Next the offset gyro, the onlytype on which I have had time even to begin in-vestigations. Its rotor axis meets both the otheraxes at right-angles, but the other two axes areorthogonal skew lines, so there are two inter-sections and one skew pair, all mutually perpen-dicular. The next has only the rotor andtorque axes intersecting and there are two pairsof skew axes, the rotor/precession and thetorque/precession, but all are still mutually per-pendicular. Third in my line comes a mon-strosity with all three pairs of skew axes, mutu-ally perpendicular. Each of these will generatesub-families with non-orthogonal axes .

I said there was a lot to do, and I am now say-ing : as you do it, be sure to keep an open mind .

We have not yet covered the situation wherebyenergy could be injected or extracted from therotor, other than via its bearings . If this is poss-ible let us speculate on what this might be. Couldit be a mechanical form of what we electricalengineers call `radiation'?

One more demonstration to complete thislecture. The apparatus is a simple wheel in abearing ring with the ring tethered near to oneend of the wheel shaft and suspended from apoint on the ceiling of this theatre some 50 feetabove us. A retort clamp stem on the benchmarks the point directly below the supportpoint. The wheel is spun and held with its axishorizontal and with the supporting cord vertical .This cord contains a swivel to prevent it impart-ing torques to the gyro .

Upon release, the gyro traces out a path asshown in plan in Fig . 4.20, the end of the shaft

56

Fig . 4.20 A gyro tethered to the roof (over 50feet high) orbiting (plan view) .

remote from the supported end acting as`leader' . The wheel spirals out from the centreuntil it appears to be in stable orbit in a circularpath about 20 feet in diameter (see Plate 4 .20) .Now the gyro axis has `drooped' to the positionshown in Fig . 4 .21 . At first we thought that thiswas evidence of frictional power loss, but it is

Plate 4.20 A 16-lb gyro orbits the theatre whensuspended on a 50-ft rope .

Fig . 4 .21 Gyro angle when in maximum orbit .

not so. Fig . 4 .22 shows that had it been otherwiseit would have needed to extract energy from therotor. Instead it chose not to raise its masscentre, as shown .

But what is happening now?-for no goodreason we can find, it has begun to spiral inwards,still with `nose leading' as shown in Fig. 4 .23 .As it does so, the axis of spin begins to rise again .Will it return to centre from where it started?No, it goes into steady orbit at about 6 feet dia-meter, completes about 6 revolutions-and thenstarts on an outward track again! Once again itreaches full amplitude, performs several orbits

Fig . 4 .22 If the gyro were to remain horizontal, howwould it lift itself through height, h?

Fig . 4 .23 Gyro illustrates the Bohr atom!

there and returns to the 6-foot orbit once again .The most enthusiastic among us might be for-

given for exclaiming : `The Bohr Atom!' It maybe some twist in the string, some imperfectionin the system, just some coincidence that means

we cannot guarantee to repeat it . But we mustkeep trying and, above all, never forget .

Beware the Jabberwock, my son!The jaws that bite, the claws that catch!

There are many heads, many claws, but what an

adventure!

THE JABBERWOCK