Physics in Your Kitchen Lab -Kikoin Mir

8/3/2019 Physics in Your Kitchen Lab -Kikoin Mir

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First published 1985 :,,,, , -Revised from the 1980 Russian ed itionSL 1c-t(,,

EDITORIAL BOARDAcademician I.K . Kikoin (chairman), AcademicianA.N. Kolmogorov (deputy chairma n), 1.Sh. Slobodetskii(scientific sec reta ry), Cand. So. (Phys.-Math.), Correspond-ing mem ber of th e Academ y of Scienc es of th e USSRA.A. Ahrikosov, Academician B.K. Vainstein, Honouredteacher of the Russian Soviet Federative Socialist Re-public B .V. Vozdvizhenskii, Aca dem ic~a n V.M. Glush-kov, Academician P.L. Kap itsa, Prof. S.P. Kap itsa,Corre spon ding mem ber of the Acad emy of Scie nces of th eUSSR Yu.A. Osipyan, Corresponding member of theAcademy of Pe dago gical Sciences of th e USSR V.G. R a-zumovsk ii, Academician R.Z. Sagdeev, M.L. Smolyanskl!,Cand. Sc. (Chem.), Prof. Ya.A. Smorodinskii, Academl-cian S.L. Sobolev, Corresponding member of the Academyof Sciences of the USSR D.K. Phaddeev, Correspondingme mb er of th e Acade my of Scle nces of th e USSRI.S. Shklovskii.

@ H a f i a ~ e n s c ~ s o~H ay x a* ,Pna oeaa penal'qua 4aaa xo-MaTeMaTHeCKOH JIUTepaTypLI, 1980@ English translation, Mir Publishers. 1985

Conte


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ContentsDetermining the Poles of a Magnetb y B . A le in ikovA Peculiar Pendulumb y N. MtnzLissa jous Figure sb y N. MinzExercisesWaves in a Flat Plate (Interference)b y A . KosourovHow to Make a Ripple Tank toExamine Wave Phenomenab y C . L . StongAn Artificial Representation of aTotal Solar Eclipseb y R . W . W o o dRelieve It or Notb y G. KosourovColour Shadowsb y B . KoganWhat Colour Is Brilliant Green?b y E . Pal'chikovAn Orange Skyb y G . KosourovThe Green Red Lampb y V. M ayerMeasuring Light Wavelength with aWireb y N. RostovtsevExercisesMeasuring Light with a PhonographRecordb y A . B on da rA Ball for a Lensb y G . Kosourov

Edit

Phstudiexpertionsonly This set uthe thereexperis a undeer nebecomof thMoisticathe mlaborbookreseaexperformeincluthe joJusan ex


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Editor's Notesand descr ipt ions of one. I t i s recommended,therefore , tha t r eader s per form the exper iment sdescr ibed themselves . The means for this arereadi ly avai l able , and i t should soon becomeobvious tha t exper imenta t ion i s a capt iva t ingpas t ime. Th e exper iments presented here need

I not b e confining; they ma y be var i ed and expand-I ed , p r ov i d i ng , i n t h i s w ay , an oppor t un i t y f o rI real scient i f ic inves t igat ion.Th e book is dedicated to Georgi i Ivanovich KO-II sourov, one of th e founding fathers of Kuant .I Kosourov, who edi ted the exper imental sect ionof th e journal i n i t s f i rs t year of publ icat ion,has contr ibuted several very interes t ing ar t iclesto th i s col lec t ion . Among the o ther author sof this book are a number of famous physicists ,as wel l as youn g researchers jus t beginning the i rcareers . W e hope this book wi l l fascinate notonly s tudent s a l r eady in teres t ed in phys ics who

I i n t end to make i t t he i r l i f ework but a l so thef r i ends to whom they demons t ra t e the exper i -ment s in a l abora tory made r ight a t home.

A Deby A . D

The A sa teman dulessnesnothingpu l l onwill leFiguIn i t s

Fig. 1aut , anontac t


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10 A. Dozorovcircuit. Naturally, lamp L , connected to thecircuit, does not light up in thi s case. If theentire device is tossed into the air, however,weight G becomes weightless and does not tightenthe string. The elastic plate straightens out andthe terminals connect, which switches on thelamp. The lamp is lit only when the device is in

l a weightless state. Note that th is stat e is achievedboth when the device is thrown up and as itreturns to the ground.The adjustment screw S makes it possible toI place the terminals so that they have a smallclearance when the device is stationary. The

device is fastened to the inside of a transparentbox, as shown in Fig. 2 .A little practical advice about construction.In order to provide for the use of a large-cell(flat) battery or a small one-cell battery, reservespace for the larger battery. Access to the bat terycompartment should be facilitated since batteryI may have to be replaced frequently. The batte rycan be secured to the outer surface of the device,and two holes for connecting wires should be

provided in the casing.Any th in elastic metal st rip can be used as anelastic plate, even one half of a safety razorblade (after fastening the blade to the stand , youwill see where to co lnect the string for the weight).The design can be simplified further, i f theadjustment screw and terminal K 1 are combinedand the plate functions as terminal K 2 (Fig. 3 ) .Figure 4 shows a design that has no adjustmentscrew at all. If you think a little, you canprobably come up with an even simplesdesign.

A

, Fig. 4


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A Cartesian Diverb y A . VilenkinA toy ship made of paper will float easily, butif the paper gets soaked, th e ship sinks. Whenthe paper is dry, it traps air between its belland the surface of the water. If the bell getssoaked and begins to disintegrate, the air escapesthe bell and the ship sinks. But is it possibleto make a ship whose bell alternately keeps orreleases air, making the ship float or sink as wewish? It is, indeed. The great French scholar andphilosopher Ren6 Descartes was the first to make

such a toy, now commonly called the 'CartesianDiver' (from Cartesius, the Latin spelling ofDescartes). Descartes* toy resembles our papership except tha t the 'Diver' compresses andexpands the air instead of letting i t in and out.A design of the 'Diver' is shown in Fig. 5.Take a milk bottle, a small medicine bottle anda rubber balloon (the balloon will have to bespoiled). Fill the milk bottle with water almostto its neck. Then lower the medicine bottle intothe water, neck down. Tilt the medicine bottleslightly t o let some of t he water in. The amountof water inside the smaller bottle should be regu-lated so that the bottle floats on the surface anda slight push makes it sink (a straw can be usedto blow air into the bottle while it is underwater).Once the medicine bottle is floating properly,seal the milk bot tle with a piece of rubber cutfrom the balloon and fastened to the bottle witha thread wound around the neck.Press down the piece of rubber, and the 'Diver'will sink. Release it, and the 'Diver' will rise,

An AuThis compin. Th

bottle, soon asmedicinthe 'DiAn Auby V . M

Most Ihe sipliquids,Americabegun hjust suc


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14 V. Mayer and N. Nazarov An Autdescribed t ha t first experiment in his book aboutRobert Wood. He wrote that there was an eleva-tion over a foot high around a puddle, and every-body knew that water would not flow uphill. Roblaid a hose on t he ground and told one of the boysto seal its end with his finger. Then he startedfilling the hose with water until it was full.Already a born demonstrator at that age, Rob,instead of leaving his end of the hose on theground, let it dangle over .a high fence whichseparated the road from the ditch. Water flowedthrough th e siphon. Thi s was apparently Wood'sfirst public scientific victory.

The conventional siphon is so simple thatalmost no improvement in its design seemspossible. Perhaps its only disadvantage is tha ti t is necessary to force the air from bends in thesiphon prior to operation. Yet even this problemwas solved, thanks to human ingenuity. Onceinventors had understood the shortcoming in thedesign, they removed i t by the simplest possiblemeans!To make an automatic siphon*, you will needa glass tube whose length is about 60 cm andwhose inner diameter is 3-4mm. Bend the tubeover a flame so that it has two sections, one ofwhich is about 25 cm long (Fig. 6). Carefully33-35mm from its end with the edge of a needlecut a small hole (I) in the shorter arm aboutfile (wet the file first), The area of the hole shouldnot be more than 0.5-1 mm2. Take a ping-pong

areamedth

Fig.

sidehe tubeThis version of the automatic siphon, invented by Ie is to.D. Platonov, was described in Z a vod ska ya L a b o ra t o r i w ,4 , N o. 6 (1935) (in Russian). ke anot


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i6 R. W. Woodinside the ball. Its initial diameter should beabout I mm.Quickly lower the arm of the siphon with th eball at the end into a glass of water. The tubewill fill almost immediately with a rising col-umn of water broken by a series of air bubbles.When the water reaches the bend, it will movedown the second arm of the siphon (Fig. 7) .and in a few moments a continuous stream ofwater will begin to flow from the end of the tube!If the experiment i s unsuccessful at first, simplyadjust the siphon slightly. The correct operationof the automatic siphon depends on the appro-priate choice of diameters of the holes in thetube and the ball. Faul ty positioning of theglass tube and the ball or an inadequate sealbetween the tube and the ball may also spoil thesiphon operation. The second hole in the ballcan be gradually enlarged with a needle file toimprove the performance of the siphon. As soonas the siphon is operating satisfactorily, gluethe ball to the tube.How does t he automatic siphon work? Look atFig. 6 again. When the ball is lowered into theglass, water floods simultaneously into opening 2and into the open end of the glass tube. Waterrises in the tube at a faster rate than in the ball.The water rising to opening I in the wall of t h ~tube seals the tube. As the ball floods with waterthe air pressure inside it rises. When equilibriurris reached, a small air bubble is forced througfthe opening I. The bubble cuts off a small col

umn of water and carries i t upward. The wa teth at follows reseals opening I , and the compressecair forces another air bubble into the tube, cut

'Vortex ting osectionmixturwater. rises toof the with wof wateEXERCI1.LShowslower r2. To bethe siphosmall glahe setupthe or iginstopper . to see th3. Determdepends 4. Make awith a ru

Vortexby R.WIn the to a lectcertain mto teacheThe clrequire mis rather

a pine boNature,

2-01544


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Vortex18 R. W. Woodwith a circular hole 25 cms in diameter i n oneend. Two pieces of heavy rubber tubing ar estretched diagonally across the opposite or openend, which is then covered with black enamelcloth t acked on rather loosely. The object of therubb er chords is to give th e recoil necessary afterthe expulsion of a ring to prepare the box fora second discharge. Such a box will project airvort ices of gr eat power, the slap of t he ring

Fig. 8

against the brick wall of the lecture hall beingdistinctly aridihle resembling the sound of a flipwith a towel. An audience can be given a vivididea of t he quas i-rigid ity of a fluid in rotat ion byprojecting these invisible rings in rapid succes-sion into the audi torium, th e impact of thering on the face reminding one of a blow wi tha compact tuft of cotton. aFor rendering rings visible I have found that byfar t he best results can he obtained by conductingammonia and hydrochloric acid gases into thebox through rubber tubes leading to two flask?in &-hioh NH,OH and HCl are boiling. Photo-graphs of large rings made in this way a re repro-duced in Fig. 8, th e side view being particular13interesting, showing the comet-like tail formerby t he stripp ing off of th e outer portions of tht

ring byThe directinstood oapparaeven dof burnimpact For bouncinplan is box, thgreater(,hat I hi~nsatiThouapparatfor lecbeautifwith tobtube abpracticestrengthin this sunlightstream succeedin the shut ter an arclaa large affair, miln elastil seconin the d2*


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20 R. W. Woodbe i l luminated by the divergent beam comingfrom the image of the arc when the shut ter wasopened. A ruby lam p was placed in f ront of t hesens i t ive fi lm. As soon as a good r ing, sym m t r ica lin form and not moving too fas t , was seen t o bein f ront of th e p la te , a s t r ing leading to the shu t -I t e r w as pu l led and th e p l a t e i l lum ina ted w i th adazzl ing f lash. The r in g cas ts a perfec t ly sha rpshadow owing to the smal l s ize and dis tance of

th e source of l ight ; the resu l t ing pic ture is repro-duced in Fig . 9. Th e r ing i s seen to cons is t ofl a laye r of smoke and a Layer of t ransp aren t ai r ,wound u p in a spi ra l of a dozen or more comp leteI turns .Th e angular ve loci ty of ro ta t ion ap pears tc

increase as th e core of the r ing i s approachedthe inner por t ions being screened f rom fr ic t ioni f w e m ay use the t e rm , by the ro t a t ing l aye r :surround ing them. T his can b e very nice ly show1by di f ferent ia t ing the core , forming an a i r r ingwi t h a smoke core . I f we ma ke a smal l vor tex bolw i th a ho le , s ay 2 cm i n d i am e te r , f il l i t w it1smoke and push v ery gen t ly agains t the d iaphragm , a fa t r ing emerges which rota tes in a ver :lazy fashion, to a l l appearances . I f , however , wclear the a ir of sm oke, pour i n a few drop s oam m onia and b rush a l i t t l e a s t rong HCl a rounthe lower par t of the aper ture , the smoke form

Vortexin a th iG iv ingwe findlhe rinvor tex Considelhin cr

C'ig. 10~ ~ s ~ r ahave besmoke-cof t he vi l i s i n sgrazes tof the rwell onsuccess l ipper edi ron wirBy ta[ ,hesmokis sho


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22 S. Shabanov and V. Shubinmanner that the existence of the ring depends inno way on the presence of t he smoke. The bestway to form these half-rings is to breathe smokevery gently into a paper tube allowing it to flowalong the bottom, until the end is reached, whena ring is expelled by a gentle puff. A large testtub e with a hole blown in th e bottom is perhapspreferable, since t,he condition of things insidecan be ~ a t c h e d . t is easy enough to get a ring,one half of which is wholly invisible, th e smokeending abruptly at a sharply defined edge, asshown in Fig. 11, requires a good deal of pract ice .I have tried fully half-a-dozen different schemesfor getting these half-rings on a large scale, butno one of them gave reslilts worth mentioning.The hot wire with the sa l ammoniac seemed to bethe most promising method, but I was unable toget t he sharp cut edge which is the most strikingfeature of t he small rings blown from a tube .In accounting for the formation of vortexrings, the rotary motion is often ascribed tofriction between Lhe issuing air-jet and the edgeof the aperture. I t is, however, friction with t heexterior air that is for the most part responsiblefor the vortices. To illustrate this point I havedevised a vortex box in which friction with theedge of tlie aperture is eliminated, or rathercompensated, by making it equal over the entirecross-section of t he issuing jet.The bottom of a cylindrical t in box is drilledwith some 200 small holes, each about 1 . 7 mmin diameter. If the box be filled with smoke anda sharp puff of air delivered at the open end,a beautiful vortex ring will be thrown off fromthe culleiider surface (Fig. 12). We may even

On Vortcover thcloth, twith itIn extwo cirof two large rirotation

0sluggishthe foran extr[t at onirlto a vcan scaoscillatecnon caoppositeto eachstill as On Voh y S. ShFormat

To sttory con


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24 S. Shabanov and V. Shubinby Professor Tai t (Fig. 23). One end of this cyIin-de r , t he membrane , i s cove red w i th a f l ex ib l emate r i a l such as l ea the r . T he o the r end , t hediaphragm, has a c i rcular opening. Two f lasks ,one conta ining hydrochlor ic ac id (HCI) , the othera m m o n i u m h y d r o x i d e (NH,OH), are placed inth e box, where they produce a th ick fo g (smoke)of amm onium chlor ide par t ic les INI-1&I). B y

Membrane . Diaphragm

III Fig. 13 Fig. 14

t app in g t h e membrane , we impar t a ce r ta in ve loc -i t y t o t h e smoke l aye r c lose t o i t . As t h i s l aye rm o v e s f o r w a r d , i t c o m p r e s s e s ~ t h enex t l aye r ,which , i n t u r n , compresses t h e l aye r fo l l ow ing~ i t ,i n a cha in r eac t i on t ha t r eaches t he d i aphragmwhere smoke escapes t h rough t he open ing andsets former ly s t i l l a i r i n mot ion. Viscous f r ic t ionaga ins t t h e edge of t h e open ing tw i s t s t he smokyai r in to a vor tex r ing.Th e edge of th e opening i s not th e ma in fac torin th e form at ion of t he vor tex r in g, however .W e c a n p r o v e t h i s b y f i t t i n g a s i eve ove r t heopening i~ the Tait's apparatus. If th e edge were

On Vorteximpor tantYet they observe aIf t he t ha t i s s e tappear onvor tex r in

Vig. i6intermit tenr~pen ing .Vor tex r~ r l rdinarink fal l f roi q ~ ~ a r i u:onvection wil l be verThe se t u)fllink can n water (n this casVor tex rn t he a i r , q qimi l a r tases vi sconen t s showP the firs

- . -


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-26 S. Shabanov and V. Shubin on vorteIvortex, however. As it develops further, the vor-tex behaves differently i n water and t he air. IMovement of the EnvironmentAround the Vortex Rings

Wh at happens to the environment when a vor-tex forms? We can answer this question withIhe right experiments.Place a lighted candle 2-3 metres 'away fromthe Tait apparatus. Now direct a smoke ring SOIFig. 17

that it passes the candle but misses the flanarrowly. The flame will either go out or flicviolently, proving tha t th e movement of t hvortex involves not only the visible part of t hring , but also adjacent layers of t he air .How do these layers move? Take two pieof c lot h, and soak one in hydrochlor ic ac id,the other in ammonia solution. Hang themabout 10-15centimetres apart. The space betwthein will immediately be filled with sm(ammoniam chloride vapour). Now shoot a smoring from the appa rat ~ls nto the vapour cloAs the ring passes through the clood, theexpands while the clolld star ts moving circulFrom this we can conclude that the air cloth: vortex. . ing is. . circulating (Fig. 17).

A simPut a drbeen stiLet the in the wWhen thtwist.Vortex R

We dein water

-- -----placed intl imetres soon deveI lrrn, breaform a (Fig. 18)


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28 S. Shaba nov a nd V. ShubinW e found tha t t he d iv i s ion o f the in i t i a l r ingin to secondary r ings was preceded by expansionsin the l a rge r ing i t se l f . How th i s c an be exp la ined?S ince the env ironmen t th rough wh ich th e inkr ing moves is nonuniform, som e of i t s par ts movefa s t e r than o the r s , some l ag beh ind . The ink(which is heavier than water) tends to col lec tin the fas te r sec t ions, where it forms swel l ingdue t o sur face tension . These swell ings g ive b i r thto new d rop le t s . Each d rop le t on the in i t i a lvor tex behaves indepe ndent ly , eventua l ly produc-ing a new vor tex r ing in a cyc le tha t repea tssevera l t imes. In te rest in gly , we could no t de te r-min e an y regu la r i ty in th i s cycle: t h e number ofr ings i n th e "four th generat ion" was d i f fe rentin each of ten experiments.W e a l so found t ha t vo r t ex r ings requ i re "liv-ing" space . W e tes ted th is by p lac ing p ipes ofd i f fe rent d iameters in th e pa th of r ings i n water .W h en th e d iame te r of the p ipe was s l igh t ly l a rge rth an tha t of th e r ing , t he r ing d i s in teg ra tedaf te r en te r ing the p ipe , to produce a new r ing

wi th a sma l l e r d i ame te r . W hen the d iame te r o fthe p ipe was fou r t imes l a rge r than the r ingd iame te r , t he r ing pa ssed th rough the p ipe w i th -ou t obs t ruc t ion . In th i s c a se the vo r t ex i s no ta ffec ted by exte rna l fac tors .Smoke Ring Sca t t e r ing

W e conducted seve ra l expe r imen t s to s tudyin te rac t ion be tween the smoke r ing and th copenin g of different dia me ter. W e also studiedthe re la t ionship be tween the r ing and a surfacca t var ious angles . (We ca l led these exper iment :sca t te r ing tes ts . )

3n Vortex RConsidape r tu re lwo ca sediaphragthe r ing and the rent re ofhand, m adoes not In the f i

1,110 d iaph~Iic lmete' l' l~ is sm al'Fa i t appaor ig ina l ront ra ins thA s imi la r co l l ides csom ewh atc:ontre cor~uwly o rmorigina l di~ rxp la inwNow le twi th a su


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30 S. Shabanov and V. Shubinsurface is perpendicular to the velocity of t hering, the ring spreads without losing its shape.This can be explained as follows. When the airstream inside the ring hi ts the surface, it producesa zone of elevated pressure, which forces th ering to expand uniformly. If the surface is ata slant relative to the original direction, thevortex recoils when colliding with it (Fig. 2 0 ) .This phenomenorl can be explained as the effect

11 , of elevated pressure in t he space between th e ring11 ", ) '

and the surf ace.

Interaction of RingsThe experiments with interacting rings werundoubtedly the most interesting. We conductelthese experiments with rings in water and in thair.If we place a drop of ink into water fromheight of 1- 2 cm and, a second later, let anothcdrop fall from 2-3 cm, two vortices moving adifferent velocities will form. The second drowill move faster than the first (u, greater than ulWhen t he rings reach the same dept h, they begito interact wi th each other i n one of three possibways. The second ring may overtake the fir:

On Vortex

one witholion the rcpel onering flowrrlore intIhe ink w

I;ig. 21i ~ r kpasse1 1 new, smI)roitk dool)served 'l'l~e secoI l ~ oirst whllrc more ~loslroy thvortices eol' I.he firsI'inally, Z l c ) , in tlrrough th


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ring expands. As before, this is a result of Tornado

even at h1. Solde

in diameteI

8 microm~nachinesI 10 the shaI

move closer.


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ba t te ry , the d i s c beg ins ro ta t ing and causes thel iqu id in th e g la s s t o c i rcu la te . Th i s c i rcu la t iond i s tu rbs the su r face be tween the wa te r and o i l ,and a cone fi lled w ith th e oi l soon forms. Th econe grows unt i l it touches the disc , which the nb r e a k s t h e o i l i n t o d r o p s , t u r n i n g t h e l i q u i dtu rb id . Af te r the mic romotor i s shu t o ff , th e o i l

d rops re tu rn to the su r face where they re faa con t inuous l aye r . Th e exper iment can thenrepeated.Figures 24a and 24b show photographs offo rma t ion o f th e a i r cone. W e modif ied theperim ent s l ight ly here by f il l ing th e glass wwate r on ly .2. An even more convinc ing model of a t ordb-can be cons tructed by solder ing a piececappe r wi re (o r a kn i t t ing need le ) abou t 25in l iength and 2 m m i n d i a m e t e r t o t h e s h a f la 'micromotor . Solder a rec tangular brass orp l a t e a b o u t 0.5 X 10 X 2 5 m m i n s iz e a t r i

r ingles to ti o check ths l ra igh ten r~l izewobbLower tI,or 15-20 Swi tch on

)n] lhe wat,n~ lds o thIl n to rnadoIppc?ar, s ig~ O I I o l d till behavYo11 can s1lrrge.s.(h n t in ue

,loc:k on ~ ~ ~ o k o c ln bio11 speed r l l t le rwate


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Felix Hess The Aerod

The tornado iwill! suck) in bodies: lying! jon:jtlbottom of the glass before the stir rer is switchcon i f their density is greater than th at of wat(which is not true of the wooden block).:Align the shaft of the motor with th e axisthe glass. You will see a cone moving down tlsha ft and air bubbles which mark its continution under the plate (Fig. 25c ) . If you place soxwell washed river sand on the bottom of t

glass, youof the torThese nlways caTh e Aerhy Felix H

Imaginemaking itit come toYet of codoes, provthrown pr;As is wet.ho aborigboomeranpnrts of thfor instancns far as tlisnppoinboomerangornngs cawnr boomof the first11sweaponboomerangI,l~rown tI)oomeranhers, are ActuallyIshis.There-An abridg111 the Nove


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Felix Hess The AerodIn Au stra l ia , a num ber of which look l ike boomeangs , so tha t the d i s t inc t ion be tween boomeranand th rowing o r s t r ik ing c lubs i s no t a sh a rp oNei ther is the dis t inc t ion between w ar boomerangs an d retu rn boomerangs. T he shap e of boomerangs can differ f rom tr ib e t o t r ib e (Fig . 26)Whether a g iven boomerang belongs to thre tu rn type o r no t canno t a lways be in fe r reas i ly f rom i t s appea rance . Re tu rn boomeranhowever , a re usual ly less mass ive and ha ve a 1obtuse angle be tween the ir two arms . A t y p ire turn boomerang m ay be between 25 and 75t ime te rs long , 3 to 5 cen t ime te rs wide0 .5 to 1 .3 cm th ick . T he ang le be tween th emay va ry f rom 80 to 140 degrees . T he wmay be a s much a s 300 g rams .The charac ter is t ic banana-l ike shape of moboomerangs has ha rd ly any th ing t o do wi th t hab i l i ty t o re tu rn . Boomerangs shaped l ike thle t t e r s X , V, S, T, , H, Y (and probabother le t te rs of th e a lph abet) can be ma dere tu rn qu i te we l l. The e s sen t ia l th ing is t h e crsec t ion of t he arm s , which should be m ore convon one s ide than on the o the r , l ike the wiprofile of an ai rp lan e (see Fig. 27). I t i s o nl y freasons of s tabi l i ty tha t the overa l l shape ofboomerang mus t l i e more o r less in a plane . Thif y ou ma ke a boom erdng o ut of on e piecena tu ra l w ood , a smooth ly curved shape fo llowithe gra in of the wood is perhaps the most o bvichoice. If you use othe r ma terials , suchplywood, p las t ics or meta ls , there are cons irably more possibil i t ies .How does one th row a re tu rn boomerang?a r u l e i t i s ta k e n w i t h t h e r i g h t h a n d b y o n e

Pig. 28


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40 Felix Hessits extremit ies and held vert ical ly upward, themore convex, or upper, side to the left . There aretwo possibi li ties: ei ther the free extrem ity pointsforward-as is th e pract ice among the Austra-lians-or it points backward. Th e choice dependsentirely on one's personal preference. Next, the

Fig. 27r ight ar m is brought behind the shorilder andthe boomerang is thrown forward in a h orizontalor slightly upward direction. For successfulthrowing, two things are important . First , theplane of th e boomerang at the mom ent of itsrelease should be nearly vertical or somewhatinclined to th e r ight , but c ertainly not horizontal .Second, the boomerang should be given a rapidrotat ion. This is accomplished by stopping thethrowing motion of the r ight arm abruptly justbefore the release. Because of its iner t ia theboomerang wil l rotate momentari ly around apoint si tuated in the thrower 's r ig ht hand. Henceit w il l acquire a forward veloci ty and a rotat ion-a l ve loc i ty a t t he same t ime.At f i rst the boomerang just seems to f ly away,but soon i t s path curves to the lef t and of tenupward. Then it may describe a wide, more orless circular loop and come down somewhere near

The Aeroth e throbefore second as a whoI t is a near ag

Fig. 28hovers soand theEvery w ith resand hovecan oftenon the wdepends thrower, as the induring a be much


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42 Felix Hesshighes t poin t can be as h igh as 1 5 meters abovethe ground or as low as 1 .5 meters. I have heardth a t w i th modern boomerangs of Aust ra l ian makedis tances of m ore tha n 100 meters can be a t ta in ed ,s t i l l fo l lowed by a perfect re tu rn , b ut I regrett o s a y t h a t s o fa r 1 hav e not been able to make aboomerang go beyond about 50 meters .I n th e foregoing genera l descr ip tion i t wastac i t ly assumed tha t the thrower was r ight -handed and used a "r ight-handed" boomerang.I f one were to look a t an ord inary r ight -handedboomerang f rom i t s convex s ide whi le i t was infl ight , i ts direct ion of rotat io n would be counter-clockwise. Hence one can speak of the leadingedge and th e trai l i ng edge of each boomerangarm. Both t he leading and th e t ra i l in g edges of anaboriginal boomerang are more or less sharp.Th e lead ing edge of a mod ern boomerang arm isb lun t , l ike t he leading edge of an a i rp lane wing.Somet imes t he a rms have a s l i gh t tw i s t , so t ha tthe i r leading edges are ra i sed a t the ends .The entire phenomenon must of course beexplained i n ter m s of th e interact io n of t heboomerang wi th the a i r ; in a vacuum even a boo-merang would describe noth ing but a parabola .This in terac t ion , however , is diff icult to calcu lateexa ctly because of th e complicated na tur e of t heproblem. Let u s nonethe less look a t the ma t te rin a s imp le way .If one throws a boomerang in a horizontald i rec t ion , wi th i t s p lane of ro ta t ion v er t ica l ,each boomerang arm w il l "wing" th e air . Becauseof th e special profile of th e arm s the air wil lexert a force on them directed from t he f lat ter ,or lower , s ide to t h e more canve x, or upper , s ide

The Aer(Fig. 29force ea r ight -f rom the

Fig. 29This formake a Fol lowt ion , onet o t h e a iupward , adds t o tpoin ts dos i te d i rebe smal lFig. 30)exper ienclef t but aaxis , wh


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44 Felix Hessits upper part to the left. Actually this turningover will not be observed because the boomerangis spinning rapidly and hence behaves like :gyroscope.Now, a gyroscope (which really is nothing morethan a rapidly spinning flywheel) has the propertythat, when a torque is exerted on it , it does notgive way to that torque but changes its orienta-tion around an axis that is perpendicular to boththe axis of rotation and,the axis of the exertedtorque; in the case of a boomerang the orienta-tion turns to the left. This motion is calledprecession. Thus the boomerang changes it sorientation to the left, so that its plane wouldmake a gradually increasing angle to its pathwere it not for the rapidly increasing forces thattry to direct the path parallel to the boomerangplane again. The result is that the path curvesto the left, the angle between boomerang planeand path being kept very small.In actuaI practice one often sees that , althoughthe plane of the boomerang is nearly vertical a tthe star t of the flight, it is approximately hori-zontal at the end. In other words, the plane of theboomerang slowly turns over wi th it s upper partto t he right; the boomerang in effect "lies down".Let us now consider the question in more detail.Because much is known about the aerodynamicforces on airfoils (airplane wings), it is conve-nient to regard each boomerang arm as an airfoil.Looking at one such wing, we see that it movesforward and at the same time rotates aroundthe boomerang's center of mass. We explicitlyassume that there is no motion perpendicularto the plane of the boomerang. With a cross

The Aeroboomeranthe centewings, bboomeranmass, whboth armdistance A precedfollowingpoint of continuorespect toairstreamedge of imaginedboomeranat the arforces onner?Let us moving ity v with tomary tocomponenthe dragportional wing is na componinfluence;perpendicvelocity"portional ,Looking clear that boomeran


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Felix Hessw i t h r es p ec t t o t h e a i r d u e t o t h e r o t a t i o n , h ow -ever , is different for each poin t . For a rotat io nalvelocity o a n d a p o i n t a t a distance r f rom theaxis of rotat ion (which passes th roug h th e boo-merang 's center of mass), this veloci ty is or.For each point on the arm one can reduce thevelocities u an d wr to one resu l tan t ve loc i ty .I t s component perpendicu la r to the a rm i s Vef f

Fig. 31 Fig. 32(Fig. 32). Of cou rse, th e va111eof V ef f or a partic-u la r po in t on th e a rm wi l l change cont inuous lyduring one period of revolution. One assumest h a t t h e c o n t r i b u t i o n s t o t h e l i f t a n d d r a g o feach part of a boomerang arm at each momentare again proport ional to (Vefr) ' .CalcnIat~ionswere made of th e fol lowing forcesand torq ues, averaged over one period of revolu-t ion: the average l i f t force L; the average to rqueT, with i t s components TI a round an ax is para l -1 1 1 1 t o a (which makes the boomerang tu rn t o the

Th e Aerodle ft ) and(which mt h e a v e raveloci tywhich s lI t tu rn s T , depen

Fig. 33arms th a tpropor t ionThe forcas a wholfor each by a rms wcompletely- -~ o wedoes a bothese aerodforce of g ra


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as Felix Heas T h e Aer1the average torque T causes the gyroscopic pre-cession of a boom erang. Le t us ta ke a closer looka t the gyroscope. If a gyroscope spins around it sax is wi th a ro ta t iona l ve loc i ty o and one exertsa to rque T on i t , ac t ing a round an ax is pe rpendic-ular to the spin axis , the gyroscope precessesa round an ax is pe rpendicu lar to bo th th e sp inax is and the to rque ax is (F ig . 34) . The angula rvelocity of the precession is called 51. A verysimple connection exis ts between 51, a , T andthe gyroscope 's moment of ine r t i a I , namely51 = T1I.o . W e have seen tha t fo r a boomerangT i s p ro p o r t io n al to o v , s o th a t th e v e lo ci ty o fprecession 51 mu st be prop ortio nal t o w vl lo ,or vlI. H enc e th e velo city of precession doesnot depend on a , the rota tio nal velocity of th eboomerang.An even more s tr iki ng conclusion can be drawn.Th e velo city of precession is prop ortio nal to vlI ,the factor of proportionality depending on thoexact sha pe of t he boomerang. Therefore one canw ri te 51 = cv, with c a c llaracteris t ic parameterfor a certa in boomerang. Now let the boomeranghave a ve loc i ty twice as fas t ; i t the n changes theorien tation of i ts plane twice as fas t . That im-plies,' ,however, th at th e boom erang flies throu ghthe same curve!Thus, rough ly speaking, the diameter of aboomerang 's orbit depends neither on the rota-tional velocity of the boomerang nor on i tsfo rward ve loc i ty . Th is means tha t a boomeranghas its pa th d iamete r more o r le s s bu i l t in .Th e dimensions of a boomerang 's f l ight pat hare proportional t o the moment of in ertia of th eboomerang, and they are smaller i f the profile of

the ar ma boomin a rmate r iais need(and ofNow equa tioThese ecomputtio n of How real booson it w01 a bodone bcontrolthrowinI have ments , of expecamerae lec t r ic1.5-volta ho l lo(see F imade ts trong Some oshown icomparBecathrowerboomergerated


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C. 1. Pokrnvsliy I A IIydrod

hy G. I .Fig. 35

F i l l ho ld ing

t ive wasca lcu la thimself betweenany ra tet ies of reasona

A Hydin a Fa

l e t i t dof th e t


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52 V. Mayeran e las t ic impact . Dur ing impact , the meniscusof th e water in the tes t tu be , which i s normal lyconcave because of capillary force, will rapidlylevel out , and a th in s t ream of water wi l l sud-denly burs t l ipward f rom the cent re . Figure 38shows the water surface before impact (brokenline) and a fter impa ct (solid l ine). The streamupward separa tes in to drops , and the uppermostdrop reaches a he ight subs tant iaI ly h igher thanth at f rom which th e tube i s dropped. This indi -ca tes tha t t he energy in the water i s redis t r ibuteddur ing impact so tha t a smal l f rac t ion of waterc lose to th e cent re of the m eniscus shoots out ofthe tube a t h igh ve loc i ty .A device tha t redis t r ibutes energy i s ca l leda mechanism. Usual ly , th is word i s appl ied tosolid parts ( levers, toothed wheels, s tc.) , al-though there are l iquid and even gaseous mech-an ism s . The w a te r i n t he tub e i s j u st one exam pleof such a mechanism.Hydrodynamic mechanisms are especia l ly im-por tant when very grea t forces tha t cannot bewi ths tood by convent ional sol id par ts are in-volved. The force of explosive material in a

car t r idge , for example , can be par t ia l ly concen-t r at ed by m ak ing a concave cav i ty i n t he ca r t-r idge , which i s l ined wi th a m eta l sheet. The forceof the explosion compresses the metal and pro-duces a thin metall ic jet whose velocity ( if theshape of th e l in ing i s correc t ) may reach th eescape velocity of a rocket.Thus , th is modest exper iment on a very s implephenomenon in a tes t tu be re la tes to one of th emost intere st ing problems of th e hydro dyna micsof ultra high speeds.

An In s t

An Inwith aby V .MI n Peleganttes t tudroppesurfacetube upcl ings sniscus the wawater tas if iout . Tha thin jthe tubYou str iki ngof a tediameteend of a t oy coverinth at endand ad jwater ithe p ipin the gNow s lover thimmedipiece o

An Instr


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54 V. MayerFigures 39 a n d 40 show drawings of photo-graphs taken a t d i f fe ren t moments dur ing th i sexperiment . They depict different s tages of theformat ion an d d i s in tegra t ion of var ious cumula-

t ive je t s . The top two p ic tures show the je tproper; the bot tom two depict the break-up oft h e je t i n t o i n d i v i d u a l d r o p s.Tr y to exp la in the resu l t s of th i s exper imentby compar ing i t wi t h the one descr ibed byPokrovsky . This se tup i s espec ia l ly in te res t ingbecause i t a l lows us to observe the ac tua l fo rma-t ion of a cum ula t ive je t, which is more diff icul tin the exper iment wi th a fa l l ing tes t tube be-cause the hu ma n eye i s no t fas t enough to reg is te rthe phenomena tha t t ake p lace dur ing impac t .Never the less , we advise you to re tu rn to theexper iment wi th the fa l l ing tuhe once moreto examine the detai ls of the formation of theje t. W i th th i s in mind wc sugges t you so lve thefol lowing problems.EXERCISESi . Determine whether the shape of the test tube bot-tom affects stream formation. Does the stream developbecause the bottom directs the shock wave in the water?To answer thi s question, solder a tin bottom of anyshape (plane or concave, for example) to a thin-wallcopper pipe. Use theee modified test tubes in the exper-iments described above to prove th at the shape of thebottom does no t influence the format ion of the stream.Thus, the results of this experiment cannot be explainedas the direction of the wave by the bot tom.2. Determine wliet,her i t is necessary for the liquidto wet the walls of the test tuhe. Place a sma ll piece ofparaffin inside a glass test tube, and melt the paraffinover the flame of a dry fuel. Rot ate the tuhe over theflame to coat the inside with a thin paraffin film. Now,

Fig. 39

Fig. 4

56 V. Mayer and E. Mamaeva Magic w


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repeat the Pokrovsky experiment with this coated testtube. The cumulative jet should not form, which meansthat the walls of the tube must be wet for the experimentto work properly.3. What other experiments can be set up to obtaina climulative jet in a tube that is stationa ry relativeto the observer?

Magic with Physicsb y V. Mayer and E. Mamaeva

Take a glass pipe, one end of which is taperedlike that of a pipe tte, and show the pipe to youraudience. Ho ld a glass of water (heated to 80 -90 "C) by its rim in your other hand, and showit to your audience, too. Now, lower the taperedend of the pipe into the glass, and let the pipefill wit h water. Close th e upper end of t he pipewith vour finger and remove the ~ i p erom the-glass " ( ~ i ~ .1).Your audience will be able to see air bubblesappear at the lower end of t he pipe. They grow,leave the walls of the pipe, and rise to th e topof t he pipe. Bu t th e water stays i n the pipe!Now, empty the pipe back into the glass byremoving your finger from th e upper en d, andwave the pipe in the air several times beforetaking some more water. Close the lipper endwith your finger again, and quickly p ull t he pipeout of th e glass and tu rn it upside down (Fig.4 2 ) .A strong stream of water over a metre hig h willburst o ut of th e pipe.Although th e secret of t his tri ck is very simpleindeed, your audience is tinlikely to guess i t,

Fig. 41

The glawhereas 20 "C). Ywater lea

M. Golubev and A. Kagalenko A Drop o


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58

The explanat ion for the powerfu l s t ream ofwater i s more compl ica ted . When hot wateren t e rs t he p ipe f rom the g l a s s , t he a i r i n t h e uppe rpar t of th e p ipe remains a t room tem pera tu rebecause the p ip e conducts hea t poor ly . Af ter youhave c lo sed the uppe r open ing and tu rned thep ipe ups ide dow n , t he ho t wa te r s t r eams down-wa rd , hea t ing th e a i r quickly . T he pressure r i ses ,and th e expand ing a i r shoo ts t he r ema in ing w a te rout through the tapered end of the p ipe .Use a g l a s s p ipe 8-12 mm in d i ame te r and30-40 cm long for th is exper iment . The smal leropening should be about 1 m m i n d i a m e t e r .Between t r icks the p ipe should be wel l cooled(you can even b low through i t ) because the he ightof t he foun ta in w i l l depend on the t empera tu redif ference be tween th e a i r and th e water i n thep ipe. T he op t ima l amoun t of wa te r i n t he p ipefluctuates from 114 to 113 of i t s volum e and caneasi ly be determined empirical ly.

A Drop on a Hot Surfaceby M . Golube v and A. Kagalenko

Turn ing an i ron ups ide down and l eve l l i ng i thor izonta l ly , le t a l i t t l e water dro p on i t s hotsur face . I f th e temp era ture of th e i ron i s s l ight lyover 1 00 OC, the d rop w ill d iffuse as expected andevapora t e w i th in n few seconds. I f , however ,the i ron i s much hot te r (300-350 "C ) , somethingunusual wi l l happen: the drop wi l l bounce be-tween 1 and 5 mill imetres of1 t l ie i ron (as a bal lbounces off t he floor) and w ill th en move over

the ho t sof such pera turecalm er t he d roply , increvaporaLarger dsmal ler changes.i n d i ameseconds)W ha t enon? Wsur face ,fract ionsto 10 0 Oa ra te thgrea ter recoi l s bouncesto boi l inthe d ropa d is tanthe v apodrop in i s r each"live" a When tha t of compresbe suppoforce that ion of tnot iceab

1. Vorobiev Sllrface T


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60

Oscilla tions , for example, compress ion, ten-s ion, or even more complex oscil la tion, maydevelop, especially in large drops (Figs. 43 an d

44). The photograph in Fig. 43 shows a darkspot in th e centre of t he drop. T his is vapourbubble . In large drops several sue11 bubbles ma y

appear. Oof a r ingmiddle . ceeds so Figure 44of oscillaKeep tducting e1. Thewithout sruns in toably red2 . Thehorizon tafor geod

3. SafeThe condinsu la tedboiling

Surfaceby I . Vor

Th e cocan be dt in g l iq uand a mdiamete rever . Foplaced wia glass ofly . T he win by th e

62 I . Vorobiev Surface T


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The coefficient of surface tension o can eas i lybe ca lcula ted f rom the r i se of th e water y a n dthe c learance be tween the p la tes d . The forceof the surface tension is F = 2 o L , where L i sth e length of th e plate (m ult ipl ied by 2 becauseth e water con tac ts both p la tes ). This force re-ta ins a layer of water th a t weighs rn = p L d y ,

Fig. 45 Fig. 46where p i s the water dens i ty . Consequenl ly ,2 o L = pLdygHence , the coefficient of the surface tension isa = 1/2pgdy ( 2 )

A more interest ing effect can be obtained bypressing the p la tes together on one s ide and leav-ing a smal l c learance on the o ther (Fig . 46).In t h i s ca se t h e wa te r w i l l r is e , and the su r facebetween th e p la tes wi l l be very regular andsmooth (provided th e glass i s c lean and dry) .I t i s easy to in fer th a t the ver t ica l c ross sec tionof the surface is a hyperbola. And we can proveth i s by replac ing d with a new expression for theZclearance i n fo rmula (1).T h e n d -- D follows

from the s im i lar i ty of th e t r iangles ( see Fig. 46) .

Here D iL i s th e f rom th ec learanceThus ,

Equa t ionbola.The p l10 c m b ys ide s l ioo

Fig. 47s t ick . Usein develoease in rgraph p apa graph drthe curveg u l a r ~unarea (Fig.

G4 V. Mayer Experimen


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If yo11 hav e a thern lon~ eter , ou can s tu dy th edepen dence of surface tensio n of w ate r tem pera -ture . You can a lso s tud y the influence of addi t iveson water tens ion.Th e force of surfac e tensio n F i s d i rec ted a tr ight angles to th e l ine of con tact between th ewater and the glass (Fig . 48). The ver t ica l com-ponent of t he force is balanced by t he weightof th e water column. T ry to ex pla in what ba-lances i t s hor izonta l component .

Experiments with a Spoonful of Brothb y V.Mayer

The next t ime you are served boui l lon for d in-ner, scoop up a big spoonfu l , don ' t swallow i t im-mediate ly . Looli carefully a t th e broth ins tead:

--c---- _- Fig. 49you wi l l see large drops of f a t in i t . Note t he s izeof thes e droplets . Now pour some of th e bro thback in to your soup bowl , and look again a t thebroth i n the spoon. The drop of fa t should havedif fused and got te n th inn er but b igger in d iame-ter . W ha t i s th e reason behind th is phenomenon?Firs t le t us see under what condi t ions a dropof f a t can l ie on th e surface of the bro th wi thoutdiffusing. Look a t Fig. 49. A dro p of li qu id 2

I( the fat) The dropvironmenvapours meet a t an increpoint 0surface teinter face gen t i a l t o1 Fi2 I =uwhere a,,media Iat th e in t1 Fi3 I = ~and 2 anI F ~ s=02Here a,,tensions.Obviouto ta l of aF12 + F , ,or the i r (after th evalues an0 1 3 = 0 1 20 1 , s in 8 ,H ere , 8 , agents to t

66 V. Mayer Experim


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of medium I. These angles are cal led angles ofcontact.I t fol lows f rom Eq. (1) th a t equi l ibr ium of t hedrop is possible if the surface tensions are relat-ed asa13< 01, -t- a23 .Since sur face phenomena in a l iquid are prac t i -cal ly indep end ent of the gaseous env ironm entover it , we as sume tha ta13= al a n d a,, = a,.W e c a ll al a n d a, surfa ce tensi ons of liqu ids Iand 2 , respec t ive ly . I n th is case these va lues re -fer to th e sur face tens ions of both t he bro th andt h e f a t .So, a d rop of fat w il l f loat o n th e surface ofth e broth wi tho ut diffusing if th e surface ten-s ion of th e bro th i s less tha n th e to ta l of t he sur -face tensions of th e fa t and t he interface betweenthe b ro th and the f a t :01< 0, -t- 01, (2)If the drop is very th in (a lmost f la t ) , and 8,wi l l be sma l l (e l = O2 = 0 ) a n d t h e e q u i l i b r i u mcond i t i on fo r t he d rop wi l l be

=a,-t-(J1,W h e n al> a, 4-al, there a re no anglesand la for which Eq. (1) would hold true. There-fore , l iquid 2 does not make a drop on the sur -face of l iq uid I in th is case, but d i ffuses on i t s sur -f ace i n a t h in l aye r .

Now le t us t r y to expla in the res ul t s of our ex-perim ent with a spoonful of b roth. Drops of f at

f loat oholds. bro th iha s chf a t a,unchanout somsion of f irs t apwater amountpla in wlayer owe redubroth sreducesresul t , To tepe r imencer whisunflowrefi l l f rpurposeover thefuse buthe watNow ful of bfa t c losand reuone sucw i t h t aconta indrops ofg lass tu5'

E~p~ r i


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Fig.

amount of water out with a rubber bulb, thedrops enlarged. When more water was removed,the drops became even larger and changed shapt3(because of the water currents). The beginnihgs

of futurFurther The ruprupture.

M. Kliya How o G


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Fig. 54

Fig. 55broke and rear ranged in to a new drop (Figs .52-55).Perhaps now you wi l l agree tha t i t i s wor th-whi le watching your soup before put t ing i t in toyour mouth!

How tby M.

Moderie ty oftransistomany olabora toof dev ehave aps ize f roweighinThe mof ten repressurediamondin yourto g rowKAl(SOsubs tans ional lyour ownlook at W hencons tance r t a in s a tu ra t et i ty of t emperasolubi l ilu t ion t

comes u

72 M. Kliya How to G


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If a satu rat ed solut ion is cooled, the excess ofthe subs tan ce wi l l p rec ip i ta te . F igure 56 showsth e dependence of p o tash a lum so lub i l i ty on tem-pera ture . Accord ing to t he g raph , if 100 grams ofa solu t ion satura ted a t 30 "C a re cooled to 10 OC,over 10 gram s of th e substanc e should precipi-

S o l u t i o n t e m p e r a t ur e Fig. 56t a te . Consequent ly , c rys ta l s can be grown bycool ing a sa tura ted so lu t ion .Crys ta l s can a l so be grown by evapora t ion .W hen a sa tu ra ted so lu t ion evapora tes , i t s vo l-ume decreases , while the amou nt of dissolved sub-stance remains unchanged. The excess of sub-s tance thu s produced fa l l s as a p rec ip i ta te . T o seeh o w t h i s o c c u r s , h e a t a s a t u r a t e d s o l u t i o n , a n dthen cover the ja r con ta in ing the unsa tura ted so-lu t ion wi th a g lass p la te and a l low i t to coo l toa temp era ture be low the sa tu ra t ion tempera ture .T h e s u bs t a nc e m a y n o t p r e c i p i ta t e w i t h t h i s m e t h -od , i n which case we wi l l be le f t wi th a supersa t -urated solut ion. Th is is because we need a se ed,a t iny c rys ta l o r even a speck of the same ub-

s tance , c rys ta l sjar or reth e numAs an a rbest of Th e sTake twone of tt u r e , anWhen dlu t ion iany of tjar . Covsolut ionte r a shi n t h e jafore selNow Firs t of undesirasels , s teI t eapot . l u t i o n i nth i s wamore . Tit aside t i o n a plower thjar . S inseed crya s t h e t ethe seeding. (Ifl y , i n tr o

74 G . Kosourov Crystals


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continue t o grow once the solu tion h as cooled, ifyou l i f t the cover and le t the wa te r evapora te .Do not le t dust enter the jar. Grow th w ill con-t in u e f o r two o r th re e d a y s .W hen growing c rysta ls t r y no t to move ortouch the jar. After a crysta l has developed, re-

move i t f rom th e so lu t ion , and dry i t care fu l lywith a paper napkin so it wil l re ta in i t s sh ine .The crysta ls develop differently , depending onwhether t he seed c rys ta l i s p laced on t he b o t tomof the jar or suspended from a thread (Fig. 57).You can even grow 'a necklace ' by ru nnin g athrea d severa l t im es over th e seed crys ta l befores u s p e n d in g i t i n th e s o lu t io n .Growing c rys ta ls i s an a r t , and you may no tbe completely successful r ight away. Do not getd isappoin ted . With a l i t t le pe rs is tence and ca re ,you can produce beautiful crysta ls .Crystals Made of Spheresby G . Kosourov

Before we can predict , explain, or unders tandth e properties of a crysta l , we mu st deter min e i t s

s t ruc tursymm etcan te ll whe theredges uncrystal idevelopsterized bfie ld , wof doubbeam, ar ies abuself.Differbe s tudiemodels oin natu rta llograpels, whica rrangeming our oretical Crystaatomic frepuls ivea t tem ptsforces pra ther gdrawn toenergy opo ten tiapo ten t iaattractivit increaThe dep

70 G. Kosourov Crystals M


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i s show n i n F ig , 58. In equ i l i b r ium , a tom s a s-sum e places of min ima l po tent ial energy. If the rea re m any a tom s , t h i s t endency l eads to t he r epea t-ed form atio n of th e most energy-efficient con-figuratio n of a sm al l group of ato ms . Th is con-figuration is cal led the unit cel l .Some substances have very complex s t ructures .Th e un it c el l of some si l icates, for exam ple,

Fig. 58 Fig. 59conta ins over 200 a tom s . O the r subs tances , m an ym eta l s , fo r exam ple , fo rm the i r c rys t a l l a t t i c eby a very s imple a lgor i thm. Natura l ly , wesha l l s t a r t f rom the s im ples t coord ina t ions . Inour exper iments a toms are represented by meta lspheres. E las t ic forces, developin g as a result ofthe con jugat ion of the spheres , serve as the repul -s ive force , and the a t t ra c t iv e force is .provided b ygrav i ty .St re tch a th in p iece of rubb er ( t h is ma y be apiece of a surgical glove) over the opening of ajar , box, or sec t ion of p ipe , a nd fas ten i t w i th arubber band. Place two spheres on the piece of

rubbe r . Tspheres wte nt ia l ena p p r o x imthe depespheres os l igh t ly , rows (Fig

Fig. 60

a t the apof which spheres wt ice ca l lea cornerby s ix sphform reguIf you taround a nthe spherebut th e space wi l l

78 G.Kosourov Crystals


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will translate itself into the original position.After six such rotations each sphere resumes itsoriginal position. In such cases crystallographerssay tha t the axis of symmetry of the sixth order,which is oriented perpendicular t o the plane of th elatti ce, passes through the centre of each sphere.This axis makes the lattice "hexagonal". In addi-tion to the symmetry axes of the sixth order,third-order symmetry axes pass through the cent-res of th e holes formed by neighbouring spheres.(The third-order symmetry axis is a straightline; each time the latt ice is rotated 120 degreesaround this axis, the original configuration re-peats. Irr egu larly shaped bodies have first-ordersymmetry axes since they return to the originalposition after a single complete revolution. Con-versely, the symmetry axis of infinite order passesthrough the plane of a circle at right anglessince the circle translates itself into the originalposition a t an infinitely small angle of ro-tation.)The following discussions will be clear onlyif you have spheres for build ing models of differ-ent crystals. Consider the holes on either sideof a row of spheres (Fig. 61). Since the number ofholes in either row equals the number of spheresin a row, an infinite lattice of spheres has twiceas many holes as spheres. The holes form twohexagonal lat tices, simi lar to those formed by thecentres of th e spheres. These three la ttices areshifted relative t o one another in such a way thatthe sixth-order symmetry axes of each la tticecoincide with the third-order symmetry axes ofthe other two lattices.The second layer of spheres fills one of the lat -

tices oftiguousThe elmay nothe sphfore, sitom layfrom plan integand fil

Fig. 61

The sof holesfind th e of the laspheres lattice llayer. Wthe holeIn this replica the fourso on. E

Crystals


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l ayer , Our pyramid wi l l be ra ther f rag i le (F ig .63) because the "at tract ive" force in o ur modelac ts downwards only , and the spheres in theholes alon g the edges are easi ly pressed ou t bythe spheres of the layers above.This kind of packing, cal led hexagonal closestpacking, is typ ica l of beryl l ium , magne sium, cad-mium , and he l ium crys ta l s a t low tempera turesand pressures over twenty f ive atmospheres. Thispacking ha s only on e system of closely packed,para l le l l ayers . The th i rd-order symmetry ax is ,which passes through the centre of each sphere, isperpendicular t o th is system of layers . T he sym-me t ry o rde i J;S thu s reduced: when th e ax is passesthrough centkes of the even-layered spheres,the la t t i ce I ias ' l the s ix th-order symmetry ; thesame axi s passes throug h th e centres of t he holesin the odd layers , and the symmetry of the oddlayers , re la t ive to th i$ ax is , i s , therefore , on lyof the th ird order . Nevertheless , thi s packing iscal led hexag onal , because i t can be viewed as twohexagonal lat t ice s of even and odd layers . Notea lso tha t th e e mp ty holes in a l l l ayers a re a r rangedone over the o ther and channels pass through the

whole hexagonal s t ruc ture , in t o which rods ,whose d iameter is 0.155 that of the diameter ofthe sphere can be inser ted . Tbe cent re l ines ofthese channels a re the th i rd-order symmetryaxes. Figure 6 3 shows the model of a hexagonals t ruc ture wi th rods placed in t he channels .Now le t us pu t the spheres from the th i rd lay erin to th e holes above th e vaca nt holes of the bot-tom layer . W e can cons t ruc t two d i f fe rent pyra-mid s (Figs. 64 and 65), depen ding on the sys tem

G. Kosourov Crystal


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of holes we select for the second layer of spheres.The faces of the f i rs t pyram id are equi la tera lt r iangles wi th hexagonal packing, which do notdi f fer f rom th e packing of the bot tom layer ofthe py ram id . I n o the r w ords , ou r py ram id i s atetrahedron, one of the f ive possible regular poly-hedrons. This packing has four families of close-ly packed layers , whose normals coincide wi ththe th i rd-order sy mm etry axes of the te t rahe dronwhich pass through i t s apexes . In such a packingthe l aye r s r epea t eve ry th i rd l aye r . The l a t e r a lfaces of th e second pyra mid are isosceles tr iangle s,and th e py ramid i t se l f i s pa r t of a cube, in tercept -ed by the p lan e formed by the d iagonals of thefaces wi th a common apex (Fig . 66) . The packingin th e la tera l faces of such a pyram id forms asqua re l a t t i ce w i th i t s row s pa ra l l e l t o t he d iag -ona ls of t he c ube face.Obviously , we have only one type of packingwith two di f ferent or ienta t ion s . I f we remove thespheres f rom th e edges of t he te t rahedro n, thefaces of th e cube wil l emerge. Co nversely, by re-m oving the sphe res t ha t m ake u p th e edges ofthe cube , w e tu rn the cube in to a t e t r ahedron .This packing, ca l led cubic c loses t packing, i scharacterist ic of neon, argon, copper, gold, pla-t inu m and lead crys ta ls . Cubic c loses t packingpossesses al l th e elements of c ubic symm etry .In pa r t i cu l a r , t he t h i rd -o rde r sym m et ry axes o fthe t e t r ahedron co inc ide w i th th e space cube d i -agonals , which are a lso th i rd-order sym metryaxes for the c.ube.This packin g is based on a c ubeof fourt een spheres. Eig ht of th e spheres formth e cube, an d s ix form the cent res of i t s faces.I f you look c losely a t the second pyram id

(Fig . 6the apea comThe eqespeciahexagoduced one anodensi tyi n s y m

Fig. 66If wea squar

ing. Anot pacbetweeners l ie we comhedral pla ter a l hexagonw i th i t slyhedroan octa

84 Ya. Geguzin A Bubble


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i s a lso the cubic c loses t packing with the facesof the cube paralle l to the plane of the base. Re-move the spheres forming the edges, and you wi l lf ind f ive spheres in the upper in te rcep t p lanewhich for m the face of an e lementary cube.These models can be used in a num ber of physi-cal exper iments . By shaking the piece of rubber ,

Fig. 68 Fig. 69for example, you ,can s imulate the heat- inducedmotion of a tom s. (You will see how 'a r ise intemper ature ' destroys the pac king of t he spheres .)Since eacl hexagon al layer occupies relat ivelyshallow holes of the next laye r , the layers areloosely bound, and s lippage develops easily. I fyou s lide one hexagonal layer against another ,you w il l see th a t easy s l ippage , in w hich th e lay -ers move as a whole, occurs in three directions.A s imila r s i tua t ion can be observed in rea l c rys-tals , which exp lains the specif ics of plastic de-formation in crystals .Models can be built f rom any kind of sphere.If you do not have ba l l bear ings , use la rgenecklace beads or even sma ll apples . Figures 68and 69 show unit cells of cu bic and hexagonal

packingPing-pomake wespecialphysics Variota n t i nthem agmodels1A Bubby Ya. GOn Simu

I n th ementa l everyona s implof the mneeds isgies, anphysicalunders tamode l bcan onei t s consbe a l e abut must r u th peof coursis a te rmodels. eas i ly c

86 Ya . Geguzin I A Bubb


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The very best model needs no explanation atal l since its clari ty gives i t the force of a proof.There are many convincing and elegant modelsin physics, particularly in solid state physics.In this article we shall discuss a 'live' model,which i ll~ls tra tes nd reflects the structu re flaws,and complicated interactions of real crystal verywell. This is not a new model. I t was conceivedby the outstanding British physicist L. Braggin the early 1940s and realized by Bragg and hiscolleagues V. Lomer and D. Naem. Therefore,we shall call it the BLN model after Bragg, Lo-mer, and Naem.What Do We Wa nt to Simulate?

The answer is clear: real crystal. Real crystalis a vast set of identical atoms or molecules ar-ranged in strict order to form a crystal lattice.Occasionally, this order is disturbed, signifyingthe presence of defects in the crystal. Anothervery important characteristic of crystals is th einteraction of the atoms forming the crysta l.We will discuss this interaction a bit later. Nowwe will simply state that they do interact!Without interaction, the atoms would form aheap of disorderly arranged atoms rather than acrystal. The maintenance of order in crystalsis a direct consequence of this interaction.Another widely used model is the so-called deadmodel of c rys tal, in which wooden or clay ballsare bound by straight wires. The balls representatoms, and the wires are the symbols of theirbonds in 'frozen' state. The model is 'dead'since it 'freezes' th e interaction sf the atoms. In

this reakinds oen t colorepresethoughabout tout falsnot depthe ordmodel space amost licurrentin repatoms on the generaling withimportath e idemolecuusefulneOur orather twe needcrystal,zen in The Int

Perhasuch infact thaatoms iconstan

Ya. Geguzln I A Bubbleabo ut th e distance between the posit ions around


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which atoms fluctuate in heat induced mo-tion. T he a mp litude of these fluctuations is con-siderably sm aller th an the distance betweenatoms.) Th e distance has a definite valu e since i fwe try to stretch i t , the atoms resist the effort andattract one another, and if we try to reduce thedistance, the atoms repel each other. The facttha t th is d is tance i s de te rmina te a l lows us toconclude that a tomic interaction is character-ized by attrac tion and repulsion simultaneously.At a certain distance between atoms (we call itdeterm inate), th e forces of attr act ion and repul-sion become equivalent in absolute values. Thea toms in the la t t ice a re loca ted a t exac t ly th isdistance.I t would be usefu l to be able to s imula te thecompetition of attractive and repulsive forces,Such a technique would revital ize atomic inter-action in crystal. The authors of the BL N modelcreated just su ch a metho d. Inst ead of wooden andclay balls, th ey used t in y soap bubbles.Th e Interaction of Soap Bubbleson Water SurfaceTwo soap bubbles on the surface of a bod yof w ater ar e not indifferent to each other: theyare first a t trac ted to one another but, after touch-ing, are repelled. Th is phenomenon can be ob-served i n a very sim ple experiment, for which wewi ll need a shal low bowl, a need le from a syr-inge, th e inner balloon from a volley ball , and anadjustable clamp to control the compression ofthe nozzle of t h e balloon. Fill the bowl wi th

soapy waof glyceblow onballoon. needle inmerse th(not deeslightlydle at re

soap bubin futu rement, trytance froimmediatthe fourthin diameOnce ttheir movour interffirst and they do non the sutic al bubb' bles of d

Now letdrives the

90 Ya. Geguzin 1 A Bubble


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of two m atch stick s lying on the surface of t he wa-ter . Since the bubbles and the matchs t icks areboth soaked by th e water , th e natur e of t he i r in-terac t ion i s genera l ly th e same. Two bubbles f loa t-ing close together form a very complex surfacew i th the w a te r , how ever , w hereas t ha t o f t hetw o m a tchs t i cks i s m uch sim ple r and , t hus , eas -ier to s tudy (Fig . 71) . The force tha t br ings the

-- 2 r ---_ _ - _ _ _ Fig. 71two f loa t ing m atchs t icks together develops as fo l -low s . W a te r soaks th e m a tchs t i cks , and th e su r-face of t h e water near th e s t icks i s , therefore ,curved. This curvature genera tes force tha t ac tsupon th e l i qu id . The fo rce i s de te rm ined by su r -face tens ion and di rec ted, in th is case , upward(we shal l assmxe the matchs t icks are complete lysoaked). Under th e ef fec t of th i s force , the l iqu idr ises a long th e s ides of th e matchs t icks , the r i sebeing more pronounced in the region between th es t icks (F ig . 71). T he l iquid appears to s t re t,ch ,and th e p res su re in t he l i qu id d rops r e l a t ive t oth e a tmospher ic pressure , by an addi t ion al pres-

20 asu re Ap =- =- where o is the coefficient ofd r 'su r face t ens ion , d i s t he d i s t ance be tw een them atchs t i cks , and r= d l2 i s t h e cu rva tu re r ad ius ofth e surface of th e l iquid . Consequent ly , the ab-solute valu e of t he pressure of t he l iqu id on th ematchs t icks in the area between them is le ss t h a s

the a tm1 f rom theforce thFrom I enon. Si

I the matc

loci ty thdecreasesdraw clothe bubbcamera ot ion. As swi tchedto w a tchto the i r puls ive fby a n incompressbubb les Soap bmodels, bles on t h

ya. Geguzin I A Bubb


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Fig. 73

simply R = 5X lo4blution ois R p %tained bdimensithis verple, tha

interactcrystalsThe Mo

The festing swith mother siin a realy posstrate th

Ya.GeguzinThe BLN model can be used to verify certain

A Bub


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corollaries to t he theory of crystal t hat is abso-lutely free of defects, i.e. the so-called ideal crys-tal. I t is almost impossible to obtain such a crys-tal in nature, but it proved rather simple andeasy to construct one made of bubbles (Fig. 75).One of the most common defects in crysta l isa vacant position at a point in the lattice, whichis not filled by an atom. Physici sts call this phe-nomenon a vacancy. In the BLN model a vacancyis represented by an exploded bubble (Fig. 76).As both common sense and experiments with realcrystals lead us to expect, the B LN model showstha t the volume of a vacancy is a littl e less thanthat of an occupied position. When a bubble ex-plodes, neighbouring bubbles move slightly in-to the hole left by the explosion and reduce itssize. This is almost impossible to detect with thenaked eye, but if we project a photograph of th ebubbles onto a screen and carefully measure thedistances between bobbles, we can see that thevacancy is somewhat compressed in comparisonwith an occupied position. For physicists this isevidence of both a quali tati ve and a quantit a-tive change.Very often, crystal contains an impurity, in-troduced in the early stages of its history, thatdeforms i ts st ructure. To solve many problems ofcrystal physics, i t is very importan t t o know howthe atoms surrounding the imp urit y have changedposition. Incidentally, the presence of an impu-rit y is left not only by the nearest neighbours butalso by the atoms a considerable distance from it.The BLN model reflects this clearly (Fig. 76).

Most crystalline bodies are represented by po-

lycrystals made of many small, randomly oriented


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crystals separated by boundaries. We expect manyproperties of the polycrystaIs (such as mechan-ical strength or electrical resistance) to dependon the structure of the boundaries, and, in fact,the BLN model bears this out. I t showed crystalphysicists that the structure of such boundariesvaries according to the mutual orientation ofboundary crystals, the presence of impurit ies a tthe boundary, and many other factors. Some partsof the polycrystals (grains), for example, mayenlarge at the expense of others. As a result, aver-age grain size increases. This process, called re-crystallization, develops for a very explicit rea-son: the greater the size of t he grain, th e less itstotal boundary surface area, which means thati t has lower excess energy linked to the bounda-ries. The energy of a polycrystal i s reduced in re-crystallization, and, therefore, the process mayoccur spontaneously (since it moves the systemto a more stable equilibrium, at which energystorage is minimal). The series of photographs i nFig. 77 illus trates a large grain 'devouring' asmaller grain inside it in successive stages.

The moving boundary between the grains ap-pears to 'swallow' the vacancies it comes across(this was predicted by theorists and carefullystudied by experimenters in real crystals). Theboundary does not change its structure in thisprocess, as the BLN model clearly illustrates(Fig. 78).Restrictions of the BLN Model

Without depreciating th e usefulness of t heBLN model, we should point out that it does

Pig. 777 -01544

B. Aleinikov Determinhave dra


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Fig. 78

one strucclosely a varietymodel hatoms cent waybe simury modifdimensiodimensiooperatiorejected.three-dimto be imnesses, hsable aiDetermby B. Al

At fithe pole1 be sure not simpdifferenmagnetithan it marked to differpole. Fohorseshoin fact, not) and

100 N. Minzt he expe r im en t i n t he day t im e , w hen a t e s t

A Peculw hich tof forc


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pa t t e rn fo r t un ing the t e lev i s ion i s b roadcast .Turn the t e l ev is ion set on , and pu t your m ag-net agains t the screen, as shown in Fig . 79. T h eimage wi l l imme dia te ly become dis tor ted . Th esmal l c i rc le in the center of th e tes t pa t tern wi l l

Fig. 79

shi f t not iceably upward or downward, dependingon th e posit ion of th e poles.An image on the te levis ion screen i s producedby a n e lec t ron beam di rected f rom ins ide th e pic-ture tube towards t he viewer . Our magnet devi -a t e s t he e l ec trons em i t t ed , and the im age i sd i s to r t ed . The d i r ect ion in w hich the m agne t i cf ie ld devia tes the m oving charge i s de termined b ythe lef t -hand rnle . I f the palm is pos i t ioned soth at t he l ines of force enter i t , the f ingers when ex-tended indica te the d i rec t ion of th e current . I nth i s pos it i on , t he thum b w hen he ld a t a r i gh tangle to the f ingers , wi l l show the di rec t ion in

pole of cordingnical" ds i t ive lycathodea re d i r eof positfore , thbe di reclear. Bw e cansouthernth e screI t i s marked th i s expneed a bcore , a rte ry in t o r , r a t eHold tht i fy i t s the corkth e curba t t e ry .

A Pecby N. M

The f athe planthe pend

lo ? N. Minzs t r a t ion of the Ea r th ' s r o ta t ion , i .e . the Fouc a u l tpendulum. The pendulum suspended on a long

A ,PecuTackth e jam


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wire osc i l la tes . A c i rc le under i t i s marked as aclock face. Since the plane of oscilla tions rela-t ive to the motionless s ta r s does no t sh if t , w hi lethe Ea r th ro ta tes on i ts ax is , th e pendulum passesthrough markings on the clock in succession.At e i ther of th e Ear th ' s po le , the c i rc le under the

Fig. 80pendulum makes one com ple te ro ta t ion in twen-ty- four hours . T his exper iment was car r ied ou tby the French physicis t L. Fouc a u l t in 1851,when a pendulum 67 metres long was suspendedf rom the cupola of t he Pan theon i n Par is .Do all pendu lums keep th e same plane of oscil-la tion? The suspension, af ter a ll , a llows oscilla-t ions in any ver t ica l p lane . To m ake the pendu-lum shown in F ig . 80a , fo ld a s t r in g in ha lf , anda t t a c h a no the r s t r ing in the midd le . T ie th e looseend of the second s tr ing t o a spoon, a pair ofscissors , or any other object, and your pendulumis ready. (The ver t ical suspension should be long-er or a t least equ al in length t o th at of the firsts t r ing . )

back (um) , aan e l l iing to th is wA sian undth at rpendulplane. s idewaNowand deous ly ,force que nt lmakesplane .penduloscillaOurcase, tminedand byfore, tbeginnThe pe r pen* Of cois stricpendulhoweveed awa

tion of th is component forces the pendulum out ofthe plane. Since the tension force varies, its per-

A PeculiaThe ravarying


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pendicular component also varies. As it swingsto the opposite side, t he pendulum pulls the otherhalf of t he horizontal string ta ut . This develops aforce that acts in the opposite direction. At the

Fig. 81 Fig. 82same time, as the experiment shows, the pendu-lum oscillates in two perpendicular planes.The curves described by our pendulum arecalled Lissajous figures, after the French physicistwho was the first to describe them in 1863. ALissajous figure results from the combination oftwo perpendicular oscillations. The figure maybe rather complicated, especially if the frequen-cies of longitudinal and la ti tudinal oscillationsare close. If th e frequencies are the same, the re-sul tan t trajectory wil l be a n ellipse. Figure81 shows the figure drawn by a pendulum whosemotion can be described as x = sin 3 t , y =sin 51. Figure 82 shows the oscillations des-cribed as x = sin 3 t , y = sin 4 t ,

and horificult to cillationscan be dLissajous

Fig. 83a perforabucket wboard undraw a cPhotogalso be mbal l whistring. Ppaper sholight andera abovhorizontapictures and 84sh

106 N. MinzChanges in th e direction of the oscil lat ionsare obvious . The change i s especia l ly sudden

t ions a weig


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i n F ig . 83 . Th e exposures of the two photographswere di f ferent , which i s obvious f rom the di ffer-ent lengths of th e t ra jec tories . T he curves seemto be inscr ibed wi thin a para l le logram, a l thoughin rea l i ty , they should be inscr ibed wi thin a rec-tangle . W e did not ge t a rec tangle s imply be-cause the lane of our camera was not s t r ic t lyhorizontal .-A reasonably correc t t ra jec tory can be obta inedin expe r im en t s w i th a hendu lum if dam pingis insignifican t . Th e oscil lat ions of a pen dulu mw i th low m ass and l a rge vo lum e w i l l dam p qu ick-ly . Such a pendu lum w i l l sw ing seve ra l t im esw i th qu ick ly d im in i sh ing am pl i tude . Nat l i ra l lychanges in th e osc i l la t ions of a pen dulum w i thsuch s t rong a t t enua t ion can ha rd ly be pho to -graphed.Lissajous f igures are common w i th perpendicu-lar osc i l la t ions . They are unavoidable , for in-s tance , in tuning osc i l lographs .

Lissajous Figure sby N . MinzTh e s imples t osc i l la t ions of a body a re those inwhich the devia t ion of the body f rom i t s equi -l ibr ium posi t ion x i s descr ibed as

x = a s in ( o t + cp)where a i s t he am pl i tude , o i s t he f r equency , a ndcp i s the i n i t i a l phase o f s sc i l l a t i on , Such s sc i l l a -

c i rcui t- -I n ths imul tal a t ions

su l ta nt be a sumx = A1

I t i s em en t f roordinatessecond mi l lus t ra te

108 N. Minz

added (solid s inusoids) . The broken line repre-sents th e resu l t ing osc i l la tion , which is no I Lissajou


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longer harmonic.More complicated trajector ies appear if twomu tua l l y perpendicula r osc i l la t ions a re added .The body in F ig . 86 moves a long such a t r a jec to-ry . I t s form depends on the ratios of f requencies ,am plitu des, and phases of th e two mu tua lly per-pendicu lar oscilla tions. As we know, such trajec-tor ies are called Lissajous f igures . The setu p usedby Lissa jous in h is exper iments is shown in F ig .87. The tun ing f o rk T' osc i l la tes in a ho r izonta lplane, whereas T is ver tical. A I ight beam pas-s ing through a lens is ref lected by a mirror a t-tached to T' towards a second mi rror f ixed on T.The ref lection of the second mirror is seen on ascreen. I f only one tuning fork oscilla tes , thel igh t spot on th e screen wil l move a long a s t ra ightl ine . If bo th tun ing forks osc i l late , the spot w i l ldraw in tr ica te t r a jec tor ies .Th e trajectory of a . b o d y w i t h t w o s i m u l t a -neous , m ut ~i a l ly e rpendicula r osc i l lat ions is de-scr ibed by a system of equationsx =A, s in (o , t f cp,),y=A, s in (02 t4- cp,),where x and y are the projection s of th e body dis-p lacement on X and Y axes.For s impl ic i ty , assume cpl = cp, = 0 and ol=- 0, = O. The nx = A i s i n o t ,g = A 8 s in w t . Fig. 87

110 N. MinzA2T h u s , y =- . Consequently, Eq. (2) de-A , Lissajouw hose elongat


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scr ibes a s t ra ight l ine segment . Slope a w i t hrespect to X ax i s i s

3n;Now let cpl = rpi + T . T h e nx =A, cos (o, t + ;) ,y = A 2 s i n (0 2 t+ v2).

Consider first th e simplest case, where Al == A ,, cp; = v 2 = 0 and ol = o2= o , th a t i sx = A c o s o t ,y =A s i n o t .

A po in t w i th x and y coordinates de termined bythe abo ve equat io ns makes a c i rc le of A radius .A nd , i n f ac t , x2+ y2= A 2 c os 2 o t +A 2 s i n 2o t ==A 2, w hich m eans tha t t he t r a j ec to ry of m o-t ion i s a c i rc le .Now let A, # A 2. Le t us p lo t a t r a j ec to ry fo rA, = 1 a n d A , = 2. At th e momen t of ma ximald i sp l acem ent , x = A, = 1 , t h a t i s , cos o t == 1 , o t = 0. Consequent ly , y = 2 s in o t = 0 .S im i l a r ly , w hen x = 0 , y equa l s tw o , and w hen

-r/z -x = -, y equa l s 1 /2 , and so on .2The g raph p lo t t ed w i th these coord ina t e s w i l lbe an e l l ipse whose major s e m i a x is i s A , a n d

I t i s = 1, w88b) . Cwe canNow

((0

= 0. Tcomex = A1y = A ,Trans fofol lowinx - , (* The fax = A , cy = A , sdescribes

i.e., a po

112 N. MinzThis curve i s par t of a parabola wi th i t s axis a longX a n d t h e a p e x a t x = A1 (Fig . 89) . Thus , we

I Lissa jousonto X,t i on on t


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have an open curve .

Fig. 89 Fig. 90Now let us check t he effects of th e frequency onthe shap e of t he t ra jec tory . W e wi l l ass ign equalam pl i tudes to t h e l a t e ra l and long i tud ina l osc il -la t ions descr ibed by sys tem (3).Let us p lot curves , for example , descr ibed bythe fol lowing equat ions :

x = A c os o t , y = A s i n 2 o t ,x = A cos a t , y = A s i n 4 o t .The eas i es t w ay t o do th i s i s t o d raw a c i rc l e ofA radius (Fig . 90) and ma rk th e points corre-n n 3nsponding to angles wt , which equ al 0 , s, T, 'n 5n 3n 7n- - - -' , , , n , ...,2n . To de t e rm ine the po in t s- - - -with coordinates x = A cos wt and y = A s in 2wt ,remember th a t for the c i rc le whose radius i se q u al t o u n i t y ( r = 1) t h e c os o t i s n u m e r ic a ll yequal to a projec t ion of th e vector radius r (o t )

r ad ius , the c i rc

Fig. 91of th e p1 te rmined

I can con

I n both cnumber 91a, b).

The fig1 - 0 1 6 4 4

th e following system of equations5 = COS 2ot,

LiasajousWhen thv , = O


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y = sin 3ot.When do open figures occur? Are there any com-mon regularities in their origin? Consider thefollowing equationsz = A1 cos po t ,y = A , sin qot.

First, note that at t he point where the curve re-verses along the same trajectory, the velocitiesof th e body along the X and Y axes become equalto zero simultaneously. The body moving alongthe curve stops at exactly this moment and thenstart s moving back. If x = A , cos pot , then

When t, sz tl = t (the difference between t ,and tl is small),pota -pot1 ru PW I - P ~ ~ Isin - 2

As a result,u, = - A,po sin pot .Similarly, for a,V , = A 2 q o cos got.

From thLissajous

The curvdition.Lissajoof an osone harmlation aptal may of t he alloscope Anyoning and a simpleof the ruother habench vithen relfigures inThe msum of parts. Thto t he bebend t o tis perpesection. Sthe oscil


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118 k Kosourovwork best. Figure 94 shows photograph obtainedexactly in this way.Now try similar experiments for yourself.

Waves iWhatThe calvelop c


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EXERCISES1. Prove that all curves described by the following systemof equations are open:2. Derive an equation for a curve with the followingparametersz= A , cos a t, y = A , cos 2 0 tWaves in a Flat Plate(Interference)by A . Kosourov

Wave propagation is perhaps the most univer-sal phenomenon in nature. Water, waves, sound,liglit and radio, even deformation transfer fromone part of a solid to another are examples ofthis phenomenon. According to quan tum mechan-ics, the motion of microscopic particles is alsocontrolled by th e laws of wave propagation. Thephysical nat ure , velocity of propagation, fre-quency and wavelength of all these waves aredifferent, hut despite these differences, the mo-tion of all waves is similar in many respects.The laws of one kind of w al e motion can be ap-plied almost without modification to waves ofanother nature. The most convenient way to stu-dy these laws is t o st udy waves on t he surface ofa bodv of water.

surface begin tolibriumsurfacethe centake tothrownnate speby th e dor phasAll wtheir afrom thexamplsure. Rof tenspropertsuch thspecificpoint tdependmediumThe have abreaks mediumwater, thi s casradiallywave vof i ts point s

120 A. KosourovOne of the main principles of elementary wavetheory is the principle of wave independence, al-so called the principle of superposition. The prin-

Waves inwithoutcap is a the bell


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ciple states that a disturbance caused by a waveat a point of observation is not influenced byother waves passing through the same point. Theprinciple of superposi tion is , in fact , a simplerule for determining the summary effect of wavesfrom different sources. A summary oscillation issimply a sum of the oscil lations caused by eachsource independently.Interference is a characteristic feature of waveprocesses. Interference is the combinat ion ofphenomena that develop in a medium in whichwaves propagating from two or more sources os-cill ate synchronously. The oscillations of somepoints of t he medium may be stronger or weakerunder the action of th e two simultaneous sourcesth an t hey would be under t he effect of eithersource in isola tion. Synchronized waves may evensuppress each other completely.Let us try to produce interference that we cansee with our own eyes. An experienced observercan easily see the interference caused by thewaves from two stones thrown in to a pond. Thismethod is unsui table for s tudy of interference,however. We need, instead, a stahle pictllre ofinterfering waves in the laboratory.The first thing we will need for this experi-ment is a vessel for water. The vessel shouldhave gently sloping walls to avoid masking wavesfrom the source with ref

Physics in Your Kitchen Lab -Kikoin Mir

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