Problems in Physics - Kapitza [pdf]

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32. ProblenI1nsP hysics
The problempsu blisheidn thisc ollectiowner e compilebdy me fors tudentosf theM oscwo Physi­
cal-Tcehnical Institutew,he re I taughta coursei n generaplh ysicsin 1947-194T9he. collection
alsoi ncludse problmes givena t examinatiofnosr postgraduastteu diesa t the Institutoef Physi­
calP roblemast the USSR Academyo f Sciences.T he problemwse res electeadn dp reparedf or
printinbgy I.Sh. Slobodetisik andL .C. Aslamazovw,h o recentlgyr aduatefdr om the Physical­
TechnicalI nstitute.
In compilintgh ese problemIs had a certaino bjectivei n mind,w hiche xplainst heir uncon­
ventionaflo rmulation.T he followinge xplanation willm ake their solutiono f greateri nterest
to the reader.
Problems olvingi s of particulari mportancei n thes tudyo f the exact sciencessu cha s
mathematicsm,e chanics,p hysics,e tc. It enabels the studentt o applyh is own knowledget o the
solutiono f practical questionHso.r eoverf,o r the teacheri t representasn excellentm eansf or
finding out whethert he studentr eallyu nderstands stuhbej ecto r whetherh isk nowledgei s a
merea ccumulatioofn facts leanredb y heart.T eachingb y meanso f problems olvinagl soh elps1 11
revealinagn d cultivatintghe creative scientifitch inkinogf they oungg enerationT.h e need
for thisw as quitee vident2 5y ears agow hen theP hysiacl-TechnicaIln stitutwea s foundeda s an
institutioonf higherl earninign tended specificalfloyr the selectioann d traininogf research
workers. It is wellk nown thatf ruitfuslc inetificw ork requirenso to nlyk nowledgea nd under
standinbgu t alsoa capacitfyo ri ndependenta nalyticaaln d creativteh inking.I n effect,t hese
problemwse re compileda uasse fulm eansf or the discoverye,v aluatioann d cultivatioonf these
qualitiedsu ringt he teachinpgr ocess.
I strovet o achievteh ise ndb y formualting the majorityo f questionisn the followinmga n­
ner. A smallp roblemi s presenteda,n d the student, usingt hek nown laws of physicsm,u sta na­
lysea nd describqeu antitavteilyt hen aturalp henomenoinn volved.T hesen aturalp henomenwae re
selecteidn termso f theirs cientifiocr practicali nterestw ithint he scoopfe the students'
levelo f knowledge.
Mosto f thep roblemasl lwo an umbero f approahces tot heirs olutioinn ordert o revealt he
studetn's individualityO.ne examplies the problme of the trajectoroyf an airplanwei tha
stateo f weightlessneisns its cabin. Heret hes tandaradp proach would be to write thee quation
327
LdK-LUoL iltJ. C;:ULLClJ.lUl.L
actingo n a pointi nsidet he plane as equalt o zero. Therei s, howeevr,a notehr, simpler
solution.I f the planef ollowst he trajectoroyf a freelyf lying body,w hich is closet o a
parabolian the Earth'sf ield,t hena bodyl ocatedi nsidet he planec an be in a stateo f weight­
lessness.T he curiousst udentc an pursuet he matterf urtehr and finodu t\ .. h at is requiredd ur­
ingt hef light in order simultaneoustloy a chievea stateo f weightlessneasts a ll points of the
plane'sc abin.A notherq uestiotnh atc an bec onsideerd is whatn aviagtionali nsturmentsa re
necsesary for the pilott o fly the plane alongt he trajectoyr requried for weightlessnesse.t c.
A charactiesrticf eatureo f our problemsi s thatt heyh ave no definitaen swerb ecauset he
studentis allowedt o proeced furthera ndf urtherw itht hea nalyssi of the problepmo sed,d epe�
ingo n his own abilitieasn d inclinatniso.
Thes tudent'asn swersp rovide a cluet o the trenda ndn atureo f his scientiifct hinkin,g
which iso f partciulari mpotrancei n the selectioonf postgraduawtoer kers.T hei ndependent
solutioonf suchp roblemsst imulatetsh ec apacitfyo r scienticf ithinkinagn dc ultivateasn
attractiotno scientifipcr oblems.
�josotf thep roblemhsa ve anot.hdeirs ticntivet rait. They do notc ontani numericavl alues
of physicaclo nstantosr parameetrs,a nd thset udenth as to chooset hemp ersonally,T hus,i n
the probleomf weightlessenss in an airplanteh e studenti s askedt o detemrinet he durationo f
weightlessnestsh,o ughi t is mentionetdh att he airplane is of a modernt ype. Heret he student
has to pick outth e airplane'cse ilnig and maximsupme edf or himself.W e did so because few
peoplei n our country appteoar c arew hethert he futuer sceitnisto r engineeirs taught to ima
gine the actualm agnitudeosf suchc ommonp hysicavla luesa s velocityc,u rrentv,ol tages,t rength,
tempearture, etc.i nc oncrettee rm.s
In solvinga scientifipcro blemt her eseacrh scientisatl waysh as to visualizec learltyh e
magnitudaen d relatives ignificcaen of thep arametesr usedf or descirbing thep henomenone xam­
ined. Thush e willb e ablet o picko ut the crucialp arameterfso r the studyo f theg ivne pheno­
menon.H encet he importancoef teachintgh ey oungg eneartiont hatt hes ymbolwsh ichr epresent
physicamla gnitudesi n thef ormulaes houlda lwaysb e visualizeidn termso f concretqeu antita­
tivev alues. In physcis, contrartyo mathematicbso,t ht hep arametersa ndt hev araiblesi n a
mathematiceaqlu ationne cessarilrye presencto ncretqeu antities.I n our problemwse therefore
train the studenttso seeko ut ther equiredm agnitudesf or themselveisn the literature.
The studetns of theP hysicla-TechnicaIln stitutseh owedi nteresitn these problaenmds
frequentlsyu bjectetdh emt o collectivdei scussi.o nIn the examiantions,t hes tudetns were
alwaysg ivne completfer eedotmo usel itearture for solvingt he problems.U suallya few (up to
5) prolbemsw ere given per examiantio,n so as to enable thes tudetns to choose2 3 of them.
Thus,t he inclniatinoso f a studentc ouldb e gaugferdo mh iss electoin of problemsF.o r post
graduatee xamiantionsn,e w andm orec omplexp roblemwse rep reparedi;n thesec ases,h owveer,
thes tudentw as allowedn ot onlyu se of liteartureb uta lsof reedomt o seeka dvice. Indeed,
the scietnistm ustc ultiavte thes killo f usingt he advice of othesr, apart from learnnig the
use of litearture. In scientifwiocr k, discussoins and consultaotnis withc olleagueasn d
instructorasr e essetnialf or success;t his)h oweve,r requires ap roepr trainnig fromt hev ery
begininngo f thes tudies.
We usuallay lloweda bouto neh ourf or the solutioonf eachp roblem.A ll problemsh ave to
be solvedi n writ.ing,b ut the capabliitieasn d character tohfe s tudetn becomee videnmto stly
1n thec ourse oaf verbald iscussoin of thew rittent ext. Theg reatetrh e taleonft they oung
more than an hour.
Today theesi s doubta s to the enormousi mportanceo f sciencef or the developmenco f
the culture and economya moofd ern stateT. he numbero f Sovients cientistsa nd researchI Vor­
kers is constantly rising, haavlirnega dye xceededh alf a million. For this reasont he training
and educationo f young scientistsi s nothingl ess than a major,i ndependenotb jectiveo f the
state.
Apart from the MoscolVP hysical-TechnicIanls titutet,h erea re a numbero f other institu­
tionso f highere ducationi n the rSSR intendeds pecificallyf or the training of research
cadres. Teachinga t thesei nstitutionusn doubtedlhya s specifict raitso f oIVn, as distinct
from the universities producec adresf or industrya nd the national I believe
that problems compartaob lteh ose compliedb y us can be used not in teaching physics
but also such as mathematics, mechanicchse,m istry,e tc. The solution
of problemsc omparabel to those given here Hill thereforei mprove future
researchers
I.A stronomicalo bservationss how that the planet Venus is entirely in cloud,
so that the Venusians unablet o observet he heavenlyb odes, Describeh olV accu-
ratelym eaSUret he length theird ay.
<
2. The Tungusm eteorites truckt he eartaht latitude6 0, and its was converted
into heats,o thati t evaporated, Assuming thwaeti gihte d 10,00t0o ns and a speedo f 50
km/s, calculatet he maximume ffectt hat the impactc an have had on the earth'sa xialr otation.
Could such a changeb e detectedb y present-dayc locks?
3.W hat shouldb e the trajectoryo f a moderna ircrafti n ordert o simulatea stateo f
weightlessness?F or how long eQuIds uch a conditionb e maintained?
4.D eterminet he limito f accuracyo f time intervalm easurementIV ith a cathode-raoys cil­
lograph.
5.T he ancients a water-pumpo peratinga s follows. A tube bent into ah elix rotates
abouta centrala xis at an angle ex to the water surface. The 10Here nd the tubies in the
water, The wateri s raisedt o a height h. Find the optimumv alue , the throughpuot f the
pump,a nd its efficiency.
6. througha lead block,b ut are of paraf-
fin, water,o r otherc ompoundc ontainingh ydrogen atoms. is this?
7.A , as falls, pullosn the front lineS. will
he then travel?
8. car burstsd uringa race, At what speeds houldt he car traveslo that the tyre
does not deflate?
9 To protectp assengersi n car crashes,a ir bags can be used,w hicha re rapidly filled
with gas and absorbt he impact. Assesst he effectivenesosf this technique. If a vehiclem ov-
ing at 100k m!h crashes,a nd a person'sh ead strikingt he bag is injured,' ''hatm ust be the
size of the bag and the gas pressurei n it?
10.B y what factorc an the heighto f an acrobat'sj ump be increasedb y using a springboard
once?
II.E xplainw hy, for a bow of a givens ize, therei s B certains ize of arrow whichy ields
the longestf light. Estimatet his size for a bow of a given shape.
330 COLLECTEPDA PERSO F P.L. KAPITZA
12.A n acrobato f "eightP stansd on a ballo f radius R and massM .T he balli s on a
horizontapll ane,a nd rollsw ithouts lippinogn thep lane. Analyset hew ay in whicht hea cro
bats houldw alko n the balli n order to makei t roll,a nd how the coefficienotf frictioonf
hiss olesi s relatedt o the rollinga cceleration.
13.W hatm ustb e the speed of a tennisb allt hatc an breakg lass?
14.A ballt hrowno n to a solids urfacer ebounsd fromi t. Estimatteh ed ependence of the
heighto f reboudn on thev iscosityo f the soli.d
IS.E stimatteh ed urationo f impactb etweena footbalaln d a wall.
°
16.A steelb allf allsf roma heighth � IIeIm on an inclinpeLda ne( angle r:J. 10),r ebounds
elasticallfya,l lsa gain,a nd so on. Describet he motion,a ssumintgh att he plane is of unlimi­
tedl engtha nd thatt he impactst akep lacew ithoutl oss.
17.E xplainw hy a personc an run on very thini ce but cannots tando n it withoutf alling
thr ough.
18 Estimatteh e ordero f magnitude of the speeda t whicha personm ustr un on wateri n
ordern ott o sink.
19.H hy cant hem ovemento f a bicycleb e controlleidn 'on hands' riding?
20.A satellitpea sseso ver Novosibirsakt 02.30, and overM oscowa t 06.00. Wherew illi t
be at 20.00 and at 21.45? (Moscowt imei s use)d.
21.A spaceshiips t ravellingf romt he eartht o Mars. Halfo f its surfacie s blackened
and absorsb thes un'sr adiationc ompletley: the otherh alfi s polishemde tal,a nd reflecttsh e
sun'sr adiationc ompletely.E xamineh OIr" adiatiopnr essrue willa ffectt he translaotniala nd
rotatoinalm otiono f thcer aft.M akea quantitatei evstimtae of the effectf or a sphericla
crafto f mass5 tonsa nd diameter3 00 cm.
22.A contaienro f Hateri n which a Hater rati s s.\Jimminigs in equilirbium on a balance.
A string is loweredf roma rod attachedt o the others ide of the balances,o as to toucht he
water.T he rat begintso climbu p the strin.g Will equilibriubme maintianed?
23.A fly is siltingo n the bottom ofb eaa kert hat is on a balanceT.h e fly fliesa way.
At whatp ointi n timew illt he balancef irstr egistetrh ed epartureo f the fly?
24.D eterminteh ed istortioonf a Liquids urfaced ue to the gravitatioonf a sphere.D is­
cusst he possibilitoyf obevring thise ffecti n order to measurteh eg ravitationcaoln statn.
25. Determinet hem inimums ize of an objectd istinugishabloen a photogrpah takne froma
satellitaet a heighto f 300 km.
26.A n astronauitn a stateo f weightlessnehsass to dig aw ell. How canh e do this?
27.A rocketw ithm ass M is motionlesasb ove the earth'ss urface. The speedo f theg ases
Leaving the rocketi s u .F indt he power of the rocket'esn gine.
28.W hatw illb e the trajectoroyf a bullte firedf orwarsd/bakcwarsd/siedwaysf roma
satelliet.
29. How cana na stronuat returnt o hiss pacecrfat if ther opej oiningh im to ita ccident-
allyb reaks.
30.H ow can thed irectiono f flight of a satellit(ew eight 100k g, orbitr adiu4s0 0 km) be
°
changedb y 2 ? Describei np rinciplet he possbilew ayso f performnig this manoeurve.
31.A s the earth� ovesi n its elliptcial orbit, its speedi s always eitheri ncreasinogr
decreaisng. Cant he accelreatiobne measuredb y meanso f a liquidl evel?
32.C alcultae thec hangei nt earth'st emperatuirfe thmeo onw eret o fallo n it. Take
the heacatp acitoyf bothb odise to be I cal/c3m deg.
PROBLEMSI N PHYSIe s 331
33.D escribe the distortioonfs t he earth'so rbit due to s�olarr aditaiAonp ressrue. E=sti­ :'
matet heir amount.
34.A loadw it h massM iss uspended
I
on a stringp assnig througha pulley,a nd
swingsw itha given amplitude, The end of
I
the string issl Dlvlmyo veda distancel
L
(Fig. I). Find the change in the period ,
andi n thaem plitudoef thes wing,a ndc al­ I
M
-r
culatet hew orkd one.
3�. How can ther elation betweent he
lenght and the periodo f a pendulumb e
foundw ithoutc alculaotni? FIG. 1
36.A cari sm oving witha cceelrationa up a hillside at angle ex . Findt he oscillaotni
period of a pendulum( length L) insidet he car.
37.S ay which yout hinka ret he simpelst andm oste xact experimentalm ethodsf or using
knownp hysical constantsa nd a standarodf length( the metre)t o genreatea unit of time( the
second)w ithouuts ing astronmoical observatioonrs thea ccelertaion duet o gravtiy (as if the
expermientw ereb eingd onei na deepm ine-shaft or on anotherp lanet.)
38.D eterminet he maximruamn geo f audibilityo f a convresation int he open air.
39. Of an eveningo,n the river,o ne mafyin d thata distantc onversation is quite clearly
audible. How doest his happen?
40. Ab ell is hangingo n a pillara t a height h. The wind speed is u. If the speedo f
sound is c, at whatp oint on the groundw ill the soundo f the bell be loudest?
41. Explainw hy thereh ave been instancesw heret he whole fronte nd of a cannonh asf lown
off when the cannonw as fired.
42. Will a gardeenr wateirng al awnw itha hosea lwayse xpereincea reactvie force?
43. Examnie whetheru ltrasonivci brtaions coudl in pricnipleb e usedi n an apparatutso
forma n image of obje.cutns derw ater.
44.L ist and descrbie all the methodso f generantgi sound.W hicho ft hesei st he most
economical?
45. Ac onveyebre ltc arriesl oads horiznotallyb,u t hasa transverssel opes o thatw ater
canr uno ff. The objectsb eing carreid lief ree on the belt,b ut thef riction isg reate nough
to stopt hem frosmli pping sideways.T he belt enter�a buildnig througha narrows liti n a
perfectyl smootwhal li na planep erpendciular to the directino of motion. Whatw illb e the
movemenotf a cylindelry ingf reeo n the belta fteri t is carried against thew all? If itw ill
in facmto ve,f ind thed irections,p eed,a nda ccelreationo f the motion, taknig as given the
slopeo f the beltt o the horioznta,l the speedo f the belt,a nd the coeffciient of friction be­
tweeni ta ndt hec ylinder.
46. Explainh ow a boy on a swingi ncreasest he ampltiudeo f his movement.
47. Whatm ovemensth ouldb e performebdy a person rotatai nhgo opr oundh isb ody?
48. Determinet he dampnig of vibratiso nof a penduluimn a rarfeiedg as.
49. Am otor-cycliisst r idingo n thew allo f a barrel-shadp reing. Findt hec onditoin for
this motiotno be stable.
50.A hollows pacei n a penduluim sf illedw ith vai scousl iquid.E stimaet quantitavteily
the effecto f thel iquido n the perioda nd dampinogf thev ibartions.
JJ2 COLLECTEDP APERSO F P.L.K APITZA
51.L ist thef actosr which affectt he acclalcryo f a watch.A ssesst heir relativiem por-
tanee.
52. The wallso f a rectangulavre ssela re slowlyb roughtt ogethewri thr elatvie speedv
from an initial distancel . The vesselc ontanis a particlweh oses peedr elatvie to the earth
is u. How doest his speedv ary witht het ime t? Assumet hatc ollisionwsi tht he wallsa re
perfectleyl astci.
53.T he point of supporto f a penduluims oscillantgi rapidlyi n a horizontalp lane. Find
the equilibriupmo sitioonf thep enduluumn derg ravity.
54. (a)S howt hatt he vibrationosf a pendulucma n be maintainebdy causingt hep ointo f
supportt o oscillatheo rizontalolry vertically.
(b)F indt he phasea nd periodo f theseo scillaotnis,a nd estimatteh e amplitudnee eded
to maintaitnh e vibrationfso r a simplep endulum.
(e)E xaminet he stabilitoyf thesep rocesses.
55.T woi denticadli scsa re placeda shortd istancea part.T he upperd isci s hungo n a
wire and canu ndegro torsionalo scillations.T he lowerd isc executefso rcedt orsionaols cilla­
tionsw itha specfiieda mpltiude,a nd a periode qualt o thato f thef reeo scillaotnis of the
upperd isc.T he discs arep uti n a vesselc ontaininag rarefied gaisn,w hicht hem eanf ree
patho f the gasm oleculeiss muchl ongert hant he lineard imensionosf the vessel.D etemrine
the forma nd amplitudoef the steadyo scillationosf the upperd iscr elativteo the lower one,
andt het imec onstanft or the attainmenotf thef inals tate.
56. Two identiciaslp hereso f knownm ass,l yingf reeo n a perfectlsym oothh orizontals ur­
face,a rej oinedb y a tautt hread.A verticarlo d movingh orizontallayt a constant speed
strieks the thr ead.I f the thr ead can be broken by a forceF' , what speed0 f the rod is
necessartyo breakt het hread?T he part of the thread whircehc eivetsh e directi mpacti s
strentghened.
57.T he soundo f an aircrafftl yingd irectltyo wardsy oua nd thena wayf romy oui sr ecorded
on tape. How cany ou determiniet s speed?
58." Whisperingga lleryw"a vesa reg eneratedI n a barrel-shapoepde nr esonatorT. he reson­
atori s rotatnig abouti ts axis. Howd oest he rotatoin affectw avep ropaagtion in the direction
of rotatioann di n the oppositdei rectoin? Do beats occuarn,d at whatf requency?
59.A bronez cylinde(r radius R, lengthl ,\ o}altlh icnkess h; h/R «{ I)i s suspendedH.h at
mustb e thes ize of the cylindeir fi ts naturarla dial oscillatiofnr equencfy 2000s- l? Des­
cribeo therp osisble typeso f oscillatioann,d e stimatteh eir freuqencies.
60.B y whate xperimentmaela nsc ouldo ne determinteh e speedo f propaagtiono f universal
gravity? whatd ifficultiepsr eventt his?
61.T he surfaceo f a riveri s an inclinepdl ane. Can a bodyf loatf reelyd mm the river
at a speede xceedintgh e maximumf low rate?
62.T herea re twoi denticalb uckets.
eachw itha holen eart he bottom( Fi.g 2).
One is standinogn a horizontatla ble,
the othero n a wedgew hoseu ppers urface
is alsoh orizontalF.r omw hichb ucket
willt hew aterl eakf asteri f its origi­
nal leveli st hes amei n both?
63.Io lyh do door-screepnrso tectd oors
fromd rifted snow?
PROBLEMSI N PHYSICS 333
64.A man is walkinga long a road at a speed u. On his chesti s a newspapero f weight P.
What is the minimumc oefficiento f frictiont o preventt he newspaperf rom slippingd own? The
air densityi s p and a wind blows againstt he man with speed v. His chest is regardeda s a
plane. How is the resultc hangedf or a convexc hest? The area of the newspaperi s S.
65. Explainw hy sprayf lies up when a stone or a raindropf alls on water. Does the height
reachedb y the spray dependm ore on the size of the stone or on its speedo f fall? I,'haits the
maximump ossible height?
66.A trolleyi s runningo n rails
with initials peed Vo .Th ere is water
in the hollowb etweent he rails. The
trolleyc arriesa verticalp ipe with a
tap, dippingi nto the water and connec­
ted to anotherp ipe with variablec ross­
section( Fig. 3). It is known that with
a trolleys peed Vo and the tap closed,
the water can rise in the verticalp ipe
to a level abovet hat of the tap. Des­
cribe the movement of the trolley,
neglectingf riction. FIG. 3
67.H ow is work done by the exertionn eededt o turn a gyroscopea bout an axis perpendicular
to that of the flywheel'
68.W hy is the fork of bicycleh andlebarss et at an angle?
69.F our gyroscopesa re placedo n the sides of a square. Their ends are linked by hinges.
The squarei s hung by one corner,a nd a load can be attachedt o the oppositec orner (Kelvin's
gyrospring).D eterminet he angular momentoufm the gyroscopess uch that the diagonali s leng­
thenedb y I cm when the side of the squarei s 30 cm and the load is Ik g. How will the unloaded
systemm ove if the squarer etainsi ts shape?
70.A gyroscopei s placedo n a swings o that its axis can turn 1n a plane throught he axis
of the swing. Describeh ow a persono n the swings houldt urn the gyroscopes o as to increase
the amplitudeo f his movement. Find the most effectivem eans of increasingt he amplitude,a nd
derivea n expressionf or the rate of increase.
71.A load m is placedi n a cylinderw ith radius R and mass M at a distancel' from the
axis. Describeh ow the cylinderw ill roll withouts lippingo n a horizontalp lane.
72.A top is rotatingo n a plane,w ith angularv elocityw , mass M, and momentso f inertia
A and B. Iei s struckb y a bulleto f mass m and speed v, which lodgesi n the top. Describe
how the top may behavea nd how its behaviourc an be used to find the speedo f the bullet.
73.A heavy hoop with light spokesi s placedi n a verticalp lane and can rotatea bouta
horizontala xis throughi ts centre. A particlew hosem ass is equalt o that of the hoop is
fixed into the rDiemt.e rminet he periodo f small oscillationosf the resultingp endulum.H ow
does the periodv ary if the pendulumi s transferredt o the mooon r is placed in a frictionless
liquid.
74.A rapidlyr otatingf lywheeli s placedo n the end of a rod 1.5 ml ong standingv ertic­
ally on the floor. Above the flywheeli s a platformo n which a personc an stand. Calculate
the size of the flywheel,f or a speedo f 3000r pm, if the rod remainss tablee ven when a person
is doing gymnasticso n the platform.
334 COLLECTEDP APERSO F P.L.K APITZA
75. A given volumeo f gas is containeidn a sphericamle talv esse.l Find the gasp ressure
forw hicht he weighto f the containeirs least.
76. Whyc anl iquidn itrogebne pouredo n the hand" ithotu fear of 'burns'?
77. A verticagll asst ubeo f circulacrr oss-sectiiosno pena t bothe nds. At thel ower
end is an electrihce atingc oilc arrynig a curretn. Becauset he air is heated,a draughti s
set up. Assumingt hatt he airf lowi n the tube is laminard,e terminteh e temperaturdei ffer
enceb etweent he air outsidea ndi nsidet he tube,a s a functioonf the lengtha nd radiuso f the
tubea ndt he elecrtic poweri nput.N eglect heatt ransfetrh rouhg theg lass.
78. Whata ret he necessariyn itiala ndf inalc onditionfso r partiall iquefacotni of a real
gas by a singlea diabatiecx pansionD? iscusst he liquefactioofn air as a numericaelx ample.
79.W hate xperimentcsa n be devisedt o establisahn absolutsec aleo f temperature below
0.5o K.
80. Estimateho w thickt he wallso f a given materiamlu stb e in order to keept he tempera­
tureo f a roomw ithin3 deg of its averagoev ert he year.
81.C alculatet he meant emperaturoef the earth'ss urfacea,s sumingt hati tr adiatesa s a
blackb odya nd thatt he eneryg of thisr adiatioins in equilibriumw itht hatf romt he Sun.
2
Assumet hatw ithv ertiacl illuminatio2n k W of solare nergyi si ncidento n I m of the earth.
82. Estimatteh eh eightof falld uringw hich ad rop of moltenl eadw ills olidify.
83.E stimatteh e timef or a pondt o freeze.
84.T here is a piece of ice on onep ano f a balance, anda beakero f watero n the other.
The systemi s in equilibriumT.h e balancei s coveredw itha largeh ooda nd the air is rapidly
evacuatedW.i ll the systems till be in equilibrium?
85. Supposet hat4 X1 09 yearsa gob otht he moona nd the earthw eres urroundebdy an atmos­
pherel ike the earth'sp resetn one. Estimatheo wt he denstiy of this atmospherweo uldv ary
during thtaitm ei f the bodiesm ovedi nt he solars ystema s theyd o now.
86. CalE stimatteh et imen eededt o formv isibled ropletisn a cloud chamber.
(b)A nalyse and estimatteh e factorsi nfluencitnhge thikcness of partcile tracksi n
a cloudc hamber.
87. Until electrciityb ecamew idelya vaialble,a simplet hermala ir motorw asu sedf or low­
powerw ork,c onsistinogf a watercooledc ylinder,a continuouslhye ateds ide-arm,a nd a piston
operatcnag flywheel.D escribeh ow andu nderw hatc onditiosn sucha motorf unctions.
88. It is known that,w henc oal is burnedi n steam boileornsl,y af ractionTl (Tj- Tz)T!j
of eachc aloriec an be converteidn tow ork, whereT ) and T2 are the temperaturoefs the steam
andt he environmentT.h e gas leavingt he combustiocnha mberh as a compositiodnif ferenftr om
thato f air. If it is mixed revresibyl witha ir, af urthera moutn of workc an be obtained.
Estimatteh e attainaeb lliimtt o the increasoef Tl. and devisep ossiblcey cles for carrynig
outt hisp rocess.
89. Estimatteh e theromdynamiecf ficiencoyf the firingo f cannon and hand-guns.
90. Ther eversbile reactioonf hydrogeino didef ormatoin takes placei n a mixtureo f hydro­
gena nd iodineg ases. Determnie the quatnityo f hydrogeino dide formed,a s a functioonf the
initial quantitieosf theh ydrogeann d iodine,a ssumintgh e equilibrium consktnoawnnt.
91.T wo paralell platesa re at a distancaep art that iss mallc omparedw itht heirt rans­
versed imenisons. Betweent hem are severla thins creenwsi thg ood thermalc onductivity.D eter­
mine the effecto f the screenos n the conductioonf heatb etweent hep latesi n two cases:
(alw hent he meanf reep atho f the gasm olecules int he spaceb etweent hep latesi s much
PROBLEMS IN PHYSICS 335
less than the distanceb etweent he screens;
(b)w hen this mean free path is much greatert han the distanceb etweent he plates.
92.I n ordert o shol,t'h at the viscosityo f a gas is independenot f the pressure,M axwell
observedt he dampingo f torsionalo scillationosf a disc. Investigateh ow this dampingw ill
vary as the gas pressuref alls.
93.A beam of moleculesi s createdi n a long horizontalt ube. Mattere nterst he beam at
normalt emperature.A t the othere nd of the tube, becauset he more slowlym ovingm olecules
undergoa greaterd eflectionb y gravity,a temperaturdei fferencem ay occur. !./hyi s this con­
sistentw ith the secondl aw of thermodynamics?
94. If a beam of moleculesi s passedt hrougha selectorc onsistingo f two paralleld iscs
rotatingo n a common axis and havinga perturest hat do not coincide,i t is knol'nt hat the fas­
ter moleculesc an be separatedf rom the beam,a s if by a Maxwelld emon. How can this be
reconciled witthhe secondl aloo f thermodynamics?
95.A n aircrafti s flyinga t about the speedo f sound. The fuselagei s heatedb y air fric­
tion. Estimatet he maximumt emperaturteo which the surfaceo f the aircraftm ay be heated.
96. (./hat is the changei n the Debyet emperatureosf coppera nd'' solidh elium-4u nder hydro­
staticc ompressiona t 1000at m? Hooke'sl aw is assumedv alid.
97.T herei s a small hole with diameter1 02- mOl in a vesselw herea vacuumo f 105- nun Hg is
to be maintained. Determinet he size of the evacuationt ube and the pOloero f the vacuump ump.
98. In ordert o determinet he chargeo n the electroni n the classic.le xperimentosf
Ehrenhafta nd Millikan,a chargedd ropleto f mercuryi s placedb etweent he horizontalp lates
of a capacitor.T he force of gravityo n the dropleti s balancedb y the electricf orce,s o that
the electronc hargec an be found. Examinet he effecto f Brol<niamno tiono f particleso n the
accuracyo f thesem easurements.
99. Determinet he limitings ize of plane circulard iscs of iron and aluminium( witht hick­
nessesm uch less than theird iameters)w hich are poorlyw ettedb y water.
100.T wo cylindricalc ommunicatinvge sselsw ith differentd iametersa re filledw ith loater
or mercurya nd sealed. How will the totala mounto f the liquidb e distributebde tweent he
vesselsu nderc onditionso f weightlessness?
101.C alculatet he time for the disappearancoef a soap bubblec onnectedt o the atmosphere
by a given capillary.
102.I f a fishing-linies loweredi nto flowingw ater,a patterno f stationaryc apillary
waves is formedr ound it. Explaint his.
103.A bent tube has one end attachedt o
a pump which deliversw ater at a constant
pressureP . The maximum supply of water
through the pump is Q Vs. A plug of mass
m is originallya t rest in the horizontal
part of the tube at a distance from the
open end (Fig.4 ). At what speed will it
emergei f the cross-sectioonf the tube is S,
the atmosphericp ressurei s PO'a nd the fric­ FIG. 4
tion betweent he tube and the plug is negligible.
104.D eterminet he speed at whicha two-dimensionawla ve propagateso n a stretcheds oap
film of givent hickness. Estimatet he range of such speeds.
336 COLLECTEDP APERSO F ? L. KAPITZA
105F.i nd the potentiala t the centreo f a metal sphereo f radius R carryinga charge q.
106.De terminet he deptho f penetrtaion of seasonalt emperaturev ariationsi nto the earth's
crust,u singt he thermalc onductivitoyf granite. Calculatea lso the amplitudeo f the seasonal
temperature fluctuatiaotn tsh e bottomo f Lake Baikal.
107.A n insulatedc opperb all of given radius,c overedw ith a knowna mounto f polonium,i s
placedi n a vacuum. Becauseo f the emission of particlesi t acquires charge. Determine
the way in which the potentiali ncreasesw ith time,a nd its limitingv alue.
108.A n insulateds phereo f caesiumm etal,h avinga givens ize,i s placedi n a vacuum,
exposedt o daylighto n one side,a nd givena chargeb y the photoelectriecf fect. Estimatet he
variationo f its potential wittihm e.
109.Fi gure5 shows an electrostatic
machineo f the droplett ype.\ <atedrr op­
lets chargedt o a givenp otentialf aU
from a tube into a hollow insulated
metal sphere. Determinet he limiting
potentialt o whicht he spherei s charge,d
as a functiono f the heighto f fall.
110.A free soap bubblei s electri­
fied to the maximump ossiblep otential,
limited by the breakdowns trengtho f
the air around it; how and by what
amounti s its radiusc hanged?
III .A clock operatedb y radioactivee nergyc onsistso f a smalla mounto f radioactives ub­
stanceo.n the rod of a leaf electroscope.T he radiationa nd the loss of chargec auset he elec­
troscopet o be continuouslcyh arged,a nd the leaves divergeO;n reachinga certaina nglet hey
toucha n earthed contaacntd fall back to theiri nitialp osition. Calculatet he designo f such
a clockw ith a periodo f I minute. Estimaittes possiblea ccuracy.
112.A smallm etalc ylindero f given size is swingingl ongitudinallwyi th frequencyw .
Assumingt hat electronsm ove freelyi n the metal,c alculatet he chargesf ormedo n the ends.
Make a similacra lculationf or an insulatingr od with permittivityE .
113.A dielectricp latei s in a uniforme lectricf ield. Determinet he torqueo n the plate.
114.Th e horizontalp lates a plane capacitora re connectedt o a batteryg ivinga con-
stant emf c. A chargedb all betweent he platesi s suspendedi n the gravitationafli eld. How
will it move when the distanceb etweent he platesi s changed?
115.Why are electromagnetimca chinesa nd not electrophorus-tyopnee s used in practicet o
generate higpho wer?
116.I f one end of a conductor at
a positivep otentiala nd the other a
negativep otential, cau rrent flow
along it from plus to minus. Why then
does the lamp not the cir-
cuit showni n Fig. 61
IlL What limitst he sensitivity
a stringe lectrometera,n d what is
valueo f the sensitivity? FIG. 6