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§ 123. TEOPEMA 06 H3MEHEHHH KHHETWJECKOA

3HEPrHH CHCTEMbl

noKa3aHHafl B § 89 TeopeMa cnpaBe)J,JIHBa )J,Jlf! JII060H H3 TO'-!eKCII- CTeMbl. CJie)J,oBaTeJihHO, ecJIH paccMoTpeTb KaKyiOHH6y.n;h TO'-!KY cHcTeMbi C MaCCOll mk, HMeiOIIJ.YIO CKOpOCTb Vk, TO )J,Jlfl 3TOH TO'-!KH6y)J,eT

d (m1p~/2) = dA~ t- dAil,

r.n;e dA~ H dA1 - 3JieMeHTapHble pa60Tbi )J,eilCTBYIOlii,HX Ha TOtiKy BHeiii-

HHX H BHyTpeHHHX CHJI. CoCTaBJiflf! TaKHe ypaBHeHHfl )J,Jlfl Kalli)J,OH H3

TO'-leK CHCTeMbl

11 CKJia)J,biBafl HX

llO'-!JieHHO, Haif)J,eM,

'-ITO

HJIH

d (Smkv~/2) = L dA ~ +L dAk

(49)

PaBeHCTBO

(49) BbipalliaeT

T e 0 p eM y 0 6

H 3 M e H e H H H

K II H e T H '-!e C K 0 ll 3 H e p r

H H C H C T e M bl B

)J, H ~ ~ e p e H·

u H a JI h H o ii

~ o p M e. I1poHHTerpuposaB o6e qacnr 3Toro paBeH-

CTBa B npe)J,eJiaX, COOTBeTCTBYIOIIJ.HX nepeMelll,eHHJO CHCTeMbl !13 HeKo-

TOporo

Ha'-!aJioHoro noJiollieHHH, r.n;e KHHeTH'-!ecKaH 3HeprHH pasHa

T0 , B

noJiollieHHe, r.n;e 3Ha'-leHHe KHHeTH'-lecKoil 3Heprnu CTaHOBHTCH

paBHhiM Tt. noJiy'-IHM

(50)

3TO ypaBHeHHe Bblpa)KaeT TeopeMy 00 H3MeHeHHll KHHeTH'-lecKOH 3HeprHH B .n;pyroii (HHTerpaJihHOH) q>opMe: U3.MeHeHUe KUHemUtteCKOU

a~tepeuu cucme.ttbt npu 1te1wmopo.M ee nepeM.eUfeHuu paeno cyhthte pa6om

307

§ 87:

§ 126. nOTEHI..UtAJibHOE CHJIOBOE nOJIE

H CHJIOBA51 CI>YH l\l.\H51

3a}:(a'HI, paCCMOTpeHHbJe B rrpe,ro,I,IJ;Ylii,HX rraparpacpax (H B § 89),

y,n;aJIOCb pelllHTb C IIOMOlil,b!O TeopeMbl 06 H3MeHeHl!H KHIIeTH'IeCKOH3HepfHH ITO TOll rrpH'IHI~e, 'ITO BO BCeX CJJy'laHX paoOTy ,n;eHCTBY!Olii,HX CHJJ

MO:li\HO 6biJIO ITO,!l;C<!HTaTb, He 3Ha5I 3apaHee 3aKOHa rrpOHCXO,!J;Hlll,ero

,n;BHJKeHHH. BaJKHo ycTaHoBHTI>, KaKoB Boo6lll,e KJJacc CHJI, o6Jia,n;a!Olli,HX 3THM CBOHCTBOM.

Pa60Ta Ha nepeMelll,eHHH M 1M 2 cHJibi F, npnJioJKeHHoil K TeJiy B!0'-1- Ke M, Bbi'IHCJIHeTCH no cpopMyJJe (44')H3

(M 2)

(M 2 )

A<M,M,)= ~ dA=

~ (Fxdx+Fydy+Fzdz).

(54)

(M 1)

(M 1)

KaK yJKe ObiJio OTMe'leHo B § 89, Bbi'IHCJIHTb CTOHlii,HH crrpaBa HH-

TerpaJI, He 3Ha5I 3aKOHa rrpOHCXO,!l;Hlll,ero ,!J;BH/KeHHH (T. e. 3aBHCHMOCTeH X, y, Z OT Bpe:vteHH f), MO/KHO JIH!llb B CJiy'laHX, Kor,n;a CHJJa ITOCTOH~Ha

HJIH 3aBHCHT T0,1bKO OT ITOJIOJKeHHH TO'IKH,T. e. OT ee KOOp,!I;HHaT X, y, z. TaKue CHJibi o6pa3y!OT CHJJosoe rroJie (eM. § 32). TaK KaK CHJia orrpe- ,!J;eJJHeTcH ee rrpoeKUHHMll Ha KOOpAHllaTHbJe OCH, TO CHJIOBOe ITOJie 3a,n;a- eTCH ypaBHeHHHMH:

Fx=fi\(X, y, z), Fy=W 2 (X, y, z), Fz=Ws(X, y, z);

(55)

Ho B o6lll,eM c.Tiycrae 11 ,!J;JJH Bbi'IHCJieHHHpa6oTbi TaKHX CHJJ Ha,n;o B cpopMyJie (54) rrepeilTH ITO,!l; 3JiaKOM HHTerpaJJa K O,!J;HOMy rrepeMeHHOMy, T. e. Hanp HMep, 3HaTb 3aBHCHMOCTH y=f1 (x) H z= f2(X). 3TH paBeHCTBa, KaK H3BeCTHO, orrpe,n;eJIHIOT B rrpoCTpaHCTBe ypaBHeHHe KpHBOH, 5IBJI5IIO· lll,eilcH TpaeKTOpHeli TOIIKH M. C1eAoBaTeJihHO, B o6lll,eM cJiy'-!ae, pa-

60Ta CHJI, o6pa3y!Olll,HX CmiOBOe rro,1e, 3aBHCHT OT BH,!l;a TpaeKTOpHH TO'IKH npHJIO/KeHHH CHJibl.

O,n;HaKo ecJIH oKaJKeTCH, 'ITOBbipatKeHHe, CTOHilJ,ee B rpopMyJie (54) no,n; 3HaKoM HHTerpaJia 11 rrpe,n;CTaBJJHIOlll,ee co6ofl 3JieMeHTapHyiO pa6o-

TY

CHJJbi F,

6y,n;eT rroJIHb!M ,n;HcpcpepeHuHaJJoM

HeKoTopoil ¢yHKUHH

U (x, y, z), T.

e.

dA =dU (x, y, z) HJIH

Fx dx+F Y dy+F z dz=dU (x, y, z),

(56)

TO

pa60Ty A<M,M,l MOJKHO

BbiiiHCJillTb,

He 3HaH

3apaHee TpaeKTopn!O

TO'-IKH M.

U OT KOOp,!l;HHaT X, y, Z,

<f>yHKUHH

,!l;HcpcpepeHUIIaJI KOTOpOH

paBeH

3JJeMeHTapHoii: paooTe, Ha3biBaeTCH cuAoeoii ¢ynKu,ueu. CnJJoBoe noJie,

,!J;JIH KOTOporo CyllJ,eCTByeT CHJIOBaH cPYHKUHH, Ha3biBaeTCH nomeHt{U· aAbflbl.M CU/WBblM nOAf.M, a CHJibl, ,n;e}ICTBy!Olll,He B 3TOM TIOJie,- nOmeH·

U,UaAbHbtMU cuJWAtu. B ,n;aJJbHelirneM CHJioBy!O ¢YHKUUIO C'-!HTaeMo,n;Ho- 3Ha'-!Hoii <\JyHKUHeH KOOp,n;HHaT.

EcJJH B cpopMyJJy (54) rro,n;cTaBHTb BbipatKeHHe dA

H3 paseHCTBa

(56), TO ITOJJY'IHM

(1>1,)

A<M,M,> = ~ dV(x, y, z)=U2-Ui,

(57)

{M,)

317

B § 88.

r,n;e U1= U (xi, Y1•

z1) H U2= U (x2, Y2• z2) - 3Ha'-leHHfl cnJioBoii <lJyHK·

UHH B TO'-!KaxM 1

H M 2 rroJIH cooTBeTCTBeHHo. CJie,n;oBaTeJihHO, pa6oma

notnelll{UClAbHOU CUAIX pasHa pa:JHocmu 31lG'ICHUU CUAOIJOU f/JyHKtfUU 8 KOHC'IHOU U llG'IClAbHOU mO'IKaX nymu U om 13UOa mpaetcmopuu 013U:J/Clj· wp1cfl mo'IKU He saeucum. I1p11 rrepeMern,eHHH no saMKHyToii TpaeKTO· pHH V2 =V1 H pa60Ta rroTeHuuaJihHoii CHJibl pasHa HyJI!o.

OcHOBHhiM csoii:cTsoM rroTeHUHaJihHoro cnJiosoro noJIH n HBJHieTc.sr

TO, '-ITOpa6oTa CHJI IlOJifl rrpH ,!l;BH:>KeHIIH B HeM MaTepHaJibHOH TO'-!KHSa• BHCHT TOJibKO OT Ha'-!aJibHOfO H KOHe'-!HOfO IIOJIO}KeHHll ~)TOll TO'-IKH H

Hll OT BH,!l;a ee TpaeKTOpHH, HH OT 3aKOHa ,!l;BH}KeHHfl He 3aBHCHT. CHJibi, pa6oTa KOTOpb!X 3aBHCHT OT BH,!l;a TpaeKTOpiiH HJIH OT 3aKOHa

,!J;BH)KeHHfl TO'-!KHIIpHJIO)KeHHfl CHJibi, Ha3biBaiOTCfl HenomeHt(UClAbflbl.M.U,

K TaKHM cHJiaM OTHOCHTCfl CHJihi tpeHufl H corrpoTHBJieHHfl cpe,n;bi. ECJIH ycTaHOBJieHo, '-ITOCOOTHOilleHHe (56) HMeeT MeCTO, TO CllJIOBaH

<lJYHKUHfl HaXO,!J;HTCfl H3 paBeHCTBa

U = ~ dA+C HJIH U = S(Fxdx+Fydy+Fz dz)+C. (58)

f1oCTOfiHHafl C 3~eCb MmKeT HMeTb JII060e 3Ha'-leHHe(KaK Blll.l,HO ll3 q>opMyJibi (57), pa6oTa OT C He saBHCIIT). O,n;HaKo o6bi'-IHOycJioBJIHBaJOT- cH C'-!HTaTb B HeKOTOpOH TO'-!Ke0, Ha3bJBaeMOH «HyJieBOH TO'-!KOll», Be- JIH'-IHHY U0 =0 H OIIpe,n;eJifi!OT C HCXO,!l;fl !13 3TOfO YCJIOBHfl.

l13BeCTHhiMH HaM rrp HMepaMH llOTeHUHaJibHbi:X CHJI f!BJifliOTCfl CHJibi Tf!)KeCTH, yrrpyroCTH H T5IfOTeHHfl ( CM. § 88). f1oKallieM, '-ITO,!l;Jifl llOJieH

3THX CHJI ,n;eftCTBHTeJibHO Cyll.l,eCTBYJOT CHJIOBble rpyHKUHH, H Haft,n;eM HX Bblpa)KeHHH. f10CKOJlbKY DO~ 3HaKOM IIHTerpaJIOB, II3 KOTOpb!X B § 88

6biJIH IlOJiy'-leHhicpopMyJibl (47), (48) H (50), CTOf!T 3JieMeHTapHbie pa60Tbi COOTBeTCTBYIOII.I,HX CHJI, TO npH,n;eM K CJie,!l;yiOII.I,HM pe3yJibTaTaM,

llCIIOJib3Yfl

paBeHCTBO (58):

1) ,!J;Jlfl

IIOJifl

C H JI hi

T

fl )K e C T H,

eCJIH OCb Z HarrpaBJieHa Bep-

THKaJihHO

ssepx,

dA =-P dz, oTKy,n;a, C'-!HTafl U=O rrpli z=O (HyJie-

BaH TO'-!Ka B Ha'-laJie KOOp,!l;HHaT),

H3XO,!J;HM

U=-Pz;

(59)

2) ,!l;Jlfl

IIOJifl

C H JI hi

y

II p y r

0 C T

II,

,n;eHCTBYIOJUeH B,!J;OJib OCH

Ox*, dA=-cx dx, otKy.n;a,

C'-IHTafl U=O rrpH x=O, Haxo,n;HM

U=-cx2 /2;

(59')

3) ,!J;Jlfl

IIOJifl

C H JI bl

T

fl r 0 T

e H H fl

d•A =mgR 2 d (1/r), OTKy,n;a,

C'-lliTafl U =0 II pH r= 00

(HyJieBaH

TO'-!Ka B

6eCKOHe'-!HOCTH), HaXO,!l;HM

r.n;e r = V

(59")

•

X 2 +Y2 +Z2

f1oJib3YfiCb Haft,n;eHHbJMH 3Ha'-!eHHf!MHrpyHKUHll U, MO)KHO ITO <lJopMyJie (57) IlOJIY'-IHTb Te )J{e Bbipa)KeHH5l ,!J;Jlfl pa60T COOTBeTCTBYIOIUHX CHJI, KOTOpbie ,n;aJOTCfl paBeHCTBaMH (47), (48) H (50)

* TaKOe ilOJie MO)KHQ Ha3B3Tb JIHHefi:HbiM; B HeM o6.'13CTb!O,B KOTOpOl:f 33Jl,3HO CU·

JIOBOe ilOJie, !IB.'l!leTC!! np!!M3!1 JIHHH!I.

318

DD' <BE', TO ,n:oJDKHO 6biTb F2>F1• CJie,n:oBaTeJibHO, CJUCileHHO CU/la e nomenl{Ua/lbHOM

no11e 60ilbtue ma,u, zae nooepx11ocmu

ypOBHfl

npoxoiJflm eyUJ,e.

OrMe'leHHhie csoil:CTsa

ll03BOJilll0T

HarJillll;HO npell;CTaBI!Tb

KapTHHy pacnpe.n:eJieHHll

CHJI B !!OTeHIJ.HaJibliOM

t:HJIOBOM noJie

c noMOIIJ.hiO noBepxHocTei'! ypoBHH. KpoMe Toro, KaK BH.D:HO H3 paBeH-

CTBa (57),

pa6oTa'llOTe!IUH3JibHOH

CHJ!bl 3aBHCHT B KOHe'!HOMC'leTeTOJ!bKO OT TOfO,

C K3KOii llOBepxHOCTH ypOBHll H H3

KaKyiO

npOHCX0)1HT nepcMCllJ.CHHC TO'!KH.

floliCHHM

cKa3aHHoe npHMepaMH.

1. ,Il,JIH O)lHOpO,n:HOfO TIOJ!ll CHJI TSDKCCTH (eM. pHC. 231), KaK BH;J.HO 1!3 <jJOpMyJihl

(59), U=const, Kor.n:a z=const. CJie,n:oBaTeJibHO, noBepxnocTl!MH ypoBHll liBJIHIOTCll ropH30HraJibHhre nJioCKCCTH. CHJia Tll:ll{eCTil P HanpaBJiena no HopMaJIH K 3THM nJio-

CKOCTliM B CTOpOHY B03paCTaHHll U H BO Bcex TO'!KaX I!OJill UOCTOliHHa.

2. ,UJIH IIOJill CHJI TSrroTei!Hll, corJiacHO ¢opMy.1e (59"), U=conot, Kor.n:a r=const.

CJie.n:oBaTeJibHO, noBepxHOCTFMH ypoBH5! liB.'liiJOTCll KOHuenTpwrecKHe c<jlephl, uellTp KOTOpbiX COBUa.n:aeT C npHTHrHBaiOIIJ.HM ueHTpOM. CHJia B KaJK.QOH TO'!Ke !!OJ!ll HanpaBJieHa no HOpMaJIH K COOTBCTCTBYIOIIJ.CH ccpepe B CTOpOHY B03paCT3Hllll U (y6biBa-

HHll r), T. e.

K

ueurpy c<jlepb!.

EcJm

B

noTeHu,HaJibHOM CHJIOBOM

noJie HaXO)l;HTCH

c H c T e M a

M a T e p

H a JI b H hi X T 0 q e K, TO

CHJIOBOH cpyHKJJ,HeH

6y)l;eT

Ta-

KaH cpyHKilHH

KoOp,!J;HHaT TO'-leKCHCTeMbl U (x1, Yt. Zt.

.,

Xn, Yn•

Zn),

)J.JIH KOTOpOH

(62)

T. e. ,!I;HqxpepeHu,HaJI KOTOpOH paBeH cyMMe 3JieMeHTapHb!X pa60T BCeX )J.eHCTBY!:Oill,HX Ha CHCTe:\-ty CHJI ITOJIH.

§ 127. OOTEHl.J,HAJibHMI 3HEPfH~.

3AI(OH COXPAHEHH~ MEXAHH'-IECI(Ofl 3HEPfHH

llJIH ITOTeHUHaJibHOfO CHJIOBOfO ITOJIH MOlliHO BBeCTH ITOHHTHe 0

noTeHu,HaJibHOH 3HeprnH KaK o BeJIH'lHHe, xaparnepn3yJ:Omefi «3anac pa60Tbl», KOTOpblM o6Jia,[J;aeT MaTepHaJibHa5I TO'-!Ka B JJ.aHHOM rryHKTe CHJIOBOrO ITOJIH. 4T06b! cpaBHliBaTb Melli,!J;Y COOOH 3TH «3anaCbi pa60Tbi», HYlliHO ycJIOBHTbCH 0 BbJ6ope HyJieBOH TO'IKll0, B KOTOpOH 6y)J;eM YCJIOBHO C'lHTaTb«3arrac pa60Tbl» paBHbiM HYJI!:O (Bbi6op HyJieBOH TO'-!KH,KaK H BCHKOfO Ha'laJia OTC'leTa, I1pOH3BO)J;HTC5I I1pOH3BOJibHO). flomerlll,UallbHOii 3Hepr!Uet1 .Mamepua!lbHOU mottKU (J aarmO.IIt noAOJ/CCHUU M Ha3bteaemcfl cKaA.RpHaJl 8CIIUttUNa IT, paeHa.R motl pa6ome, KomopyiO npous8eoym CUAbl nO/Ifl npu nepe.AteUtptuu mottKU U3 nOAO:JfCCHUfl M 8 HyAeeoe

il=A(MO)·

J.13 onpe,!l;eJieHHJI cJie)J;yeT, '-ITO rroTenuHaJihHaH

3HeprHH II 3aBHCHT

OT KOOp,!I;HHaT X, y, Z TOIIKH M, T. e. '-ITOll=ll (x,

y, z).

Ey)J;eM B )J;aJibHeiirneM C'IHTaTb HyJieBhre TO'IKH AJIH cpyHKUHH II (x,

y, z) H U (x, y, z) cosrra)J;aiOill,HMH. Tor11.a U0 =-0

H no cpopMyJie (57)

A,10 =U 0-U=-U, r)J;e U- 3Ha'INIHecHJiosoii cpyHKUHHBTO'-!KeM noJIH. TaKHM o6pa3oM,

IT (x, y, z)=-U (x, y, z),

T. e. nomeHt(UaAbHa.R :mepcu.R 8 AI06ot1 mottKe CUA080eo noA.R paeHa 3HatteHUIO CU/108011 c/JyHKf1UU 8 3mot1 mOttKC, 83.RnlOMY C o6pamliblM 3/iaKOM.,

320

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