(формат)Тарг

§ 116. TEOPEMA OS U3MEHEHUU fJIABHOfO MOMEHTA I(OJIUliECTB ,ll.BU)l(EHU~ CUCTEMbl (TEOPEMA MOMEHTOB)

TeopeMa MOMeHTOB, ,n:OKa3aHHafl .li:Jlfl O}l.HOH MaTepHaJibHOH TOqKH

(eM. § 85), oy,n:eT crrpaBe,n:JIIlBa ,n:Jifl Kam.n:oii: ll3 TOqel{ CllCTeMbi. CJie}l.O- BaTeJibHO, _eCJill paCCMOTpeTb TQqKy CllCTeMbl C MaCCOll tnR,, llMeiO~yiO

CKOpOCTb VR,, TO }l.Jifl Hee oy,n:eT

d -- ----

dT [m0 (mkvh)] = m0 (F%) +m0 (Ft),

r.n:e F% n Fi1 - paBHo,n:eii:cTByiO~He ncex BHernHHX H BHyTpeHHHX CHJI, ,n:eiicTBYIO~llX Ha ,n:aHHYIO TOqKy.

CocTaBJiflfl TaKne ypanHeHHH ,n:mi scex ToqeK cncTeMbi H CKJia,n:br-

bafl nx rroqJieHHo,	rroJiyqnM
d	-	-	-	-	-
dT [~m0		(mkvk)] =	~m0	(Fk) +~m0	(Fj,).

Ho rrocJie,n:Hflfl cyMMa no CBOHCTBY BHyTpeHHHX CHJI cncTeMbi panHa Hymo. Tor.n:a, yqnTbiBaf! paBeHCTBo (30), Haii}l.eM oKoHqaTeJibHO

(35)

0o,1yqeHHOe ypaBHeHHe BbipamaeT CJie}l.yiOIII,yiO T e 0 p e M y M O- M e H T 0 B .li:Jlfl CIICTeHbi: npOU3800HaJi no 8peMeHU om 2/1{]8fl0W MOMeHma KO!luttecme 06U;){CeHufl cucmeMbt omHocumellbHO HeKomopoco rtenooeu;){CHOco [1eHmpa pasHa cyMMe Aw.weHmoe ecex eHeumux cull cucme.Mbt omHocume!lbHO moe-o ""e l{eHmpa

npoeKTHPYfl o6e qacTll paseHCTBa (35) Ha HeiTO}l.BlllliHhie OCll Oxyz, TIOJiyqiiM:

dKx _ ~	(F-e) dKy _ ~	(F-e)	dKz _ ~	(F-e)	(36)
dt - "-'mx	'' ---a-r-- "-'my	k	' dt - "'mz	k •

YpaBHeHI1f! (36) BbipalliaiOT TeopeMy MOMeHTOB OTHOCHTeJihHO JIIOOOH

HenO}l.BlllliHOH OCll•

.UoKa3aHHOH Teope'110H WllpOKO TIOJ!h3YIOTC5I rrpll ll3yqeHHH Bpa~a­ TeJibHOrO .n:nnmeHIIfl TeJia, a TaKme B Teopnii rnpocKorra n B Teopnn y.n:apa. Ho 3HaqeHne TeopeMbi 3THM He orpaHnqnsaeTCfl. B KHHeMaTHKe Ob!JIO TIOKa3aHo, qTO }l.Bll)KeHIIe TBep.n:oro TeJia B oo~eM CJiyqae CJiaraeTCfl ll3 llOCTynaTeJihHOrO ,n:BllllieHIIfl BMeCTe C HeKOTOpbiM TIOJIIOCOM

H spaiii,aTeJihHoro .n:snmeHHfl BOKpyr 3Toro noJIIOca. EcJIH sa TIOJIIOC BbiOpaTh u.er-ITp MaCC, TO TIOCTyrraTeJihHaf! qaCTb ,n:Bll)KeHll5I TeJia MOllieT

OhiTh n3yqeHa c noMOIII,hiO TeopeMbi o .n:snmeHHH u.eHTpa Mace, a spa- ~aTeJihHafl - c TIO'I10~hl0 TeopeMbi MOMeHTOB. Eho noKa3hiBaeT namHOCTh TeopeMbi }l.JIH H3yqeHHfl ,n:BmKeHIIH CBOOO.li:HOrO TeJia (JieTfl~llH

C3\10JieT, CHapH,n:, paKeTa; CM. § 132) II, B qaCTHOCTII, .li:JI51 ll3yqeHll5I llJIOCKOTiapaJIJieJibHOrO }l.BllllieHllfl (CM. § 130).

OpaKTHqecKafl U.eHHOCTb TeopeMbl MOMeHTOB COCTOHT eiii,e B TOM, qTo OHa, aHaJIOrllqHO TeopeMe 00 ll3MeHeHllll KOJIHqecTBa ,n:BHllieHHH, I103B0-

292

§ 117. 3AKOH COXPAHEHM.SI fJIABHoro MOMEHTA

I(OJIHI.JECTB ,ll.BH)I(EHHSI

J13 TeopeMbl MOMeHTOB MOLKHO TIOJiyqHTb TaKHe CJie}l.CTBll5I.

I. flyCTb cyMMa MOMeHTOB OTHOCllTeJibHO rreHTpa 0 BCeX BHelliHllX cnJI, }l.eficTBYIOlllllX Ha cncTeMy, paBHa HYJIIO

'2.m 0 (F'k)=O.

Tor,'I.a H3 ypaBHeHim (35) cm;n.yeT, qTo npll3TOM J(0 =const. TaKHM o6pa30M, ecAu cyJIHta MO.M.eHmae omnocumellbHO oannoe-o u,enmpa ecex npuAO'JICCHHblX K CUcmeMe 8HCUlHUX CUA paeHa HYAIO, mo 2Aa8HblU MOMeHm KaAwtecme aeu'JICeHUR cucmeMbt omnocumeJ!bHO Bmoco u,enmpa 6yaem ttucAeHHa u no nanpael!eHu/0 nocmoRnen. OpnJioLKeHne noro pe3yJibTaTa K cJiyqaiO }l.BHLKeHHH nJiaHeThr ohiJIO paccMoTpeHo B § 86.

2. flycTb BHelliHllC CHJihi, .n.e:fiCTBYIOIU;He Ha CllCTeMy, TaKOBbi, qTO cyMMa HX MOMeHTOB OTHOCHTeJihHO HeKOTOpofi HeTIO)l.BllLKHOll OCll Oz pasHa nymo

'2.mz(Fk) =0.

Tor.n;a 1!3 ypaBHeHnii (36) cJie,n;yeT, qTo npn 3TOM Kz=const. TaKHM o6pa30M, ecAU cyMMa MOMeHmoe ecex aeiicmey10wpx na cucmeMy enemnux cuA omnocumeAbHO KaKaii-nu6yob ocu paena HYJ!to, mo cAaeHbtii MOMenm KaAuttecme oeu'JICeHUfl cucmeMbt omnocumeAbHO Bmoii ocu 6yaem eeAuttuHou nocmORHHOU.

3TH pe3yJihTaThi BblpaLKalOT COOOH 3a1WH CaxpaHCHUfl cJ!afJHOcO AW·

.M.eHma KOJ!uttecme aeu'JICeHufl cucmeAtbt. 113 Hnx cJie}l.yeT, qTo BHyTpemnre CllJibl II3MeHllTb r JiaBHblll MOMeHT KOJillqecTB )J;BllllieHHfl CllCTeMbi He MoryT.

C JI y q a if B p a lll a 10 lll e fi c H c n c T eM hi. PaccMoTpnM CllCTeMy, BpalllalOIU;YIOCfl BOKpyr HeTIO)J;BllLKHOH (HJill npOXO)J;Hlllell qepe3 ueHTP Mace) ocn z. Tor.n;a no cpopMyJie (32) Kz=J zW. EcJIH B 3TOM cJiy-

qae 'Smz(Fk)=O, TO

J zW=COTISf.

0TCIO,'I.a npllXO)l.llM K CJie,n;yiOIU;llM Bb!BO)J;aM:

a) eCJill CllCTeMa HCU3Atefl5leMa (aoCOJIIOTHO TBep}l.Oe T~JIO), TO J z=

=const n, CJie}l.oBaTeJibHO, w=const, T. e. TBep}l.oe TeJio, 3aKpenJieHHoe Ha OCII, BpalllaeTCfl B 3TOM CJiyqae C IIOCTOHHHOH yrJIOBOll CKOpOCThiO; 6) eCJill CllCTeMa U3MCH5leMa, TO TIO}l. .n.eifCTBHeM BHyTpeHHllX (HJIU BHeUIHHX) CHJI OT,ll.eJibHb!e ee TOqKll MOryT y}l.aJif!ThCH OT OCH, qTO Bhl3biBaeT yseJinqeHHe J n HJIH npn6mnKaThCH K ocn, qTo npnBe}l.eT K yMeHbllleHHIO J z• Ho TIOCKOJibKY J zW=const, TO npn yBeJIHqeHHll MO·

294

MeHTa HHepUMH yrJIOBa11 CKOpOCTb CHCTeMbl 6y,n;eT yMeHhiiiaTbCH, a

nprr yMeHhiiieHrrH MOMeHTa HHepuHrryseJirrqrrsaThCH. TaKnM o6pa30M, .n:eiicTBHeM BHyTpeHHHX CHJI MO)KHO H3MeHHTb yrJIOBYIO CKOpOCTb CHCTeMbl, T8K KaK DOCTOHHCTBO K z He 03HaqaeT B005UJ,e DOCTOl'!HCTBaW.

PactMOTPHM	HeKOTOphle npHMepbl.
a) 0 ll hi T hi	C D JI a T ljJ 0 p M 0 H )K y K 0 B C K 0 r 0. ,[{Jill ,::(eMOHCTpaUHH

SaKOHa CoxpaHeHHll MOMeHTa KOJIHIJeCTB ,l(BmKeHHll y.u;o6HO IIOJib30BaThCll llpOCTiiM

llpH6opOM, Ha3b1Ba€Mb!M «llJiaT!jJOpMOH )I\yKOBCKOrD». 3To KpyrJiall rOp1130HT8JibHaH I!JiaT!jlOpMa Ha mapHKOBb!X OIIOpHb!X llO,l(lllHllHHK3X, KOTOpall MO:lKeT C ~!aJ!biM TpeHHeM BpaJJJ;aThCll BOKpyr BepTI!Ka.zibHOll OCH Z. )J.JIH qeJioBeKa, CTOl!JJJ;ero Ha TaKOH

nJiaT!jlopMe, :E mz(F'k)= 0 H, CJie.u;oaaTeJibHO, J zro=const. EcJIH IJeJioBcK, pa3se.u;ll pyKil B CTOpOHbl, COOOJJJ;HT ce6e TOJIIJKOM BpaJJJ;~HHe BOKpyr BepTHKaJib!IOH OCH, a 3a-

TeM OllYCTHT pyKH, TO BeJ!HlJUHa J z yMeHblllaeTCll H, CJie/I.OBaTeJibHO, yrJIOBall CKQpOCTb spam:eHml B03pac'i'eTTaKI:!M. cnoco6oM )'BeJIMIJeHHllyrJioaml: cKopocTH m~~tpcOKO IIOJih3YIOTCH B OWieTe, llpH Dpbi>KKaX B B03.!lyXe (caJibTO) H T. ll.

)laJiee, IJeJIOBeK, CTOHJJJ;HM Ha nJiaT!jl.opMe H€llOJl.BH:lKHO (Kz=0), MO:lKeT llOBepHYThCll B mo6y10 cTopoHy:, apamall BhiT!IHYTYIO ropnsoHTaJibHo PYKY B nporrrsono-

JIO:lKHOM	HanpaaJieHHH.	YrJIOB&ll	CKOpOCTb IJeJIOBeKa npH STOM 6y.u;eT T&KOH, 'lTOObl
B cyMMe	BeJIH'lllHa Kz	cHCTeMhl	ocraJiacb paaHoii HYJIIO.
6) p	a C K a lJ H B a II 1'1 e		K a lJ e J1 e H. ,UasJieHHeM HOf (CHJ13 BHyTpeHHllH)

qe,10BeK, CTOllW:HH Ha K8lJ€JillX, paCK3IJ8Tb llX He MO:lKeT. Ci!.€JI8Th STO MO)!(HO CJie/l.yiO-
lliHM o6pa3oM. Kor)la KaiJeJIH HaXO/I.llTCll B JieBoM aepxHeM noJio:lKeHHH A0 , 'leJioBeK
npHce.aaeT. llpn npoxol!K/I.eHMH qepes sepTHii8Jih oH 61rcrpo Bblnp!!MJilleTcll. Tor.aa
M8CCbi llpH6JIH:lKaiOTCll K OCR BpaJJJ;eHH!I Z, BeJIHIJHHa J z yMeHblllaeTCll ll yrJIOBall CKQ-
pOCTb (I) CKa'IKOMB03p3CT&eT. 3TO yBeJIHIJe!lHe (J;) npHBO)lHT B KOHelJHOM CIJeTe K TOMy,
'ITO KaiJeJIH no.u;HHMYTC11 Bhillle HaiJaJibHoro ypoBHll A0 • B rrpasoM	sepxHeM noJio-
:lKeHHH, Korna ro=O, IJeJioBeK OIIHTh rrpnce).laeT M lz yaeJiniJHBaeTcll,	HO w ocraeTCll

paBHOH HYJIIO; npH	npOXO:lKJl.eHHll qepe3 BepTHKaJib Oil CHOBa Bb!IIpllMJil!eTCll 1:1	T• .1{.
B pesyJihTaTe pa3Maxn Ka'leJieH 6ynyr aospacTaTh.		napa-
llpollCXOil,llJJJ;He	Tipll STOM BbiHY:lK/leHHble KOJI€C3Hllll KaiJeJieii: Ha3biB<HOTCll

J.tempu'leCKUJ.tU, TaK KaK OHII COBeplliiiiOTCli He DO,Il /l.eHCTBHeM nepnOJl.HlJ€CKH M€Hll10llleHcl! CHJ!bl (CM. § 96), a BCJle)lCTBHe nepHO)lll'leCKOfOH3MeHellHll napaMeTpOB CHCTe-

J..Ibl: ee MOMeHTII HHepll,HH	li I!OJIO:lKeliHll	ll€HTpa Tll:lKecTH.
B) p e a K T H B II bl ll	M 0 M e II T B H H T a. ECJill pacCM&TpHBaTb Kopnyc Bep•
TOJieTa (m.tecre c naurareJieM), ero BHHT n		ortipachlBaeMyiO Macey aoanyxa Kak: a,u,Hy
CllCTeMy, TO CH.1bl B3aHMO)leHcTBllll Me:lKJl.Y		)lBHraTeJieM H BHHTOM ll Me:lK,I\y BliHTOM

H B03JlYlliHOH Cpe)lOH 0YilYT ,!\Jill 9TOH CllCTeMb! BHyTpeHHHMH H He MOfYT H3MeHHTb ee cyMMapHbii1 MOMeHT KOJJHIJecTB ,i;BH:lKeHHll, paBHhlll Jl.O nycKa JJ;BHfaTeJill HYJIIO.

DOSTOMY Kopnyc aeprOJieTa )IOJI:lKeH na'laTh spaw:aTbCli B cropoay, npornBOITOJIOJKHyiO HanpaBJieHHIO Bpam,eHH!I BllHTa H 803/lYIIIHOH cpe)lbl. )leii:CTBYIOJJJ;llH npH 9TOM Ha BepToJieT BpaJJJ;aiOJJJ;llH MOMeHT Ha3hiBaiOT peaKmUB!iblM MOMeHTOl\1. 4To6hi npe)lOTBpaTHTb peaKTHBHOe BpaJJJ;eHHe Koprryca O)(JWBHHTOBOrD BepTOJieTa, Ha ero XBOCTOBOK <JaCTH ycTaHaBJIHBaiOT COOTBeTCTBYlOJJJ:HH pyJieBoii: BHHT. Y MHOfOBHHTOBOfO BepTOJieTa BHHTbl ,l(eJiaiOT BpaJJJ;aiOJJJ;HMHCll B pa3Hble CTOpOHbl.

llOi!BJIEmHe peaKTllBHOfO MOMeHTII MO:lKHO llCllOJib30BaTh )lJill SKCIIepHMei!TaJihHOrD orrpe)leJieHHll BpaJJJ;aiOJJJ;ero MOMeHTa aBHaiJ;HOIIHOfO /lBHraTeJI!I, TIIK KIIK STH MOM€HTbl paBHbl .u;pyr .u;pyry DO MO)I.yJIIO, a peaKTllBHbiH MOMeHT MO:lKHO H3MepHTh, ycTaHOBHB )lBHraTeJib C Bpalli8KllliHMCll BHBTOM Ha COOTBeTCTBYIOW,llX BeCaX.

fJiasa XXV

TEOPEMA OB H3MEHEHHH

KHHETHlJECKOfl 3HEPfHH CHCTEMbl

§ 121. KHHETJP.JECKMI 3HEPfiUI CHCTEMbl

I(unemut.tectwii 31iep2Ueii cucmeJrtbl nasbleaemcst ctWAstpliast eeAut.tuna

T, paertafl cy.MAte KUrtemUt-tecKux artepcutl ecex motteK cucmeJrtbt:

T = "J:.mkvU2.

(41)

KmreTHtJ:ecKaH 3HeprHH HBJIHeTcH xapaKTepHCTHKoil H rroCTyrraTeJihRoro, H Bpalll,aTeJibHOfO )l.BHJKeHHH CHCTeMbi. f JiaBHOe or.TJJiqHe BeJIH'l:H- Hbl T oT BBe)l.eHHbiX paHee xapaKTepHcTHK QH Ro cocToHT B TOM, qTO KHHeTHtJ:eCKaH 3HepmH HBJIHeTCH BCJIH'l:HHOHCKaJIHpHOll H rrpHTOM cy- I.!J,eCTBeHHO ITOJlO)f{HTeJibHOil. Tio3TOMY oHa ne 3aDHCHT OT HanpaBJieHHii: )l.BH.tKeHHH tJ:aCTefi CHCTeMbl H He XapaKTepmyeT H3MeHeHHH 3THX HanpaBJieHHH.

OTMeTHM elll,e cJie)l.yiOw,ee BalliHoe oocTOHTeJibCTBO. BHyTpeHHHe

CHJibl )l.CHCTBYIOT Ha qacTH CHCTeMbl ITO B3aHMHO rrpOTI1BOITOJIO.IKHbiM HanpaBJICHllHM. Tio ~noli npHqHHe OHH, KaK Mbi Bll)J,CJI!l, lie H3:-ACI!HIOT BCK-

TOpHbiX xapaKTepHCTHK QH K0 . Ho ecJIH rro)J. )J.eiicTBHeM BHyTpeHHHX CHJI OYL\YT H3MCHHTbC5l MOJl.YJIH CKOpOCTeH TO'l:eKCHCTCMbl, TO rrpH 3TOM 6y.n.eT H3MeHHTbCH H BeJIHqHHa T. C.Jie)J.OBaTeJihHO, KHHeTHtJ:ecKaH 3Hep-

301

(J) -

rmr cucTeMbi oTJJH'IaeTcHOT BeJilPHIH Q H Ko eme n TeM, qTo Ha ee H3Me-

HeHHe BJJHHeT ,n;eifcTBHe H BHeiiiHHX, H BHyTpeHHHX CHJJ.

£CJJH CHCTeMa COCTOHT H3 H€CKOJibKHX TeJI, TO ee KHHeTUtieCKaH

3HeprHH paBHa CYMMe KHHeTuqeCKHX 3Hepri1H 3TUX TeJJ.

Haf:i,n;eM cpopMyJJhi ,n;JIH BbJqHcJJeHHH KHHeTnqecKoii 3HeprHH TeJia B pa3Hb!X CJJyqaHX ,!l;BU:>KeHHH.

1. fl 0 C T y I1 a T e JJ b H 0 e .li. B H :1K e H H e. B 3TOM CJJyqae BCe

TOqKH TeJJa ,!J;BH:IKYTCH C 0,/J;HHaKOBb!MU CKOpOCTHMH, paBHb!MH CKOpOCTH

ueHTpa Mace. CJJe,n;oBaTeJJbHo, ~JJH	mo6oif ToqKn vk =Vc n	lflopMyJJa
(41) ,n;aeT
HJJH
T noCT =	Mvi:;;2.	(42)

TaKHM o6pa3oM, KttHemu'leCKafl :mepzufl meAa npu nocmynameAbHOM oeu:J~CeHuu pasHa noAoeuHe npouseeaeHufl MacCbl meAa Ha Keaapam cKopocmu tteHmpa Mace.

2. B p a m a T e JJ h H o e ,n; B H :>K e H n e. EcJJn TeJJo apamaeTCH BOKpyr KaKOH-HH6y,n;h OCU Qz (eM. puc. 295), TO CKOpOCTh moOOH ero TOqKU V11.=ffihk, r,n;e hh- paCCTOHHHe TOqKu OT OCU BpameHHH, a

yrJJOBaH CKOpOCTb TeJJa. fiO,!l;CTaBJIHH 3TO 3HaqeHHe B cpopMyJJy (41) H Bb!HOCH o6mne MHOlliHTeJJU 3a CK06KU, I10JJyqHM

T ep = 'Lmkffi2 hf/2 = (Smkh'k) ffi 2 /2.

BeJinqnHa, cToHmaH B cKo6KaX, npe,n;CTaBJJHeT co6of:i MOMeHT HHep-

uun TeJJa OTHOCHTeJJbHO OCH Z. TaKUM o6pa30M, OKOHqaTeJJbHO Hafi,n;eM

T ep = Jz(J) 2/2,

(43)

T. e. KUHemw-tecKaft :mepcuft meAa npu epaw.ameAbHO.M aeu:J~CeHuu pasHa noAoeuHe npouseeoeHufl MOMeHma uHept{uu meAa omHocume.AbHO ocu epaw.eHufl Ha Keaapam eco yZAoeou cKopocmu.

3. fl JJ 0 C K 0 I1 a p a Jl Jl e JJ b H 0 e ,!J; B H )K e H U e *. fipH 3TOM ,!J;BH:~KeHHH CKOpOCTU BCeX TOqeK TeJJa B Ka:>K,!l;b!H MOMeHT BpeMeHH pac-

npe,n;eJJeHbl TaK, KaK ecJJH 6bJ TeJJO Bpaii.J.aJJOCb BOKpyr OCll, nepneH)I,H- KYJJHpHOH nJJocKocTn ,!J;BH:~KeHHH n rrpoxo,n;Hmef:i qepe3 MrHoBeHHblii

ueHTP	cKopoCTef:i	P (puc. 303). CJJe,n;oBaTeJihHO, no cpopMyJJe (43)
		T ll:'!OCK = Jpffi2 ;2,	(43')
r,n;e J p	- MOMeHT	HH€pi.J.HH TeJia OTHOCHTeJJhHO	Ha3BaHHOH BbJ!lle OCH;

(J) - yrJIOBaH CKOpOCTb TeJJa.

BeJJnqHHa J p B cpopMyJJe (43') 6y.n;eT rrepeMeHHoil, TaK KaK noJJo.meHHe I.J.eHTpa p rrpH ,!l;BH:>KeHHU TeJJa BCe BpeMH MeHHeTCH. BBe,n;eM BMeCTO

J p I10CTOHHHhtH MOMeHT HHepi.I.HH J c OTHOCHTeJJbHO OCU, npOXO)I,Hmeii qepe3 I.J.eHTp MaCC C TeJJa. flo TeopeMe fl:oifreHCa (eM. § 103) J p=J c+

+Md2 , r,u:e d=PC. Tio,n;CTaBHM 3TO Bbipa:>KeHHe ,!J;JJH J p B (43'). YtiHThl-

* 3TOT cJiyiJaH MO)KeT 6h!Tb TIOJiyiJell KaK IJaCTHblil H3 paCCMOTpeHIJOfO B CJie)zy.l()o meM nyHKTe o6111.e1o cJiyiJall .U.BH)KeHHll TBep.n:oro reJia.

302

BaH, tfl'OTOtJKa p - MrHOBeHHhJH ueHTp CKOpOCTeH H, CJie,n;oBaTeJihHO,

wd=ro ·PC=Vc, r,n;e Vc - CKOpOCTh	ueHTpa Mace C,	OKOHtJaTeJibHO
naii,n;eM	+ J cuP/2.	(44)
T nJioc~< = MvV2

CJie,!J;OBaTeJihHO, npu nAOCKOnapaA-1eAbHO.M aeuJ/CeHUU KUHemutteCKafl rmepeUR tne/la pa8Ha 8Hepeuu nocmyname/lbflOCJO 8euJ/CeflltR CO CKOpOCmbiO l{ellmpa M.aGC, CAOJ/CeHHOa C KUHemU'1£CKOa 3Hepeuea epawpmellbfl020 8euJJCeHUR eoKpye t~enmpa Atacc.

PHC. 303	:c
	PHc. 304
4 *. 0 6 rn. u i1	c .r1 y tt a i1 ,IJ; a 11 )!{ e H 11 H. EcJJH aw6paTh

ueHTp Mace C TeJia B KatteCTBe nomoca (pHc. 304), TO ,n,amKeHHe TeJJa B OOlll.eM CJiyttae oy,n;eT CJJaraThCH !13 llOCTynaTeJihHOfO CO CKOpOCTbiO

Vc nomoca u apamareJJhHoro BOKpyr MrHoBeHHoil ocH CP, npoxo.n;HJUeii: ttepe3 3TOT nomoc (eM. § 63). TipH 3TOvt, KaK noKa3aHo B § 63, cKopoCTh vk JIIOOOH TOtJKH TeJJa Bk CJJaraeTCH !13 CKopocTH Vc noJIJOCa H CKOpOCTH,

KOTOPYIO TOttKa noJiyttaeT npu apameHHII TeJia BoKpyr noJIIOCa (BoKpyr ocu CP) H KoTopyiO Mbi o6o3HaiiHM vk, T. e. v;. =vc+~. TipH 3TOM no Mo,a;yJIIO vk_=ffihk, r,n;e hk- paccroHHHe TOtJKH B" oT ocn CP, a ffi -

yrJ10BaH CKOpOCTh TeJia, KOTOpaH (CM. § 63) He 3aBHCHT OT BhJOOpa llO· JIIOCa. Tor,n;a *

v: = V~ = (vc +~)~ = Vt+Vk2 + 2vc."vi,.

Tio,n;cTaB.'lHH3TO 3HatteHue v~ a paaeHcrao (41) u YIIHTbiBaH, ttTo vk= =ffih~t, Haii:,a;eM

T = (1:mk) vV2 +(1:mkh%) ffi 2 /2 +vc1:mkv(,,

f)I.e OOIUHe MHOlliHTeJJH cpa3y BbiHeCeHbl 3a CKOOKH.

B nOJJyiieHHOM paaeHcrae nepaaH CKo6Ka .n;aeT Macey M TeJJa, a BTOpaH paBHa MOMeHTY IIHepUliH J CP TeJJa OTHOCHTeJihHO MfHOBeHHOll

* 113 onpe,n:eJieHHll cKa.1l!pHoro npOH3Be.:~eHHll .n:Byx BeKTopoB cJie.n:yeT, 'ITO fl=iJ. V~UVCOS 0°=V2 , T. e. CKa.1llpHblH KBa)lpaT BeKTOpa paBeH KBa.npaTy ero MO_ny,'lll.

JroT pesyJibTaT 3,n:ech HCllO.'lb30BaH;Mb16y)l.eM llOJib30BaThCll HM 6e3 OfOBOpOK H B

,~ta.'IbHeiuueM.

303

§ 110).

ocu CP. BeJIWIHHa *e 'Smuvk=O, TaK KaK ona rrpe.n.CTaBJIHeT co6oii KOJIH'IecTBoJI.BH}KeHIUI, rroJiyqaeMoe TeJioM npu ero apamemm BOKpyr ocH CP, npoxo.n.mJLeii qepe3 ueHTp Mace TeJia (eM.

B pe3yJibTaTe oKoH'laTeJibHO rroJiy'IHM

T = Mv~/2+ lcpffi2 /2.

(45)

TaKHM o6pa30Vl, KUHemUt-teCKaJl :mepcUfl meAa 8 06U{e.M cAyttae aeu-

~eflUfl (B '-!aCTHOCTH, H I1pH IIJIOCKOI1apaJIJI€JibHOM }l.BIDKeHIIH) paeHa KU!lemtttteC/WU :mepi!UU nocmynameAbfW20 OBU~eflUfl CO CKOpocmbfO l{eflmpa .Mace, CAO~e/-UlOU C KUHemutteCKOU :mepzueii 13paU{ameAbH020 013U~e­ HUfl eoKpyz ocu, npoxoaflU{eii ttepes WHmp Mace.

Ecmr 3a nomoc Il3HTb He neHTp	Mace C, a KaKyr-o-HHOYJI.b ,n,pyryJO
TO'IKYA TeJia H MriiOBermaH ocb AP	rrpH 3TOM He 6y.n.eT Bee npe:..m npo-

XOJI.IITb 'Iepe3UeHTp Mace, TO }l.JIH 3TOH OCH 'Smh.Vk=/=0 H cpopMyJihl BH)l.a (45) Mbl He IIOJIY'IHM.

PaccMoTpHM npnMepbi.

3a,~~;aqa	136.	Bhi'!HCJIIlTh Klll!CTJI'!CCKYJO 9Heprmo		KaTliW:erOC5! OC3 CKOJ!h)KCHHll
CIIJ!OlllHOfO	UHJI!!H,ll,pH<!CCKOrO KOJieca Maceo£'! M, CCJIH			CKOpOCTb ero UCHTpa		paBHa
t'c (rM. pHc. 308, a).
p c Ill e H II		e	KoJieco COBCplllaCT IIJIOCKOIIapaJIJieJihi!Oe ,ll,Bil)KCHH€. no			¢op-
MyJie (44)	HJIH	(45)
0tl!T3CM KOJieCO CIIJIOillHbiM O.QHOpO.QHblM UHJIHH,ll,poM; TOf,ll,a (CM. § JQ2)						Jc=
=MR212, r,Qe R -			pa,n,Hyc KOJieca. C ,Qpyroil: CTOpOHbl, TaK KaK		TO•IKa 8 SIDJI5!CTt'll
1J.JI5! K0,1eCa	MfJIOBCHHblM UCHTpOM CKOpOCTeH, TO Vc=W·BC=wR,				OTKy.n:a CfJ=VcfR.
ilO,I\CTaBJISISI BCC			9TH Slla'!CHH5!, I!aH.QCM

T = Mv~f2+ MR 2t•U4R 2 = (3/4) Mv~.

§ 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