(формат)Тарг

T. e. npu nocmynameAbHOM nepertocrtoM iJouO/Certuu a6coAromnoe yCiwpertue mOLtKu paerto zeoMempuLtecKotl cyMMe omnocumeAbrtozo u nepertoc-

noco	ycKopertuii.				PesyJihTaT			3AeCh aHaJiornqeH	TOMy, KOTopbiH		AaeT
TeopeMa 0			CJIO)!{eHHH				CKOpOCTefr.
B hi q H C Jl				e H H e			0 T H 0 C H T e Jl h H 0 ro,			IT e p e H 0 C H O-
r o	n	K o p n o JI n c o B a						y c K o p e H n li.	0THOCHTeJihnoe		ycKo-
penne,		llOCKOJihKY				np H
ero	Haxo~eHHH ABHJKeaJ
HHe	nOABH)!{HbiX				ocefr		BO
BHHMaHHe			He	npHHHMa-
eTC5!, BhlqHCJI5IeTC5I 06biq-
HhiMH		MeTOAaMH KHHeMa-
THKH TOqKH (§ 40, 43). Oe-
peHOCHOe ycKopeHHe						Bhi-
qnCJI5IeTC5! KaK ycKOpeHHe								PHc.		190
TOqKH,HeH3MeHHOCB5!3aH-

HOH c nOABH)!{HbiMH ocwvm,

T. e. KaK ycKopeHne ToqKH HeKoToporo TBepAoro TeJia, no cpopMyJiaM, noJiyqeHHbiM AJI5! ycKopeHHH ToqeK TBepAOrO TeJia B § 51, 58, 62, 63, KopnoJincoso ycKopenHe BhrqucJI5IeTC5I no cpopMyJie (g4). MoAyJih

KopnoJincosa ycKopeHH5!, ecJIH yroJI Me)!{AY seKTopaMH w n V0 -r	o6os-
naqHTh qepes a, 6yAeT paseH
aKop = 21 w I·IV0 -r Isin a.	(g6)
HanpaBJieH BeKTOp aKop TaK )!{e, KaK H BeKTop w X V0 n T. e.	nep-
neHAIIKYJI5IpHo rrJiocKocTH, npoXOA5IW.eii qe~s ~KTOphr w H voT•	B TY

CTOpoHy, OTKyAa KpaTqafrruee COB\ielll,eHHe W C V0 -r BHAHO npOHCXOA5Ilii,HM npOTHB XOAa qacoBOH CTpeJIKH (p HC. }gO, a).

Hs pnc. 1go, a BHAHO TaK)!{e, qTo nanpaBJieHHe BeKTopa aKOP MO)!{HO onpeAeJIHTh, cnpoeKTHposas BeKTop V0 r Ha nJioCKOCTh IT, nepneHAHKYJI5IpHyiO w, H nosepHys :ny npoeKUHIO O:r:I. Ha goo B CTopoHy nepeHOCHoro Bpalll,eHH5!.

EcJIH OTHOCHTeJihHa5! TpaeKTOpH5! - nJiocKa5! KPHBa5! n nepeMelii,a-

eTC5I BCe BpeM5! B CBOefr nJIOCKOCTH, TO yroJI a=gQo (pHC, }gO, 6)			H B
&TOM cJiyqae	no	MOAYJIIO
		aKop = 21 w 1·1 Vo-r I·	(g6')
KpoMe Toro,	KaK	BHAHO H3 pHc. 1g0, 6, nanpasJieHrre G:ov MO)!{HO B
3TOM CJiyqae	HaHTH, nosepHyB BeKTOp OTHOCHTeJihHOH CKOpOCTH		0:,-r
Ha goo B cTopoHy		nepeHocuoro spaw.eHH5! (T. e. no XOAY HJIH npoTHB
qacOBOH CTpeJIKH,		B SaBHCHMOCTH OT HanpaBJieHH5! Bpalll,eHH5!).

Ha pHc. 1g1 AJI5! HJIJIIOCTpaunrr npnBeAeHHhiX npaBHJI noKa3aHo HanpasJierme KopHoJincosa ycKopeHHH wapnKa M, ABH)!{ylii,eroc5! BAO.llh

Tpy6KH AB B cJiyqa5Ix, KOrAa Tpy6Ka spaw.aeTC5! B nJioCKOCTH qepTe)!{a (pnc. 1g1, a) n Kor;~,a ona npn spaw.ennn orrnchiBaeT KoHyc (pnc. 1g1, 6).

l13 cpopMyJib! (g6) BII,.QHO, qTO KOp HOJIHCOBO ycKopeHHe MOLKeT o6palii,aThC5! B HYJib B cJieAYIOI11HX cJJyqafiXl

163

1) KOr,I~a

(1}=0, T. e. KOr,I~a nepeHOCHOe ABHLKeHIIe HBJIHeTCH llOCTY· naTeJibHh!M (cpOpMyJia (95)] HJIH eCJIH nepeHOCHaH yrJIOBaH CKOpOCTb B ,D.aHHhiH MOMeHT BpeMeHH o6par.u;aeTcH B HYJih;

a) g B

OJ

I aKoo 1

		Pnc.		191
2) KOrAa	V0 TH=0, T. e. KOfAa			OTHOCHTeJibH3H	CKOpOCTb B AaHHbiH
MOMeHT BpeMeHH		o6par.u;aeTC5!	B	HyJih;
3) KOfA3 a=O, HJIH a= 180°,			T. e. K0fA3 OTHOCHTeJihHOe ABHLKeHHe
npOHCXOAHT	no	HanpaBJieHHIO,		napaJIJieJibHOMy	OCH nepeHOCHOfO

Bpar.u;eHH5!, HJIH eCJIH B )l.3HHbiH MOMeHT BpeMeH!l BeKTOp V0 -r napaJIJieJieH :JTOH OCH.

Pa3,IJ.eJI Tpendi

):(HHAMHKA TOlfKH

fJiasa XV

BBE,L\EHHE B )J.HHAMHI(Y. 3AI(OHbl )J.HHAMHI(H

§ 73. OCHOBHbiE noH~TH~ H OnPE,ll,EJIEHH~

P,UHG.MUKOU Ha3biBaeTCH pas,n;eJI MeXaHIIKII, B KOTOpOM II3y'laeTCSI )J.Bll)KeHIIe MaTepua,'IbHbiX TeJI ITO,!\ ,n;eiicTBIIeM CIIJI.

,UBII2KeHIIe TeJI C 'IHCTOreoMeTpH'leCKOH TOqKH 3peHHH paCCMaTpH-

BaJIOCb B KHHeM:aTITKe. B )l;HHaMHKe, B OTJIH'lHe OT KHHeMaTIIKH, npK II3y'leHIIII)l;BH2KeHIIH TeJI npHHHMaiOT BO BHHMaHIIe KaK ,n;eHCTBYIO~He

Ha HHX CHJibi, TaK H HHepTHOCTb CaMIIX MaTepiiaJibHbiX TeJI. 00H5.!THe 0 eiiJie, KaK 06 OeHOBHOH Mepe MeXaHH'leeKOfO)J.eHCTBHH,

OKa3hiBaeMOfO Ha MaTepiiaJibHOe TeJio, 6hiJIO BBe,n;eHO B CTaTHKe. Ho B eTaTHKe Mbi He KaCaJIIIeb Bonpoea 0 B03M02KHbiX II3MeHeHIIHX ,n;eHCTByioUJ,IIX eiiJI C Te'leHIIeMspeMeHII, a npH perneHHII sa,n;aq C'lHTaJIIIBee CHJlbl noeTOHHHhiMII. Me2KAY Tel\! Ha ABH2KYUJ.eecH TeJio Hap HAY c noeTOHHHwvm CHJiaMII )l;eHCTBYIOT 06hi'IHO CHJibi nepeMeHHbie, MO)l;YJIIT H HanpaBJieHHll KOTOPhiX npH ABH2KeHIIH TeJia nsMeHHIOTCH. Dpii 3TOM nepeMeHHhiMll: MOfYT 6biTh II sa,n;aHHbie (aKTIIBHb!e) CHJlhi *, H peaKUJIH CBH3eH.

KaK noKa3hiBaeT orrhiT, nepeMeHHhre CHJihi MoryT onpe,n;eJieHHbiM oOpasoM :;asiiceTb OT speMetm, noJiomeHHn TeJia II ero CKopoCTH. B qaeTHoeTH, OT BpeMeHH 3aBIIeHT eHJia THfii 3JieKTpOB03a npii rroeTeneHHOM BhiKJ!IO'leHIIHHJIH BKJ!IO'leHHHpeocTaTa HJIII CIIJia, Bbi3biBaiOUJ,a5I KOJie6aHIIH 11YH,n;aMeHTa npii pa6oTe MOTOpa e llJIOXO IJ.eHTpiipOBaHHhiM BaJIOM; OT ITOJ!OllieHHH TeJia 3aBHeHT Hh!OTOHOBa CHJ!a THfOTeHHH IIJIK eiiJia yrrpyroeTII rrpymHHbi; OT eKopoeTII 3aBIIeHT eHJihl COnpOTIIBJieHIIll cpe.nhr (no,n;po6Hee eM. § 76). B saKJIIO'leHIIeOTMeTIIM, 'ITOsee BBe,n;eHHbie B eTaTHKe ITOHHTIIH II IIOJiy'leHHbieTaM pe3yJihTaTbi OTHOCHTeH 1J paBHOH Mepe II K rrepeMeHHbiM CIIJ!aM, TaK KaK YCJIOBHe nOCTOHHCTBa CIIJI HIIf,I\e B CTaTIIKe He HeiiOJ!h30BaJIOeb.

l1HepTHOeTh TeJia npOHBJIHeTCH B TOM, 'ITO OHO eoxpaHHeT eBoe )J.BIIllieHHe IIp!I OTeyTCTBIIII ,n;eileTBYIOW.HX CIIJI, a KOf,I\a Ha Hero Ha'lHHaeT )J.eHeTBOBaTh CHJia, TO CKOpOCTII TO'leKTeJia H3MeHHIOTC5I He MfHG-

* AKmU61iOU 06b!'!HOHa3blBa!OT CHJiy, KOTOpaS!, Ha'laB,u:eiiCTBOBaTb Ha ITQKOHW.eeCS! TeJio, MoJKeT rrpnaecnr ero B ,U:BIDKeHne.

180

JlsnJKe-

BeHHO, a nocTenemw H TeM Me,n;JieHHee, qeVI 6oJibllle rmepTHOCTb ::noro TeJia. KoJinqecTBeHIIOli Mepoli: HHepTHOCTH MaTepnaJibHoro TeJia 5IBJI5IeTC5I G_)n3nqeCKa.H BeJinqnHa, Ha3biBaeMa5I .Maccou TeJia "', B KJiaccrrqecKofi MeXaHHKe MaCCa m paCCMaTpi1BaeTC5I KaK BeJIHqHHa CKaJI5IpHafl, TIOJIO)I{HTe.TibHaf! H TIOCT05IHHa5I )l;Jifl KaJK)l;OfO ,n;aHHOfO TeJia.

KpoMe cyMMapHoii MaCChi ,n;BHJKeHHe TeJia 3aBHCHT ew.e B o6w.eM cJiyqae OT <lJOpMbl TeJia, TQqHee OT B3aHMHOf0 paCTIOJIOJKeHH5I 06pa3y!Oll.I,HX ero qacTHU, T. e. OT pacnpe,n;eJieHHfl Mace B TeJie.

4T06bi npH nepBOHaqaJibHOM H3yqeHHH )l;HHaMHKH OTBJieqbCf! OT

y'leTaG_lopMbi TeJia (pacnpe,n;eJieHHfl Mace), BBO)l.5IT a6cTpaKTHoe noHHTHe o .MamepuaA.bH.Otl motfxe, KaK o TOqKe, o6Jia,n;aiOll.l,eii Maccoii, H Ha'IHH3IOTH3y'IeHHe)l;HHaMHKH C )l;HHaMHKH MaTepnaJibi-IOH ToqKH.

J.b KHHeMaTHKH H3BeCTHO, qTo )l.BHJKeHHe TeJia CJiaraeTC5I B 06W.eM cJiyqae H3 noCTynaTeJibHOro n Bpall.l,aTeJibHOro. DpH pellleHHH KOHKpeTHbiX 3a,n;aq MaTepHaJibHOe TeJIO MOJKHO paCCMaTpHBaTb KaK MaTepnaJibHYIO TO'IKYB TeX CJiyqaf!X, KOr)l.a TIO yCJIOBH5IM 3a,n;aqJ1 )l;OTiyCTHMO He npHHHMaTb BO BHHMaHHe Bpall.l,aTeJibHYIQ lJaCTb )l.BHJKeHH5I TeJia. HanpnMep, MaTepnaJibHOii ToqKoli MOJKHO cqnTaTb nJiaHeTy nprr H3Y- 'IeHHHee )l.BHJKNIH.H BoKpyr CoJIHUa HJIH apTHJIJiepnlicKnii CHap.H,n; npn onpe,n;eJieHHH ,n;aJibHOCTH ero noJieTa n T. n. CooTBeTCTBeHHo noCTynaTeJibHO )l;BHIKYIUeecf! TeJIO MOIKHO BCef)l.a paCCMaTpHBaTb KaK MaTepnaJibHYIO TOqKy c Maccoii, paBHOli Macce scero TeJra. CnpaBe)l.JIHBocTb 3THX yTsepJK)l.eHnii 6y)l.eT o6ocHOBaHa B § 107.

J.J3yqaTb )l;HHaMHKy Mbl HalJHeM C )l;HHaMHKH MaTepnaJibHOll TO'IKI!, TaK KaK eCTeCTBeHHO, qTQ H3yqeHHe )l.BHJKeHHfl O)l.HOH TQqKH )l.OJIJKHO npe)l.IIJeCTBOBaTb H3y'IeHHIO)l.BHJKeHH5I CHCTeMbl TOqeK H, B 'IaCTHOCTH, TBep)l.OfO TeJia.

§ 74. 3AKOHbl )].HHAMHKH. 3A)].A4H )].HHAMHKH

MATEPHAJibHOfl T04KH

BocHoBe ,n;rrHaMnKH JieJKaT 3aKOHhi, yCTaHOBJieHHhle nyTeM o6o6ll.l,e- HH.H pe3yJibTaTOB ueJIOfO p5I,D;a OTibiTOB H Ha6JIIO)J,eHHH, TIOCB5IIIJ.eHHbiX H3y'IeHHIQ )J,BHJKeHHH TeJI, H npOBepeHHbie 06IIIHpHOH 06ll.l,eCTBeHHOnpOH3BO)J,CTBeHHOll npaKTHKOfi 'IeJioseqecTBa. CncTe\1aTH'IeCKH3aKoHbl ,n;nHaMHKH 6hiJIH snepshre H3JioJKeHhi 11. HbiOTOHOM B ero KJiaccnqecKO\I

CO'IHHeHHH«MaTeMaTHqeCKHe HaqaJia HaTypaJibHOH <lJHJIOCOcpHH», H3)J,aHHOM B 1687 r. **. QpopMyJIHpOBaTb 3TH 3aKOHbl MOJKHO CJie,n;yiOIUHM

o6pa30M.

D e p B bi ll 3 a K 0 H (3aKOH HHepUHH): U30AUpOeaH.H.aJ'l Om 8Heut-

H.UX eosaetlcmeuu ;uamepuaAbflaJ'l mottKa COXpaH.f/i!m ceoe COCmOJlfiUe noKOJ'l uAu paeH.oMepH.ozo npf'l.MOAUH.euH.ozo aeuJJCeH.uf'l ao mex nop, noKa npuAoJJCeH.H.ble CUAbl ne sacmaeJim ee us.Menumb amo cocmoJinue.

HHe, COBeplUaeMOe TQqKQll npH OTCYTCTBHH CHJI, Ha3biBaeTCfl )J,BHJKeHHeM

no UH.ept{UU.

* Macca HB.'IHCTCHeme Mepoil: rpaBHTaUHOHHbiX CBOHCTB TCJia (eM. § 76). "'*EcTb npeKpacHhiH pyccKHII rrepeBo,ll. c,l(eJiaHHbiH A. H. KpbiJIOBbiM. CM .. CoU-

paHHC TPYAOB aKa,ll. A. H. KpbiJioBa, T. VII. M.- Jl., 1936.

181

3aKOH IIHepUIIII OTpa>KaeT O)l.HO II3 OCHOBHbiX CBOHCTB MaTepim - npe6hlBaTb Heii3MeHHO B )l.BmKeHHH. Ba>KHO OTMeTIITb, qro pa3BHTHe ,lf[IHaMIIKII KaK HayKH CTaJIO B03MO>KHbiM JIIIlllb TIOCJie TOfO, KaK faJIIIJieeM 6biJI oTKpbiT 3TOT 3aKoH (1638 r.) II TeM caMhiM onposepmyra rocno)l.CTBOBaBlllafl co spe:v~en ApiiCToTeJifl roqKa 3peHII5I o TOM, 'lTO ,!J.BII>KeHIIe TeJia MO>KeT npoiiCXO)l.IITb TOJibKO nO)!. )l.eHCTBIIeM CIIJlbl.

Cy!UeCTBeHHb!M 5IBJIHeTC51 Bonpoc 0 TOM, no OTHOllleHHIO K KaKOH CIICTeMe oTcqera cnpase)I.JIIIB 3aKOH IIHepUIIII. HbiOTOH npe,!l.nOJiaraJI, qro cymeCTByeT HeKoe HenO)l.BII>KHOe (a6COJIIOTHOe) npOCTpaHCTBO, nO OTHOllleHIIIO K KOTOpoMy 3TOT 3aKOH BblnOJIHHeTCH. Ho no cospeMeHHhiM B033peHII5IM npOCTpaHCTBO3TO <lJOpMa cymecTBOBaHII51 MaTepiiii, II KaKOfO-TO a6cOJIIQTHOf0 npOCTpaHCTBa, CBOHCTBa KOTOpOrO He 3aBIIC5IT OT )l.BII>KY!Ueiic51 B HeM MaTepiiH, He cy!UeCTByeT. Me>K)l.Y TeM, noCKOJibKY 3aKOH IIMeeT OnbiTHOe npOIICXO>K)l.eHIIe (e!Ue faJIIIJieii yKa3aJI, qTo K 3TOMY 3aKOHY MO>KHO npiiHTII, paccMaTp1IBa5I )l.BJI>KeHIIe lllapiiKa TI(} HaKJIOHHOH nJIOCKOCTII CO BCe y6hiBaiOIUIIM yrJIOM HaKJIOHa), )!;OJI>KHbl CyJUeCTBOBaTb CIICTeMbi OTCqeTa, B KOTOpbiX C TOll IIJIII IIHOH CTeneHb!O np1I6JIII>KeHII51 )l.aHHhiH saKOH 6y)l.eT BbiTIOJIHHTbCH. B CBH31I c 3TIIM B MexaHIIKe, nepeXO)l.H, KaK o6hrqHo, K nayqHoil a6crpaKUIIII, BBO)l.5IT TIOH5ITIIe 0 CIICTeMe OTcqeTa, B KOTOpOii cnpaBe,!I.JIIIB 3aKOH IIHepU,IIII, nocryJIIIPYIOT ee cymecTBoBaHIIe II na3biBaiOT unepl{UaAbHoii cucmeMoii

omcttema.

Mo>KHO JIII )l.aHHYIO peaJihHYIO CIICTeMy orcqeTa npii pellleHIIII rex IIJIH IIHbiX 3a)l,aq MeXaHIIKII paCCMaTpiiBaTb KaK IIHepuHaJibHYIO, ycTaHaBJIIIBaeTC51 nyTeM nposepKII TOfO, B KaKOU Mepe pe3yJibTaThi, noJiyqeHHbie B npe,ll,nOJIO>KeHIIII, qTo 3Ta CIICTeMa 5IBJI51eTC51 IIHepu,IIaJibHOH, TIO)l.TBep>K- )I.aiOTC5I OTibiTOM. flo )l.aHHbiM OnbiTa )l,Jl51 HallleiiCOJIHeqHOH CIICTeMbl IIHepUIIaJibHOH C BbiCOKOH CTeneHb!O TOqHOCTII MO>KHO cqHTaTb CIICTeMy OTcqeTa, HaqaJIO KOTOpOH HaXO,li,IITC51 B UeHTpe COJIHU,a, a OCII HanpaBJieHbl Ha TaK Ha3biBaeMbie HeTIO,li,BII>KHble 3Be3)1.bl. flpii pelllelfiiii 60JiblUIIHCTBa TeXHIIqeCKIIX 3a)l,aq IIHepu,IIaJlbHOH, C ,li,OCTaTOqHOll )I.JI5I npaKTIIKII ToqHOCTb!O, MO>KHO cqJITaTb CIICTeMy OTCqera, >KeCTKO CB513aHHYIO C 3eMJieii. Cnpase)l.JIIIBOCTb 3Toro yrsep>K,ll,eHIIH 6y)l.eT o6ocHOBaHa B § 92.

B Top o ii 3 a K o H (ocHoBHOH 3aKOH )l.IIHaMIIKII) ycraHaBJIIIBaeT, KaK II3MeH5leTC51 CKOpOCTb TOqKII npii )l,eHCTBIIII Ha nee KaKOH-HII6y,ll,h

CIIJihi, a IIMeHHO: npouseeaenue MaGCbl MatnepUaAbHOU mottKU na ycKopenue, KOtnOpOe Ofia no!lyttaetn no() OeUctneUeM aaHHOU CU!lbl, paeHO no MOayAIO amou cu!le, a nanpaeAenue ycKopenufl coenaiJaem c nanpaeAenueM CU/lbl.

MaTeMaTIIqecKII 3TOT saKou Bbrpa>KaeTcH seKTOpHbiM pasencrsovr

ma=F.

(1)

flpii 3TOM Me>KAY MO)l.YJIHMH ycKopeHH51 II CIIJibi HMeeT MeCTO 3aBHCIIMOCTb

ma=F.

(1')

Bropoil saKOH )l.IIHaMIIKH, KaK II nepsbrfr, HMeer Mecro TOJibKo no OTHOllleHIIIO K IIHepu,HaJibHOH CIICTeMe OTcqeTa. lb 3TOf0 3aKOHa Henocpe,!I.CTBeHlfO BII,!I.HO, qro Mepoii IIHepTHOCTII MaTepiiaJibHOH TOqKII 51BJI51•

182

eTCH ee Macca, TIOCKOJibKY npH )J,eHCTBHH )l,annOH CHJihl TOqKa, Macca KOTOpOH OOJibiiie, T. e. 6oJiee nnepTnaf!, TIOJiyqnT Menbmee ycKopenne H Hao6opoT.

ECJIH na TOqKy )J,eHCTByeT O)l,nOBpeMeHnO neCKOJibKO CHJI, TO OnH, KaK 3TO CJie.I{yeT H3 3aKona napaJIJieJiorpaMMa CHJI, 6y)l.yT 3KBHBaJienTnbi

O)l.HOH cnJie, T. e. paBHO)l.eiicTBYIOIII.eli. R, pasnoii reoMeTpnqecKoii cyMMe )l.aHHbiX CHJI. YpasHeHne, BhlpaJKaiOIII.ee ocHOBHOii 3aKon )l.HnaMHKH, npHHHMaeT B 3TOM CJiyqae BH)l,

(2)

3TOT JKe pe3yJibTaT MOJKnO TIOJiyqHTb, HCTIOJib3Y51 BMeCTO 3aKona

napaJIJieJIOrpaMMa 3aKOH. H.e3G8UCU!r!OCmU aet1Cm8Uft CU;2, COfJiaCHO KOTO-

poMy npn O)l,HOBpeMeHHOM )l.eiicTBHH Ha TOqJ{y HeCKOJibKHX CHJI KaJK)l,a51 H3 nHX C006lll,aeT TOqKe TaKOe IKe ycKopenHe, KaKOe Ona C006lll,HJia 6bi,

.I{eHCTBYfl O.I{na.

T p e T n ii 3 a K o n (3aKoH pasencTsa .I{eiicTBHfl n npoTHBo)l.eficr-

BHH) ycTanaBJIHBaeT xapaKTep MexannqecKoro B3auMo)l.eiicTBH51 MeJK)l.Y MaTepnaJibHbiMH TeJiaMH. )lJif! )l.Byx MaTepnaJibHhlX ToqeK on rJiaCnT:

aee MamepuaAbflbte motiKU aet1cmey10m apyz H.a apyza CCUAGMU, pa8H.bl.MU no MoOYAIO u H.anpaeAeH.H.blMU BOOAb np!tMOU, coeaUH.ft10!1.{et1amu mottKU, 8 npomueonoAOJICH.bte cmopoH.bt.

3THM 3aKonoM Mbi yJKe nOJib30BaJIHCb B cTaTnKe. On nrpaeT 6oJibiiiYIO pOJib B ,I{HnaMHKe CHCTeMbi MaTepnaJibHbiX TOqeK, KaK ycTanaBJIHBaiCIII.HH 3aBHCHMOCTb MeJK)l,y )l,eHCTByiOill,HMH na 3TH TOqKH BHyTpeHniiMH cn.'IaMn.

flpn B3aHMO)J,eHCTBHH )J,ByX CB060,I{HbiX MaTepnaJibHhlX ToqeK, OHH,

COfJiaCnO TpeTbeMy H BTOpoMy 3aKOHaM .I{HnaMHKH, 6y)l.yT )l.BHfaTbC51 C

ycrmpeHH5lMH,	o6paTnO npOnOpii,HOHaJibnbiMH	HX MaCCaM.
3 a )1, a q H	)1, H H a M H K H • )lJif! CB06o,I{nOH MaTepHaJibnOH TOqKH
3a.I{aqaMH )l,HnaMHKH 51BJ151IOTC5l CJie)l.yiOlll,He: 1)		3Ha51 3aKOn )l,BHJKeHHfl

roqKn, onpe)l.eJIHTb .I{eiicTBYIOIII.YIO na nee cnJiy (nepeaJt saaatta auH.a- MUKu); 2) 3na51 )l,eHCTBYIOlll,He Ha TO•IKy CHJibl, onpe)l.eJIHTb 3aKOH )l,BH-

JKenH5l roqKn ( emopa!t, HJIH ocH.oet-taR, saaatta auH.aMuKu).

)lJIH HeCB060)l,nOH MaTepnaJibHOH TOqKH, T. e. TOqKH, Ha KOTOpyro HaJioJKena CB513h, BbiHYJK)l.aiOlll,a5I ee )l.BHraTbCfl no 3a)l.aHHOH nosepxHOCTH HJIH KpHBOH, nepBaf! 3a)l,aqa ;zr;nnaMHKH 06hlqno COCTOHT B TOM, qro6bi, 3na51 .I{BHJKenne TOqKH II )l,eHCTBYIOIII.He na nee aKTHBnbre CHJibi, onpe.I{eJIHTb peaKUHIO CB5l3H. BTopaH (ocnosna5l) 3a)l.aqa )l,HnaMHKH npu HeCB060.I{nOM )l,BHJKenHH pacna)l,aeTCH na )],Be H COCTOHT B TOM, qT06bi, 3Ha5l ,I{eikTBYIOIII.He na TOqKy aKTHBnbre CHJibi, Onpe.I{eJIHTb: a) 3aKOH )l,BHJKenn5l TOqKH, 6) peaKII,HIO naJIOJKenHOH CB5l3H.

§ 75. CHCTEMbl E)l.HHHU

nJI51 H3MepenHH BCeX MexanHqecKHX BeJinqnn OKa3biBaeTC5l )l,OCTaToqnbiM BBeCTH He3aBHCHMble )l.pyr OT Apyra e)l,HnHI.J.bi H3MepenH51 KaKHX-Hn6y.I{b Tpex BeJinqnH. )lByM5I H3 nHX npHnHTO cqnTaTb e)l,HHHUbl )l.JIHHhi n speMeHn. B KaqeCTse TpeTbeii OKa3biBaeTC5l HaH6oJiee YA06HbiM Bbr6paTb e)l.nnuuy H3MepeHI15l nnn MaCChi, HJIH CHJihl. TaK KaK

te3

9TH BeJIHqHHhi CB5I3aHhl paBeHCTBOM (1), TO npOH3BOJlhHO e,li.HHHL(y H3Me-

peHHfl KaJK,li.Oll H3 HHX Bbi6paTh HeJih3f!. 0rCIQ,ll.a BbiTeKaeT B03MOJK•

HOCTh BBe,ll.eHH5I B MeXaHHKe ,li.ByX npHHl(HTIHaJihHO OTJHl'IHhiX,ll.pyr O'f

,11;pyra CHCTeM		e,li.HI!Hl(.
n e p	B hi H	T H n	c H c T e M	e )1. H H H l(. B 3THX CllCTeMaX 3a
OCHOBHble	npmiHMaiOTC5l		e)l.IIHHl(hi	,li.JIHHh!, BpeMeHH H MaCChi, a CHJia

H3MepHeTC51 npOH3BO,li.HOH e,li.HHffl(eH.

K TaKHM CHCTe'viaM OTHOCHTCH MeJK)l.ynapO,li.Haf! cncTeMa e,li.HHHL(

H3MepeHH51 cpH3HqecKHX BeJIHqHH (ClJ), B KOTOpOH OCHOBHhiMH e,li.HHHIJ,aMH H3MepeH1-l5I MeXaHHqecKHX BeJIHqHH 5IBJI5IIOTC51 MeTp (M), KHliOrpaMM Macchi (Kr) H ceKyH.ll.a (c). E)l.HHHueil JKe H3MepeHHH cHJihi 5IBJif!eTC51

rrpOH3BO,li.Haf! e)l.HHHua - 1 HhiOTOH	(H); 1 H - 3TO	CHJia, coo6IIJ,aiO-
maR Macce B 1 I{f ycKopenHe 1 Mlc 2	(1H=1 Kr·Mic2).	0 TOM, qTo co6oii

npe,ll.craBJIHIOT 1 M, 1 Kr H 1 c, H3BeCTHO H3 Kypca cpH3HKH. Me:tK.li.Y- napO,li.HaH cHCTeMa e,li.HHHIJ. (Cl1) BBe,ll.eHa B CCCP KaK npe)l.nOqTHTeJih-

Haf! C 1961 f. H B ,ll.aHHOM	Kypce	Mhl TIOJih3yeMC5l eiO.
B T o p o il T H n c H	c T e M	e .ll. H H H u. B 3THX CHCTeMax 3a

OCHOBHhie npHHHMalOTCH e,li.HHHUhi ,li.JIHHhi, BpeMeHH H CifJihi, a MaCCa

ll3MepHeTC5I npOH3BO,li.HOH e,li.HHHUeH.

K TaKHM CHCTeMaM OTHOCHTCH HMeBrnaH 6oJihUIOe pacrrpocrpaneniie B TeXnHKe CHCTeMa MKfCC, B KoTopoii: OCHOBHbiMH e,li.HnrruaMH HBJIHIOTCH MeTp (M), KHJIOrpaMM CHJihi (Kf) H CeKyH,ll.a (c). E,li.HHHUeH H3MepeHHH MaCChi B 3TOH CHCTeMe oy,ll.eT 1 Kf •C 2fM, T. e. Macca, KOTOpOH CHJia B } Kf

coo6maeT ycKopenHe 1 Mlc 2 •
CoOTnornenHe MeJKAY	e,li.HHMUaMH cnJihi B cncTeMax Cl1 n MKfCC
Tarmso: 1 Kf=9,81 H	HJIH	1 H=0,102 Kf.
B 3aKJIIOqenHe OTMeTHM,		qro neo6xO,li.HMO pa3JIHqaTh nonf!THH pC13-

Mept-wcmb BeJIHqHHhi H eaunuup ee U3Mepe1iUJl. Pa3MepHOCTh onpe,ll.eJIHeTCH TOJlhKO BH,li.OM ypaBHeHHH, Bhipa:tKaiOIIJ,ero 3HaqenHe ,ll.aHHOH BeJIHqlfHhi, a e)l.HHHL(a l13MepeHH51 3aBMCHT eiiJ,e OT BbiOOpa OCHOBHbiX e,li.HHHU. HanpHMep, ecJIH, KaK 3TO npHHHTO, o6o3HatJ:aTh pa3MepHOCTh ,li.JIHHLI, speMenu H MaCchi cooTBeTCTBeHHo CHMBoJiaMH L, T n M, TO pa3MepHOCTh cKopocnr LIT, a e,li.HHHueii H3MepeHHH MO:tKeT OhiTh 1 Mlc, 1 KMIII H T. ,ll..

§ 76. OCHOBHblE BH)l.bl CHJl.

DpH perneHHH 3a,ll.aq ,li.HHaMHKH Mbl oy,ll.eM B OCHOBHOM pacCMaTpH-

BaTh CJie,ll.yiOIIJ.He TIOCTOf!HHbie HJIH nepeMeHHbie CHJibi (3aKOHhl H3MeHe· HHH nepeMeHHbiX CHJI, KaK npaBHJIO, ycTaHaBJIHBalOTCfl OTihiTHhlM nyTeM).

C H Jl a T H :tK e C T H.	3'foTIOCT05IHHa5I		CHJia p, ,ll.eHCTBYIOIIJ.aH Ha
JII06oe TeJio, naXO,li.HIIJ,eecf!	s6JIH3H	3eMHOH	rrosepxnocTH (rro,ll.po6nee
eM. § 92). Mo,ll.yJih CHJihi TH:tKecTrr		pasen secy TeJia.

OnhiTOM ycranosneHo, qTo no,ll. )J,eHCTBHeM CHJibi P Jno6oe TeJio np11 CB060,1J;HOM IIa,ll.eHim na 3eMJilO (c neoOJibiiiOH BhiCOThi ll B 6e3B03.ll.Yl11HOM npocrpaHcrse) HMeer o,u;no H To :tKe ycKopenne g, na3hiBaeMoe ycrwpe-

184

CH /=6,673 ·10-

(5)

11 M~IKr 2

·c ).

HUeM C6o6oiJH020 naiJeHUH, a HIIOr)J,a ycKOpeHUeM		CUJlbt tnHJICecmu *.
Tor.u.a H3 yp asHeHMH ( 1')cJie)J.yer,	qro
P=mg HJIH	m=Pig.	(3)

3-ru paBeHeTBa nosBoJUIIOT, 3Haf! Macey rena, onpe)J.eJIHTb ero sec (MO)J.YJib )J.eUCTBYIOill.ell Ha Hero CHJlbl T5DKeeTH) HJIH, 3Haf! Bee TeJia, onpell,eJIHTb ero Macey. Bee TeJia HJIH CHJia TfiJKecTH, KaK H seJIHqHHa g,

H3MeHHIQTCfl C H3MeHeHrf€M llllfpOTbl H Bb!COTbi Ha)J, ypOBH€M MOpH; Macca

LKe	fiBJifleTe5l )I,.'Ifl }l.3HHoro TeJia BeJinq.HHOll					HeH3MeHHOll.
C H JI a		T p e H H fl. TaK 6y)J.eM KpaTKO				Ha3biBaTb	CHJIY rpemrH
CKOJihJKeHHfl, )J.eHCTBYIOill.YlO (npH OTeyTCTBHH JKli)J.KOll CMa3KH) Ha
ABHIKYill.eecH		TeJio. Ee		MOAYJib onpeJI,eJIHeTCH paBeHeTBOM (eM. § 23)
				F=fN.			(4)
T}l.e f - K03¢q)HUHeHT				TpeH.H5I, KOTOpb!H	6y)J.eM cqHTaTb		TIOCT05IHHblM;
N -	HOpMaJibHafl		peaKUHfl.
	C H JI a	T fl r	o T e H H fl. 3To CHJia,		c Koropoii )J.Ba MarepHaJibHbiX

TeJia npHTflrHBalOTCfl )J.pyr K )J.pyry TIO 3aKOHy BCeMHpHOrO Tf!roTeHHfl, OTKpbiTOMY HLIOTOHOM. Cl!Jia rHroTeHH5I 3aBHCI1T or paecroHHHfl H AJIH

)J.BYX MaTepllaJibHbiX ToqeK C MaccaMH tn1 H tn2, HaXO)J.fllUHXCfl na paCCT05IHHH r Jl.pyr OT Apyra, Bb!paJKaeTCfl paBeHCTBOM

F =fm1m2/r\

r)l.e f - rpaBHTauHOHHaf! nocT05IHHa51 (s

C ll Jl a y TI p y r 0 C T H. 3Ta CHJia TOMe 33BHeHT OT paCCT05IHHH.

Ee 3HaqeHHe MmKHO onpe)J.eJIHTb HCXO)J.H H3 3aKoHa fyKa, corJiacHo

IWTopoMy	uanpHJKemre	(cn.Tia,	orHeceHHaf!		K eJI,UHHUe		nJioru.a.n.H)
nponopuuoHaJibHO Ae¢opMaUHH.			B	qacTHOCTH,	AJIH	CHJibi	ynpyroerlf
np Y2:KHHbi	noJiyqaeTcfl 3HaqeHae
		F=c'A,					(6)
r.n.e /.. -	y.n.mmeHHe (HJIH	C)KaTHe)		npyJKHHbi;	c -	raK H33biBaeMbiH
K03¢q>HuHeHT JKeerKocru rrpyJKHHbi (s CH H3Mep5IeTef! B HIM).
C ll JI a B 5I 3 K 0 r 0		T p e H H H. TaKaf!			CHJia, 33BHe5IIUaf! OT

CKOpOCTH, AerieTByeT Ha TeJIO npH ero MeAJieHHOM J!,BHMeHIIH B oqeHb BH3KOH Cpe:.(e (amr npH H3JIHqHH IKH)l.KOH eMa3KH) H MOJKeT 6biTb BbipaJKeHa paBeHCTBOM

				(7)
T)l.e V - CKOpOeTb TeJia; J.t- K03tPrpHUHeHT				COTipOTHBJieHHfl. 3aBHeH-
MOCTb BH)J,a (7) MOJKHO TIOJiyqHTb HCXO)J,fl H3			3aKoHa Bfi3KOrO TpeHHH, OTK-
pb!TOro	HbiOTOHOM.
C H JI	a	a 3 p 0 A H H a M H q e C K 0	r	0 (r H A p 0 )l, H H a M H-
q e C K 0 r 0)		C 0 ll p 0 T H B Jl e H H fl.	3'fa CHJia TOMe 33BHCHT OT

* 3aKOH CBOOO)J:HOfO rra,u:eHHH TeJJ 6b!JI OTKpbiT raJJHJJeeM. 3naqeHHe g B pa3Hb!X MeCTaX 3eMHOH ITOBepxHOCTH pa3JJHqHo; OHO 3aBHCHT OT reorpaqmqeCKO!'llllllpOTbl MeC-

Ta H Bb!COTI>I ero Hall: ypoaneM MOpH. Ha umpore Moc¥Bhl (Ha ypoane MOpH) g=

=9,8156 Mfc2.

185

CKOpOCTH H ,ll.eHCTByeT Ha TeJIO, ,li.BHIKYUJ.eeCH B T3KOH, HanpnMep, cpe,ll.e, KaK B03,li.YX HJIH BO,ll.a. 06blqHO ee BeJinqnHy BblpaJKaiOT paBeHCTBOM

R=0,5cxpSv2,

(8)

f)J,e p - TIJIOTHOCTh cpe,ll.bi; S - TIJIOlll,a,ll.h npOeKIJ,IH1 TeJia Ha IIJIOCKOCTh, nepneH,li.HKYJIHpHyiO HanpaBJieHHIO )J,BHJKeHH51 (IIJIOlll,a,ll.h MH,ll.eJIH);

Cx - 6espa3MepHblll K03¢qlHIJ,HeHT COIIpOTHBJieHH51, onpe)J,eJIHeMblll 06blqHO 3KCnepHMe~T3JlhHO H 3aBHC51llJ.HH OT <lJOpMhi TeJia H OT TOfO, KaK

OHO OpHeHTHpOBaHO npH ,li.BHJKeHHH.

l1 H e p T H a 5I H f p a B H T a I1, H 0 H H a 5I M a C C bl. nJI51 3KCIIepHMeHT3JlhHOf0 onpe,ll.eJieHH51 MaCChi )J,aHHOfO TeJia MOLKHO HCXO,ll.llTb H3 3aKOHa (1), Ky)J,a MaCCa BXO)J,HT KaK Mepa HHepTHOCTH H Ha3blBaeTC51 TI03TOMY llHepTHOH M3CCOH. Ho MO:iKHO HCXO)J;HTh H Ii3 33KOHa (5), Ky,ll.a Macca BXO,li.HT KaK Mepa rpaBHTaUHOHHhiX CBOiicTB TeJia H Ha3biBaeTC51 coorsercrsenno rpasnraunoHuofi (nJin TH:iKeJiofi) Maccofi. B npnHunne HH OTKy)J,a He CJie,ll.yeT, qro HHepTHa51 H rpaBHTaiJ,H0HHa51 MaCChi npe)J.- CT3BJI51IOT C060H O)J,Hy H TY :iKe BeJinqnHy. 0,li.HaKo ueJihiM p51)J,OM 3KCIIepnMeHTOB ycranosneHo, qro 3HaqeHHH o6enx Mace cosna,ll.aiOT c oqeuh BblCOKOH CTeneHhiQ TOqHOCTH (no OIIhiTaM, npo,ll.eJiaHHhiM COBeTCKHMli tPH3HK3MH (1971 r.),- C TOqHOCTh!O ,li.O 10-12 ). 3ToT 3KCIIepHMeHT3JlhHO ycraHOBJieHHhiH <!JaKT Ha3hiBaiOT rrpHHUHIIOM 3KBHBaJieHTHOCTH. 3iiHIIITeHH * TIOJIO:iKHJI ero B OCHOBY CBOeH 06UJ.ell TeOpHH OTHOCHTeJihHOCTH (TeOpHH T51fOTeHH51).

11CXO)J;51 H3 H3JIO:iKeHHOfO, B MeXaHHKe TIOJlh3YIOTC51 e,li.HHhiM Tepl'vlliHOM «Macca», onpe,ll.eJI5151 Macey I{aK Mepy nHeprHocrn TeJia n ero rpasnraunoHHhiX CBOHCTB.

fJiasa XVI

,ZJ;H«P«PEPEHU.HAJihHhiE YPABHEHIUI )l,BH)I(EHH~ TOlii<H.

PEWEHHE 3A,ZJ;ALJ ,ZJ;HHAMHI<H TOLJI<H

§ 77. )l.H«<>«<>EPEHU.HAJibHblE YPABHEHIUI )l.BH)I(EHH}I

MATEPHAJibHOfl TOlJKH

nJI51 peiiieHH51 3a,ll.aq ,li.HHaMHKH TOqKH		6y)J,eM	ilOJih30BaThC51	O,li.HOII
H3 CJie)J,yiOUJ.HX )J,BYX CHCTeM ypaBHeHHH.
Y p a B H e H H 51 B ,ll. e K a p T 0 B hi	X	K 0	0 p )1, II H a T a	X. 113

KHHeM3THKH H3BeCTHO, qro JlBH)I{eHHe TOqKH B Dp51MOyfOJlbHbiX ,neKap-

TOBblX KOOp,li.HHaTaX 3a,naeTC51 ypaBHeHHHMH (CM.	§ 37):
x=f1 (t), y=f 2 (t), z=f a (t).	(9)

3a,ll.aqH ,li.HHaMHKH TOqKH COCT051T B TOM, qT06hi, 3Ha51 ,li.BH:iKeHHe roqKn, T. e. ypasnemm (9), onpe,ll.eJIHTh ,ll.eficrsyiOlll,yiO Ha roqKy cnJiy

* AJJb6epT 9fmnJTefm (1879-1955) - Bbl,ZJ:aJOIII.HiiCH yqeHbiii-¢H3HK, C03.ZJ:aTeJJb crreuHaJibHOll TeOpH!i OTHOCHTeJibHOCTH (peJJHTHBHCTCKali Mexam1Ka) ll 06III.eii TeopHH OTHOCHTe.'lbHOCTH.

t86

(11).

HJIH, Ha060pOT, 3Ha5! .n:efiCTByiOill.He Ha TOqKy CHJibl, onpe.n:eJIHTb 3aKOH ee ABHmeHHH, T. e. ypasHeHHH (9). CJie.n:osaTeJibHO, AJIH perneHHH 3a.n:aq AHHaMHKH TOqKH Ha.n:o HMeTb ypaBHeHHH, CB5!3biBaiOmHe KOOPJJ:HHaTbi

X, y, Z 9TOH TOqKH H ,n;eiicrByiOlli,YIO Ha Hee CHJiy (HJIH CHJibi). 3TII ypaBHeHH5! H .n:aeT BTOpOH 33KOH ,ll;HHaMHKH.

p acCMOTp HM MaTep HaJibHYIO TOtiKy, ,ll;BHMylll,yiOC5! no.n: ,n;eiiCTBHeM CHJI

F1, F 2 , ••• , Fn no OTHOIIIeHHIO K HHepuHaJibHOH cHcTeMeOTcqe-raOxyz. DpoeKTHpyH o6e qaCTH paBeHCTBa (2), T. e. paBeHCTBa ma="L.Fk, Ha

ocn x, y, z H yqHTbiBafl,	qTo ax=d2xldt2 H T. ,n;., noJiyqHM:
d2x	d2 y	d2z	(10)
m dt2="L.Fkx•	m dt2 ="2.Fky•	mdtZ="L.Fkz•
ITJIIT, o603Haqa5J BTOpbie npOH3BO,!I;Hbie TIO		BpeMeHH JJ:BYMH	TOqKaMIT,
			(10')

3-ro H 6y.n:yr HCKOMbie ypaBHeHHH, T. e. OUcpcpepeH.t{UaJlbflbLe ypaeHeH.UJt

OBU;)!CeHUH motiKU a npHMoyzoAbfibtX iJeKapmoeblx KoopiJuH.amax. TaK KaK

,n;eiiCTBYIOlli,He CHJibi MOfYT 33BHCeTb OT BpeMeHH t, OT TIOJIOMeHH~

TOqKH,	T. e.	OT ee KOOpJJ:HHaT X, y, Z, H	OT CKOpOCTH, T. e. OT Vx=X,
V 11 =y, Vz=i,		TO B 06lll.eM CJiyqae npaBa5!	qaCTb KaMJJ:OfO H3 ypaBHeHH~
(lO).MOMeT 6biTb cpyHKUHeii Bcex 9THX nepeMeHHbiX, T. e. t,				X, y, Z, X,
y, Z	OJJ:HOBpeMeHHO.
Y p a B H e H H 51 B ll p 0 e K U H 51 X H a 0 C H e C T e C T B e H·
H 0 r 0	T p	e X r p a H H H K a. )lJIH	nOJiyqeHHH 9THX	ypaBHeHHH

cnpoeKTHpyeM o6e qacTH paBeHCTBa ma="L.Fk Ha OCH M'l:nb, T. e. Ha KaCaTeJibHYIO M1: K TpaeKTOpHH TOqKH, rJiaBHYIO HOpMaJib Mn, HanpaB-

JieHHYIO B cTopoHy sorHyTOCTH TpaeKTOpHH,				H 6HHOpMaJib Mb (eM. B
§ 42 pHC. 122; Ha HeM Oxyz- OCH,			no OTHOilleHHIO K KOTOpbiM ,!I;BH-
)KeTCH	TOIIKa). Tor.n;a, yqHTbiBaH, qTo		(eM.	§ 43) a,=dvldt,	an =v2/p,
ab=O,	noJiyqHM
	du	u2		O="L.Fkb'	(11)
	mrrr=L.Fk-r•	m-p="L.Fkn•

YpaBHeHITH (11), r.n:e v=ds/dt, npe.n:cTaBJIHIOT co6orr iJucpcpepeHt{ua-

llbH.bte ypaenemm iJeu;)!CeH.uH moliKU e npoeKt{UHX H.a ocu ecmecmeeH.H.oeo mpexzpaH.HUKa.

§ 78. PEIUEHHE nEPBOA 3A~AL[H ~HHAMHKH

(OUPE~EJIEHHE CHJI no 3A~AHHOMY ~BH)I(EHHIO)

ECJirr ycKopei-me JJ:BIDKyl.IJ,eiicH TOqKH 3a,n;aHo, TO .n:ei'!crsyiOlll,aHCHJia HJIH peaKUHH CBH3H cpa3y Haxo,n;HTCH no ypasHeHHHM (1) HJIH (2). DpH 9TOM JJ:Jlfl BblqHCJieHHfl peaKUHH H3JJ:O JJ:OnOJiliHTeJibHO 3H3Tb aKTHBHbie CHJibi. Kor,n;a yCKopeHHe Henocpe.n:cTseHHO He 3a.n:auo, HO H3BecTeH 3aKOH ,ll;BHMeHHfl TOqKH, TO JJ:Jlfl onpe,n;eJieHHH CHJibl MOMHO BOCnOJib30BaTbCH ypaBHeHIT51Mil (10) HJIH

187