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K.moqaeM, liTO uptmp tnfW!Cecmu n.lloll{aau mpeyzoAbHUKa Aea~cum e mott- Ke nepecetteHUfl eco .MeauaH. Dpn 3TOM, KaK H3BecrHo,
CE=BE/3. |
(68) |
3. 11, e H T p T 5I JK e C T 11 ll Jl 0 IIJ; a ~ H K p y I'0 B 0 r 0 |
C e K- |
T o p a • PaccMorpHM Kpyrosofl ceKrop OAB paJI:Hyca R c ueHrpaJibHbiM yrJioM 2a (puc. 111). Pa3o6heM MhicJieHHo nJIOrn;aJI:h ceKropa OAB pa,n,uycaMH, npose~eHHbiMII H3 ueHrpa 0, Ha n ceKropoB.
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8 |
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PHC. 111 |
Pnc. 112 |
PHC. 113 |
npH HeorpaHII'leHHOMyseJIHqeHHH "!HCJia n, 3TH CeKTOpbl MOJKHO pacCMaTpiiBaTb KaK TIJIOCKHe rpeyrOJibHHKH, IJ,eHTpbi THJKeCTH KOTOpblX JieJKaT Ha .n.yre DE pa.n.uyca 2R/3. CJie)I:osareJibHO, ueHTp THJKeCTH ceKropa OAB cosna~aer c ueHrpoM THJKeCTH .n.yru DE, noJioJKeHHe KOToporo Hafi~eTcH no <PopMyJie (67). 0KoHqareJihHo noJiyqHM, qro uptmp mfla~eecmu nAoU{aau Kpyzoeozo ceKmopa JJ-ea~eum rta ezo ocu cuJ.utempuu Ha paccmoflrtuu om l{ertmpa 0, paertol~t
2 R sin rx |
(69) |
Xc=3-rx-· |
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(70)
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5. 11, e H T p :T H JK e c T 11 o 6 b e M a n o JI y III a p a. 3TOT ueHrp C JieJKI1T Ha oc11 Ox (ocii CHMMerp1111, p11c. 113), aero Koop~11Hara
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(71) |
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¢opMyJihl, onpe~eJI5IIOIIJ,IIe KOop~IIHaTbl IJ,eHTpOB T5I.JKeCTII ,11.pyr11X O)I.HOp0,11.HhiX TeJI, MOJKHO HaHTI1 B pa3JIII'IHbiXTeXHHqeCKHX cnpaBO'IHHKaX.
Pa3.n.eJI BTopoii
KMHEMATHKA TOIJKH H TBEP,li.OfO TEJIA
fJiasa IX
KHHEMATHKA roqKH
§ 36. BBE)J.EHHE B K.HHEMATHI(V
Kurte.Mamwwu H33biBaercH pa3.n,eJI MexaHHKH, B KoropoM HsyqaIOTCH reOMeTpiiiiecKHe CBOifcTBa ,!l,BH:tKeHHR TeJI 6e3 yiieTa HX HHepTHoCTH (MaCChi) H .n,efiCTBYIOIII.HX Ha HHX CllJI.
KHHeMaTHKa npe.n,cTaBJIHeT co6ou, c o.n,Hoft cropoHhi, sse.n,eHHe s ,!l,HHaMHKy, TaK KaK ycTaHOBJieHHe OCHOBHhiX KHHeMaTHIIeCKHX nOHHTHU H 3aBHCHMOCTefi He06XO,!l,HMO ,!l,JIH II3yiieHHH ,!l,BHLKeHHH TeJI C yqeTOM
.n,eHCTBHH CHJI. C .n,pyrofi CTOpOHhl, M€TO,!l,hl KHHeMaTHKH IIMelOT H Ca- l\40CT05ITeJ1bHOe npaKTHIIecKoe 3Haqenue, HanpHMep, npH H3yiiemm nepe.n,atJ: ,!l,BH:tKeHHH B MeXaHH3MaX.
I1o.n, ,!l,BHLKeHHeM Mhl nOHHMaeM B MeXaHHKe H3MeHeHHe C TeiieHHeM speMeHH nOJIO:tKeHHH ,!l,3HHOfO TeJia B npocrpaHCTBe no OTHOIUeHHIO K
.n,pyrHM TeJiaM.
l(JIH onpe,n;eJieHHH nOJIOLKeHHH ,!l,BHLKYIII.erOCH TeJia (HJIH TOIIKH) B pa3Hhle MOMeHThl BpeMeHII C TeJIOM, no OTHOIUeHHIO K KOTOpOMy H3yiiaetCH ,!l,BH:tKeHHe, LKeCTKO CBH3biBaiOT K3KYIO-HH6y.n,h CHCTeMy K00p,D;HH3T, o6pa3YIOIII.YIO BMecre c 3THM TeJioM cucme.My omcttema. B .n,aJihHefirneM 6y.n,eM fOBOpHTb 0 ,!l,BH:tKeHHII TeJia (HJill TOqKH) no OTHOIUeHHIO K ,!l,3HHOH CHCTeMe OTCIIeTa, no.n,pa3yMeBaH no.n, 3THM ,!l,BH:tKeHHe no OTHOIUeHIIIO K TOMY TeJiy, C KOTOpblM 3Ta CHCTeMa OTcqera CBH3aHa. I13o6pa:tKaTb CHCTeMy OTCIIeTa 6y,n;eM B BH.n,e tpeX KOOp,!I,HH3THhiX OCeH (He noKa3hiB3H TeJIO, C KOTOphiM OHH CBH3aHhl). Bbi60p CHCTeMhl OTCIIeTa B KHHeMaTHKe npOII3BOJieH (onpe.n,eJIHeTCH IIeJih!O HCCJie,!l,OBaHH5I), H B OTJIHIIHe OT ,ll;H- HaMHKH (CM. § 74) BCe KHHeMaTHIIeCKHe 3aBHCHMOCTH, noJiyiieHHhie npH H3yqeHHH .n,BmKeHHH B KaKOH-HH6y,n;h O,!l,HOH CHCTeMe OTCIIeTa, 6y.n,yr cnpase.n,JIHBhl H B JII060H .n,pyrofi CHCTeMe OTCIIeTa.
l(BHLKeHHe TeJI COBeprnaeTCH B npocrpaHCTBe C TeiieHHeM BpeMeHH. I1pocrpaHCTBO B MeXaHHKe Mhl paccMaTpHBaeM KaK TpeXMepHoe eBKJIH- ,!l,OBO npocrpaHCTBO. Bee H3MepeHnH B HeM npoH3Bo.n,HTCH Ha ocHoBaHHH MeTO,!l,OB eBKJIH,!l,OBOH reoMeTpiiii. 3a e,!l,HHIII1Y ,!l,JIIIHhl npH H3Mepemm paccTOHHHH npHHHMaeTcH 1 M. BpeMH B MexaHHKe ciiHTaeTCH yHIIBepcaJihHhiM1 T. e. npOTeK310Il!.HM O.Ll,HHaKOBO BO BCeX paCCMaTpHBaeMhiX
95
cncTeMax OTcqeTa. 3a e.n.HRHUY speMeHH ~:pnHHMaeTCH 1c. PasMepHoCTh }I,JIHllhi o6osHaqaeTcH CHMBo.noM L, a speMemr - CHMBOJIOM T.
EBKJIH)l.OBo npocTpaHCTBO H ymmepcaJibHOe speMH OTpaJKaiOT peaJihHhre CBOilcTBa npocTpaHCTBa H BpeMeHH Jlllillh npHOJIHJKeHHO. 0}1,HaKO, KaK IIOKa3biBaeT OIIhlT, )l.JIH )l.BHJKeHHH, KOTOpb!e H3yqaEOTCH B MeXaHHKe (.ll.BHJKeHHH CO CKOpOCTHMH, )l.aJieKHMH OT CKOpOCTH CBeTa), 3TO IIpHOJIHJKeHHe }l,aeT BIIOJIHe )l.OCT3TOqHyEO }I,JIH rrpaKTHKII TQqHOCTb.
BpeMH HBJIHeTCH cKaJIHpHoii:, nerrpepbiBHO nsMeHHIOIII,eiicH semi'lHHOH. B 3a.n.aqax KHHeMaTHKH speMH t npHHHlviaiOT 3a HesasHCHMoe nepeMermoe (apryMer-n). Bee .n.pyrne nepeMeHHble BeJinqHHhi (paccTOHHHH, CKOpOCTH H T. )]..) paCCMaTpHBaiOTCH K3K H3MeHHIOIIJ.HeCH C TeqemreM BpcMeHH, T. e. K3K t1JyHKUHH BpeMeHH t. 0TcqeT BpeMeHH Be)l.eTCH OT HeKOTOpOrO HatW/lbHOcO MO.Mertma (t=O), 0 BhlOOpe KOTOporo B K3)K)J.OM cJiyqae ycJioBJIHBaiOTCH. BcHKHii .n.aHHhiH MOMertm epeMertu t orrpe}l,eJrHeTcH qncJioM ceKyH}I,, rrporne.n.umx OT HaqaJibHoro MOMeHTa .n.o }l,aHHoro; pa3HOCTb MeJK)l.y K3KHMH-HHoy}l,h )J.BYMH llOCJie}J,OBaTeJibHbiMH MOMeHTaMII BpeMeHII H33hiBaeTC51 npoMeJICymKOM epeMertU.
J1oqepiiHYThie H3 OIIb!Ta H IIO)l.TBepJK)l.eHHh!e npaKTHKOH OCHOBbl,
•Ha KOTOph!X CTpoHTCH KHHeMaTHKa, .n.a10T aKCHOMhl reoMeTpHH. HnKaKHX )l.OllOJIHHTeJihHhiX 3aKOHOB H.'IHaKCHOM )l.JIH KHHeMaTIIqecKOfO H3y'IeHHH )l.BHJKeHHH He TpeoyeTCH.
]lJIH perneHHH 3a)l.aq KHHeMaTHKH Ha)l.O, qTQObl YJ3yqaeMOe )J.BHJKeHue
ObiJIO KaK-TO sa,n;aHo (onncaHo).
Kurte.Mamuqectw 3aoamb aeu:J!Certue UJ?U 3a!<OH aeuJICeHUfl tne/la ( mottKu) - srtaiium saiJamb no.llOJICertue amoco me.lla (motaw) omrtocume.llbHO iJamwu cucmeMbt ommemae .ll106ot1 AtOMertm epeMertu. Y cTaHOBJienne Ma-
TeMaTHqecKHX CIIOCOOOB 3a)l,aHHH )J.BHJKeHHH TOqeK HJIII TeJI HBJIHeTCH O)l.HOli H3 BaJKHhiX 3a)J.aq KHHeMaTHKH. fi03TOMY H3yqeHHe )l.BHJK(:'HH5I JIIOooro oobeKTa 6y,n;eM HaqnHaTh c ycTaHoBJieHHH crroco6os 3a)J.aHHH
::noro )J.BHJKeHHH.
0CHOBHaH 3a)l.aqa KHHeMaTHKH TOqKH H TBep)l.OfO TeJia COCTOHT B
TOM, qTOOhl, 3HaH 3aKOH }J,BIIJKeHHH TOqKH (TeJia), ycTaHOBHTh MeTO)l.bl onpe,n;eJieHHH BceX KHHeMaTHqeCKHX BeJIHqHH, XapaKTepH3yiOIII,HX }J,aHHOe )l.BHMeHHe.
HsyqeHHe KHHeMaTHKII HaqHeM c nsyqeHHH )l.BHJKeHHH npoCTelirnero OObeKTaTOqKII (KHHeMaTHKa TOqKH), a 3aTeM rrepeH)J.eM K II3yqeamo
KHHeMaTHKH TBep)J.OfO TeJia.
HerrpephiBHaH JIHIIHH, KoTopyiO onncbmaeT )l.BHJKYIII.aHCH ToqKa OTHOCHTeJihHO )l.aHHOH CHCTeMhl OTCqeTa, Ha3biBaeTCH mpaeKmopueii TO'IKH. Ecm1 TpaeKTopuel1 HBJIHeTCH rrpHMaH JIHHHH, )J.BHJKeHHe ToqKH Ha3biBa·
a eCJIH KpHBaH - KpU60/lUHeUHblM.
§ 37. CllOC06bl 3A,[J,AHIUI ,[J,BH)I(EHIUI TOlJI(H
JlJIH 3a)l.aHHH )l.BHJKeHHH TOqKH MOJKHO IIpHMeHHTb O)l.HH II3 CJie)l.ylO· lUHX Tpex CllOCOOOB: 1) BeKTOpHbiH, 2) KOOp)l.HHaTHh!H, 3) ecTeCTBeHHhiH.
1. BeKTopHbiH crroco6 sa.n.aHHH )l.BIIJKeHuH
T 0 q K H. I1yCTh TOqRa M )l.BHJK(:'TCH ITO OTHOllleHHIO K HeKOTOpOH CHCTeMe oTcqeTa Oxyz. I1oJioJKeHHe 3Tofr ToqKn B JII06ol1 MoMeHT speMeHH·
96
MOlliHo orrpe,n;eJIIITb, 3a.n;as ee pa.n;nyc-BeKrop r, rrpose.n;eHHbiH n3 HaqaJia Koop.n;nHar 0 B roqKy M (pnc. 114).
BeKrop r6y.n;er c retieHHeM BpeMeHH H3Me· HHTbCH H no Mo,n;y;rno, II no HarrpaBJieHII!O. CJie.n;onareJibHO, r 5IBJIHeTCH nepeMeHHbiM BeKTOpOM (BeKropOlll-tPYHKUHeH), SaBIICHIUIIM OT apry· MeHra t:
(1)
PaBeHcrso (1) n onpe.n;emrer 3arw1-1 .n;snJKeHHH rotiKH B neKropuofi <PopMe, raK KaK ono nosnc~:_mer n mo6oii MOMenr npeMeHII rrocrpoHTb
COOTBeTCTBYIOIUHH BeKrop r H HaHTf! f10JIO/Kelme ,ll,BHJKyill,e~ICH TOtiKH. feoMerpiiqecKoe Mecro KOHUOB neKropa ,-:; r. e. ro.n;orpa<jl 3Toro BeK-
ropa, orrpe.n;emrer rpaeKropmo .n;snHzyrueilcH TO'IKtr.
AHaJIHrHqecKH, KaK msecrHo, seruop sa.n;aercH era rrpoernuiHMII Ha Koop.n;nnarnhre oca. B npm,wyroJibHbiX iJ,eKaproBbiX KOOpp,nHarax ,D;JIH BeKropa r 6y,n;er: fx=X, fy=!J,_f:-=Z (CM. pnc. 114), r,n;e X, !J, Z-}J.e- KaprOBbl Koop,n;IIHaThi rotiKII. Torna, ecJIH z
j, k
KOOp,!1IIHaTHbiX OCeii, IIOJIYtiHM )J;JIH f |
Bbipa- |
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JKeHHe |
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(2) |
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CJ1e,n;osareJibHo, 3aBHcn:vrocrb (I) |
r or t |
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6y,!!,er H3BecrHa, ecJIH 6y.n;yr 3a,n;aHbi |
Koop- |
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,n;HHaTbi X, !J, Z TOtiKH KaK tPYHKUHH BpeMe- |
PHC. 114 |
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HH. TaKofr crroco6 3a,n;aHHH .n;BHlliemm TOtiKH |
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(Koop.n;IIHaTHhrii) paccMorpHM HHllie. |
BeK- |
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rop fMOllieT 6b!Tb 3a,n;aH, I<&K !13BeCTllO, H HHbiMH CIIOC06aMH, HarrpHMep ero MO,ll,yJieM H yrJiaMH C OCHMH HJIH npoeKUH5I\1H Ha OCH .I\pyrHX CHCTeM Koop.n;nnar. .Umr noJiyqenHH o6ruHx cpopMyJI, He 3il.BHCHII.l,HX or roro, KaK KO!IKpeTHO sa,n;aH BeKrop r, 6y.n;eM HCXO,ll,HTb 113 BeKTOpHOfO 33KOHa ,ll,BHllieHH5I, npe.n;craBJieHHoro pasencrsm1 (1).
2. |
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c rr o c o 6 |
3 a .n; a n H H .n; B H lli e- |
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r o q K H. |
DoJiollieHue |
TO'IKII MO)KHO |
HerrocpeiJ,CrBeHHO orrpe- |
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,n;eJirrTb ee ,n;eKapTOBbJ}vi.H KOOp,ll,HI!aTaMH X, !J, |
Z, |
KOTOpb!e IIpH ,ll,BHllieHHH |
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TOqKH 6y,n;yr C Te1IeHHeM BpeMeHH H3MeH5ITbC5I. l-J:ro6LJ 3HaTb 33KOH ,ll,BHJKeHHH TOtiKH, T. e. ee TIOJIOllieHHe B rrpocrpaHCTBe B JII060H MOMeHT npeMCHH, na.n;o snarh sHatiemur Koop;J.HHar roqKn ,n;JIH KaJK,n;oro MOMenra BpeMeHH, T. e. 3H3Tb 33BHCHMOCTII
(3)
YpaBHeHHH (3) rrpe.n;cran:rrH!OT co6oii ypaeHeftUfl aeua~CeHUJl mofiKU e npRMoyc:o.llbHbtx ae~wpmoebtx Koopowwmax. 0HH orrpeiJ,eJIHIOT saKoH ,ll,BH)KeHHH TOtiKH r!pH KOOp,ll,IIHaTHOM Cr!OC06C 3a,U:aHH5I )J;BHlliC!IHH *.
* 3a,'(aTb)\BH)KeHHe TO'JKHMO)KJIO, liO.'lb3YHCbH 1\pyrHMH CHCTeMaMH KOOpi\Ilii3T, llallpHMCp HO.l!llpHblMH (CM. § 47), c¢epH'l.eCKHMH H T. )l,.
4 N• 2173 |
97 |
Ec.rm .n.BmKemre ToqKI1 npmiCXO.ll.11T Bee Bpe:vrH B OIJ,Hoii H roii /Ke rrJiocKOCTI1, ro, np11HHB ary rrJIOCKOCTb 3a ITJIOCKOCTb Oxy, noJiy'HIMB 3TOM CJiyqae )l.Ba ypaBHeHI15I )l.BI1JKeHHH:
x=fdt), y=f2(t). |
(4) |
HaKoHeu, rrpn npHMOJIHHeiiuoM .n.mnKeHim roqrm, ecmr B.ll.OJib ee rpaeKTOp11H HarrpaBHTb KOOp.ll.11HaTHY!O ocb Ox, )J:BH)I\eHHe 6y.n.er orrpe-
JJ.e.TIHTbCH O)l.HHM |
ypaBHeHI1e\1 (3atWHOM np:U!IO/lUHeUHOZO d8U:JJCeHUfl |
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tnO'llW) |
x=f(t). |
(5) |
YpaBHeHHH (3) |
n (4) rrpe.n.craBJIH!OT co6oii |
o.rr.HoBpe'lleHHO ypaBHe- |
HHH rpaeKTopii11 roqKH B rrapaMerp11qecKoii cpopr.re, rne poJib napm.terpa HrpaeT BpeMH f. 11CK.lJ!Oqi1B II3 ypaBHeHI1}I l{BHJKeHHH BpeMH f, MO:ti(HO Haiirn ypaBHeHne rpaeKropnn B o6hrquoii <PopMe, T. e. B Bl1)l.e, .ll.aiOI.UeM 3aBHCHMOCTb MeJK)l.y KOOp,ll.11HaTaMH TOqKH.
UpHMep. Ilycrb ,ll,BH1KeHHe TOIJKH B mJocKocrll Oxy ,n;ano ypaBHeHJun.•H:
x=2t, y=1212 , (a)
r,n;e x, y BbJpaJKeHbi B caHTH\Ierpax; t - B ceKyH,n;ax.
Do 9THM ypaBHeJJHllM MOJKHO Hai!TH, IJTO B MO~reuT BpeMeHII f=Q TOIJKa H3'i0,!\IITC5l
B ilO.TlQA,eHHH |
M 0 (0, 0), T. e. B Ha'laJJe K00p,!\1IH3T, B r,!OMeJIT t1 = !c - B ilOJJOJKeJU!H |
M1 (?.,12) H r |
,n;. TaKHM o6pa30M, ypaBHerurH (a) .=~e,rcnnlTeJJbHO onpenemnor no"10JKe- |
HHe T04Kl! D JJI060H l\10MeHT Bpe\JetiH. ll,aBUll t pa3hL!e 3Ha1JeHHll H H306pa1K3ll COOTBeTCTB) IOlljHe IIOJIOJKeHHll T04KH Ha pHCYHKe, ~1Q.1,eM ilOCTpOHTb ee rpaeKropmo •
.lJ.pJTHM nyreM rpaeKTOpHIO MO)KHO HaiiTH, HCK.1l04HB f !13 ypaBHeH!lli (a). Jh nepBOfO ypaBilCHHll HU>.O;J,EM t=xi2 H, il01\CT3B,1l!ll 910 3Ha4eHHe t BO BTOpoe ypaBue-
IIHe, noJTyqaeM y =3.\ 2 CJii:J:OBareJJbHO, |
rpaeKropner1 TO'IKH HBJIHercl! napa6ona c |
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BepU!HHOil: B |
H3'!3J1e KoOptl,HHaT H |
OCbJO, |
nnpaJT.'Je,lbHOH OCII Oy. Ll,pyme |
npnMepbl |
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onpetJ,eJieJlHll |
rpaCKTOpHH TO'!KH CM. B § 41. |
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3. |
E c T e c T B e H H hi ii |
c n o c o 6 |
3 a n a H n H |
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H u H |
To q K 11. |
EcrecrBeHHbPvl (11JIH TpaeKTopHbiVI) |
crroco6o:vt3a)l.aHmi |
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A |
,n:BHJKeHHR yJJ.o6:-ro |
nO.'Ib3oBaTbCH B |
rex cJiy- |
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qaHx, KOr,ll.a |
TpaeKTOpHH )l.BmKyW,ei'ICHT0'1KH |
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z |
H3Becrua |
sapam~e. |
Dycrh |
Kpu:aal'! AB HBJIH- |
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eTCH |
rpae1nop w:ff ToY:r~u A1 |
npH ee .n.B;uKeHI1H |
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OTHOCIITeJibl-!0 CHCTe\!bl OTCqeTa Oxyz (pHC. |
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115). Bbr6epe:.1 HG :noi'I rpaeKropuu |
KaKyro- |
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0'/ ----- |
HI1oyD.h uerro:I,BHIKHYIO roqKy 0', Koropyro |
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B |
np11MeM 3a I-Ia"'aJio orcqera, |
11 ycraHOBI1t.I Ha |
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rpaeKropun |
no.1oJKHTe.1hrroe |
n |
orpnuare.JJb- |
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PHC. 115 |
Hoe HarrpaBJieHI1H orcqeTa (KaK na rwop.n.unar- |
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HOH oc11). Tor;J,a rro.'Io/;;:enueroqKu M Ha rpa- |
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eKTopnu |
6yJJ;eT O)l.H03HaqHO |
orrpe.n.e.'IHTbCH KpHBO.'IliHeliHOl! KOOj))l.H- |
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HaTOll s, |
KoropaH |
panHa |
paccroHHHto or roqKII 0' .n;o |
roqKI! M, 113Me- |
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peHHOMY |
B;:J,OJib )l.yrl1 TpaeKTOp11H |
H B3HT0\1Y |
C COOTBeTCTBY!OI.UH~ |
3Ha- |
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KOM. Tipn ,ll.BHJKeHHH roqKa M nepeMer.uaercH B rroJIO:tKeHiur M 1 , M 2, |
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CJie,ll.OBaTeJibHO, paCCTOHHHe S |
oy)l.eT C Te"!eHI!eW BpeMeiHI |
H3MeHHTbCH. |
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4ro6br 3HaTh no"'IoiKeimeToqKII M Ha rpaer<Topmr B JI!Oooii MO.\teHT Bpe-
MeHn, Ha)l.O 3l:!aTb 3aBHCHMOCTb |
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s=f (t). |
(6) |
98