(формат)Тарг
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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
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KOOp,!1IIHaTHbiX OCeii, IIOJIYtiHM )J;JIH f |
Bbipa- |
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JKeHHe |
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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).
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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, |
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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 |
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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) |
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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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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 |
.n. B 11 JK e- |
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H u H |
To q K 11. |
EcrecrBeHHbPvl (11JIH TpaeKTopHbiVI) |
crroco6o:vt3a)l.aHmi |
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,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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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'/ ----- |
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HI1oyD.h uerro:I,BHIKHYIO roqKy 0', Koropyro |
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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 |
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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
YpasHel!ne (6) |
H Bhipa:>KaeT 3atWH oeu:J~Cenu;z tnOIIKU M Bl\OJib Tpa- |
eKTopmi. |
• |
TaKHNI o6pa30\I, |
'IT06hi3J :J.aTh ,nBmKeHne TO'IKHecTeCTBeHHbiM crro- |
co6oM, na.no 3a,l\aTb: 1) TpaeKTopn:D TO'IKH;2) HaqaJio OTC'leTaHa TpaeKTvpnn C yKa3aJII!e'vlIIOJIOJ!OITeJibHOrO H OTpHUaTeJibHOfO HanpaBJiem:;l OTC'leTa; 3) 3aKOH ,nBmKeiiMI TO'IKH B,n0•1b TpaeKTOplll! B Blf)l.e s=f (t).
3al\!eTHM, 'lfOBeJIH'IHHaS B ypaBHeHHH (6) OIIpe,neJIHeT IIOJIO)KeHHe ,nBnMyr.uei:icH T04IOI, a He rrpof•.neHHhiH e10 rryTh. HanpHMep, ecmi ToqKa, .nsuraHcb H3 Ha4aJia 0', ,1\0XOJ,HT .no noJioJKeHnH M 1 (puc. 115),
a 3aTel\l, nepeMei.UaHCb B o6paTHOM HarrpaBJiei-IIHJ, IlpHXO)IJIT B IIOJIO.fKeHPe AI, TO B 3TOT \!O~.teP.T ee I<Oop)l.HHaTa s=O'M, a rrpoi1,neii!IhiH 3a
Bpe;,tH JlBmKeHHH |
nyTh 6:.;r,eT paseH O'M1 +M1A1, T. e. ne |
pasen s. |
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§ 38. |
BEI(TOP |
CI(OPOCTH |
TOlJk.H |
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0.J;HOH H3 OCIIOBHbiX |
KHHeMaTlflieCK!IX |
XapaKTepHCTHK |
)l.BHJKe- |
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HHH TO'IKH HBJIHeTCH BC'KTO!)HaH BeJIH4IIHa, |
Ha3biBae~taH CK0fJOCn1b/O |
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mo111\U. |
Bse)l.eM cHaYa.la rroHnT>re o cpen.Hci1 CKopocTH TOYKII 3a KaKoti- |
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HH6y)l.h |
npOMe)KYTOK BpeMeHH. I1yCTb ,nBH.lKYI.UaHCH TO'IKa HaXO,!l;HTCH |
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Pnc. |
l 16 |
B MOM:eHT Bpel\IeHH t |
B liOJIOI!<eHl!H ;\1, orrpe,neJIHe:VIO'.lpa,nnycoM-BeKTO- |
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p0\1 r, |
a B MO\leHT tl |
rrpHXOJ,l!T B IIO.'IOlliei-IHeM1> Oi1pe,neJIHeMoe BeKTO- |
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pO\! t 1 |
(pnc. 116). Tor.J,a nepeMelQeHHe TO'IKH3a npo'<IeMyToKBpeMeHH |
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At=t1-t orrpe,neJHieTCH B~Kropo\t |
JH.\11, Koropr:,p"f 6y,ne'II Ha3hiBaTh |
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8CKtnOpO"H nepeAte~WHW! nlO'IIW. 3TOT BeKTOp HarrpaBJTeH IIO XOp,!l;e, eCJIH T04Ka ,ll,Bl'1KeTC5I KpiiBOJIHHeilHO (Dl1C. 116, a), H B)l,OJib CaMOi'r TpaeKTOpH!I AB, KOr,L(a .J:B:Ii-KeHne HBJIRercH rrpHMomme{IHbm (p11c. 116, 6). l1s Tpeyro.1hiiHKa 01H \11 Bli.J:Ho, qro r+.wiH1 =G"; cJie,!J;oBaTeJibHo,
MM1 = rl-r=~r.
0THo::neHHe BeKTopa nepeMer.uemiH T04KH K cooTBeTCTBYIOI.UeMy IIpOl<ieJKyTKY spe\reHH :I,aeT BeKTOpHyl-0 Be,TIH4IIHy, Hi13b!BaeMy10
1tCU ITO \10,!l;yJIJ-O 11 HanpasJieHIUO CKOpOCtnb/0 tnO'IXU Sa rrpoMe.tKyTOK BpeMeHH Af:
(7)
4* |
!79 |
HarrpasJieH BeKTOp Vcp TaK )Ke, KaK n BeKrop MMl! T. e. rrpn KpnsoJIHHeiiHOM JI:BH)KeHHH BJI:O.Tib xopJJ.bi MM1. B eTopony JI:BH)KeHHH TO'IKH,
a IIpii rrpHMOJIHHeHH0\1 )l.BH)KeHIUI- B)I:OJib eaMOH TpaeKTOpHII ( OT )l,e- JieHHH Ha ~f HarrpaBJieHHe BeKTOpa He H3McHHeTeH).
QqeBH)l.HO, 'ITO '!eM MeHhllie oy)I:eT rrpoMe)KyTOK BpeMeHII ~f, )l.JI51
KOTOporo Bbi'IHeJieHaepe)l.HHH |
eKopoeTb, TeM BeJIH'IHH.aVcpoy.n.eT TO'I- |
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nee xapaKTepH30BaTb )l.BI!iKefiHe TO'IKH. 4TOObi |
IIOJIY'IHTb TO'IHYfO |
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XapaKrepHeTHI<y |
)l.BHiKeHIIH, |
BBO)I:HT |
IIOHHTHe 0 |
C!WpOCtnU |
tnOiiKU 8 |
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iJaHHbtu MOJteHm epe.tteHu. |
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CKopoerbfO _2'04KHB )I:aHHhiii MOMeHT speMer-m t |
HasbrB~TCH |
seKTop- |
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naH BeJIH'IHHaV, |
K KOTOpoii erpeMHTeH epe)l.HHH eKopoeTb Vcp IIpH eTpeM- |
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JieHHl-1 rrpoMe)KyrKa speMenH 11t K nymo: |
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v= hm (vq,) = |
ltm |
M. |
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tl.t--..0 |
M->-0 |
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Tipe)I:eJI ornoUJeHHH L\rl M rrp11 M -+ 0 npeJI:eTaBJIHeT eo6oii rrepsy10 |
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npOH3BOJ\IIyfO OT BeKTOpa rITO apryMeHTY f |
H OOO:JHa1-IaereH, J{aK H npo- |
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l13BO)l.H3H OT eKaJIHpHOH tPYHKUHH, CHMBOJIOM drtdt. 0KOHqareJibHO IIOJiyqaeM
(8)
I1TaK, eeKtnOp CKOpOCtnU tnO'IXU 8 aaHHbLU MOMeHm Bpe.MeHU paeet-t nepeou npou3eoiJHou om paiJuyca-eeKmopa moqKu no epe.MeHu.
TaK KaK npe,7JgJihiibiM narrpasJieHHeM ceKyrueii MM1 HBJIHeTeH Kaeare.rrbHaH, 10 eeKmop cKopocmu moliKU 6 iJaHHbtu MOAteHm epeMeHu Ha- npae.llet-t no Kacame.llbHOu K mpaeKmopuu mottKu 6 cmopoHy iJem/CeHUJl.
<J>op111yJia (8) I10Ka3biBaeT TaKtKe, 'ITOBeKrop eKopoeTH V paBeH OTHOUJeHHIO 3JieMeHTapnoro nepeMerueHHH TO'IKH dr, HarrpasJieHHoro no
KaeaTe/IbHOH K rpaeKTOpHH, K eooTBeTCTBYIOIUeMy rrpoMe)KyTKY BpeMeHII cit.
f1p11 npHMOJIHHe}moM )l.BHJKeHHH ser<rop eKopoern vsee speMH narrpaBJien B)I:OJib npHMOH, ITO KOTOpOH )I:BH)KeTeH TO'IKa, H MOJKeT H3MeH5!TbeH JIHillb 'IHCJieHHO; npn KpHEOJIHHeHHOM ,I\BH)KeHIH1 KpoMe 'IHeJIOBOfO 3Ha'IeHH5! Bee BpeMH !13MeHHeTeH 11 HarrpaBJieHHe BeKTOpa eKOpoeTII TO'lKH. Pa3Mepnoerb eKopoeni LIT, r. e. )l.JIIIHa/spe\1H; B Ka-
'IeCTBe e)I:HHHU H3MepeHHH rrpHMeHH!OT oobi'IHO Mle HJIH KM/'1. Bonpoc o6 orrpe.n.eJieHHH MOJI:YJIH eKOpoeTH oy)I:eT paeeMOTpeH B § 40 II 42.
§ 39. BEI(TOP YCI(OPEHIHI TOlli(M
YeKopeHHeM TO'IKH Ha3biBaeTeH BeKTOpHaH BeJIH'IIIHa, XapaKTepH3YfOJUaH 113MeHeHHe e Te'IeHHeMBpeMeHH MO)I:yJIH H narrpaBJieHHH CKOpoeTH TO'IKH.
DyeTb B neKoTopbrii MOMeHr speMeHH ~ JI:BH.tKYIUaHeH TO'IKa Haxo- )l.HTeH B II0•10.tKeHHH M H HMeeT eKOpOCTb V, a B MOMeHT f1 rrpHXO,I\HT 1\
100
llOJIO)KeHHe Ml H HMeeT CKOpOCTb vl (pnc. 117). Tor,n:a sa npoMemyTOK BpeMenn M=t1- t cKopocTb TOIIKII noJiyiiaeT rrpnpaur.enne L1v=v1-V.
,UJI5l IIOCTpOeHH5I BeKTOpa f1v OTJIO)KHM OT TO'IKHM BeKTOp, paBHbHi Vlt
H IIOCTpOHM rrapaJIJieJiorpaMM, B KOTOpOM |
}l:HaroHaJib!O |
oy,n:eT Vlt a |
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OJJ;HOH H3 CTOpOH i). Tor,n;a, |
O'lel3HJJ;HO, BTopa5! CTOpona H oyJJ;eT H306pa- |
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maTb BeKTOp f1v. 3aMeTHM, |
'ITO BeKTOp |
M |
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L1v acer,n:a narrpaaJien a cTopoHy Bor- |
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HYTOCTH TpaeKTOPHH. |
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0THO!Ileime IIpHparn:eHIUI BeKTopa |
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cKopocTn Sv K cooTBeTcTayiOW:eMy npo- |
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MemyTKY BpeMenn L1t onpe,n:eJI5leT eeK- |
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mop cpeiJHeco ycKopeHuJl mottKU 3a .3TOT |
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npoMemyToK BpeMenn: |
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acp= i1V;M. |
(9) |
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PHc. |
117 |
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BeKTop cpe,n:nero ycKopenn5l nMeeT To me HarrpaBJieHne, 'ITOH BeKTop Sv, T. e. nanpaaJien a cTopony aornyToCTH TpaeKTopmr.
YcKopenueM TO'IKHB )J.aHHbiH MOMeHT Bpe'\1eHHt Ha3biBaeTCH BeK-
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TopnaH aeJiwmna a, K KOTopofi cTpe'\1HTCH cpe)J.nee ycKopemre acp npn cTpeMJieHim rrpoMemyTKa DpeMeHH L1t K nyJIIO:
- . |
!:!.v dv |
a= hm |
Tt=ctr |
t.t-+0 |
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HJIH, C yqeTOM paBeHCTBa (8), |
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(10)
CJie,n:oBaTeJibHo, eeKmop ycKopeHUJl mottKU e iJaHflbtu MOMeflm epe.MeHu paeeH nepeou npou3eoonoi1 orn eeKmopa cKopocmu uAu emopou npou3eoiJflou om paouyca-eeKmopa mottKu no epeMeHu.
Pa3MepnocTb ycKopemm LIP, T. e. )J.mma/(apeMH) 2 ; B KatieCTse e,n:HHHUbi H3MepennH npHMeHHeTcH o6brqno M/c.2 •
lb 4JopMyJibl (10) CJie,n:yeT TaK)Ke, qyo BeKTOp ycKopeHHH TOqKH ~ paaen OTHOlllcHmo 3JieMeHTapnoro npupanrennH BeKTopa CKopocnr dv J{ COOTBeTCTBYIOW,eMy npoMemyTKY BpeMeHll dt.
Hafi,n:eM, KaK pacrroJiaraeTCH BeKTop ano OTHOllleHliiO K TpaeKTOpHH TOtiKH. DpH rrpHV!OJIJ1HeHH01! JJ:BH)KeHHH BeKTOp aHanpaBJieH B)J.OJib rrpRMOH, IIO KOTOpofi JJ;BHmeTCH TO'!Kd.EcJIH_TpaeKTOpHeH TOtiKII HB_:IHeTCH IIJIOCKa5I KpHBaH, TO BeKTOp ycKopeHH5l a, TaK )Ke KaK H BeKTOp acp• JiemHT B ITJIOCIWCTH 3TOH Kpl1BOH H HarrpaBJieii B CTOpOHY ee BOfHYTOCTH. ECJIH TpaeKTOpHH He 5lBJIHeTC5I IIJIOCKOH Kp!IBOH, TO BeKTOp acp HarrpaBJieH B CTOpOHY BOrHyTOCTH TpaeKTOpHII H Jie)KHT B IIJIOCKOCTH, rrpO· XO)J.HW:eii IIepe3 KacaTeJibHY!O K TpaeKTOpHH B TOtiKe M H npHMYIO, napaJIJieJibHYIO KaCaTeJibHOH B COCe)J.HeH ToqKeM1 (pHc. 117). B rrpe}\eJie,
101
IWrJI:a TOtii<a M 1 3Ta rrJio.::KoCTb saHHMaeT no.riomeHHe TaK H33b!Bae~1:0ll conpuKaCatOU{eiicJl nAOCKOCmU, T. e. IT.10CKOCTH, B KOTOpOH npoHCXOJ(HT 6ecKOHetiHO 1\iaJib!H ITOBOpOT KacaTeJibHOH K TpaeKTOpllli npn 3Jie'<1eHTapHoMnepeMern.eHim .D.BIDI<yw.e:ics:r TO'!KH*. CJieJI:oBaTeJib-
Ho, B o6rn.eM cJiyqae 6eKmop ljCKOpeHUJl alleJ.:CUtn 6 conpuKaCaiOU{CUCH. nAocKocmu u nanpa6AeH B cmopoHy 6omymocmu Kpu60ii. orrpeJ(eJieHHH MO.D,YJil'IycKopeHI1H oy.D,eT paCCMOTpeH B § 40
§ 40. OUPEAEJIEHHE CKOPOCTH H YCKOPEHH51 TOI.fKH llPH KOOPJ],HHATHOM CUOCOiiE 3AJ],AHH51 J],BHJKEHH51
HafmeM, KaK BbitiHCJIHIOTCH CI<OpOCTb n ycKopeHne TOtiKH, ec.rm e~ )l.BHmesne sa.naHo ypaBHeHIU!MH (3) Hmi (4). Borrpoc o5 orrpe.neJiemm TpaeKTop!m B 3To:vr cnyqae ObiJI yme pacc:.rorpen B § 37.
<lJop:VlYJibl (8) orrpeJ(e.lJHIOW,He 3Ha'leHHHV II a, COJ(epmaT npoii3BOJ)-Ibie no BpeMeHH oT BeKTopon r H V. B paBeHCTBax, co,'l;epmaIIJ.HX rrpOI13BO)J,Hble OT BeKTOpon, rrepeXO!l; K 33BIICII?-!OCTHM MemJ(y II\ rrpoeK,l!Ul:VIH ocyrn.ecrBJIHeTCH c rroMOUJ.biO cneJI:yiOrn.ei1 Teope:vrbl: npoCKtfUJl npou360ot-wii om eeKmopa rta ocb, Nenoo6UJICHyto 6 oaw-wii cucmeMe Otncttema, pasHa npOU3600flOU Om npoeXI{UU otU/JcpepeHI{Upye.MOZO eeKmopa Ha my JJCe ocb, T. e.
ecJIH |
- dp |
dox |
dp,1 |
rlpz |
( 11) |
q= CiT' TO |
qx= dT' |
q:J= dT' |
qz=Tt • |
||
I. |
0 n p e .n e JI e H II e |
c K o p |
o c T II T o q K II. |
BeKTop CKO- |
|
pocnr TO'lKIIV=dr)ut. 0TCIOJ.3 Ha OCHOB3Hill! ¢op:.\1YJ1 (11), Y'HlTb!BaH,
'ITOrx=X, ry=lj, rz=Z, H3HJ(eM:
dx |
dq |
dz |
(12) |
Vx=dt' |
VY=dt' Vz=(it |
||
W!II
( 12')
r.ne TO'!KaH3.ll. oyKBOH eCTb CHMBOJI .il.HcpcpepeHUIIPOBaH!11'!no Bpe:.\IeHIL TaKHM o6pa3oM, npoetcl{UU cKopocmu mo'tKu Ha ;~oopouHa:nta:te ocu pOfJHbt nepebtJtt npou3600HbtM om coom6erncmeytOw,ux Koop8w-wm rnc<tKU no 6pcMeHu.
3HaH rrpoeKUHH CKOpOCTH, Haft.n;eM ee :\WJI:y.!Jb H narrpaBJieHIIe (T. e.
~' y, KOTOpbie BeKTOp Vo6pa3yeT C IWOp.D,HHaTHblMH OCHMH) ITO <PopMyJiaM
v= Vv;+v~+v;; |
} |
(13) |
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cos a= vxfv, cos~= r.'ylv, |
cosy= Vz/V. |
||
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* ,UJJH npocTpaHCTBemwil: KPHBOH (HarrpHMep, n.1H BllHTosoii .'ll!H!!H) s Ka>Knoii TO'!Ke Kpl!BOH 6yneT CBOll COITp!!KacaiOID.aHCll ITJIOClWCTb. Jl:MI n.~OCl{Or'\ KpHBOH C0-
11pHKaCaiOID.allCll llJIOCKOCTb COBrranaeT C llJIOCKOCTbiO 3TO!f Kp!!BOH H HBJIHeTCH o6rn.eii IVIH DCCX ee TO'IeK.
102
2. 0 IT p e J( e JI e H H e y C K 0 p e H H H T 0 q K H. !WpeHHH TOqKH a= dVfdt.
T. e. npoeKt{UU yc1wpeHtL"l mottKU na KoopiJunamrtbte ocu pasnbl nepebw n,JOU360dHbV4 om npoeKt{Uii cKopocmu UAU amopbu.t npou3eodHbU.t om cwmeemcmey10u~ux KoopauHa:n mottKu no apeAtenu . .Mo,J,yJIL H HarrpaBJie-
H:re ycKopeHHH Hai!JJ,yTCH H3 cpop!11YJI
|
a= Va~-,at -f-a~; |
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cos l'r = az/a, |
} |
(15) |
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cos a 1 = axfa, cos ~1 = ayfa, |
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r,J,e ah |
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1, ''hyrJILI,- |
o6pasye:vrhle |
BeKTopoM yCKopeHHH c Koopnu |
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B |
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hc:THbll\iH |
OCH:\1!!. |
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11TaK, eCJI!I J(BH:>KeHlie TO'IKH33,!l,3HO B )l,eKapTOBbiX llpHMOYfOJIHib!X |
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KOOp,:l;!!HaTaX ypaBHeHHHvlil (3) HJIH |
(4), |
TO CKOpOCTb TOtiKH orrpe.'l,eJIH- |
||||
eTCH no ¢op\JyJia'1 (12) H (13), a ycKopenne- no q)opMyJraM (14) n
(15). |
Dpn 3TOV1 a c.riytiae .'J.Bmxemm, rrponcxoll.HUJ.ero B OJJ.HOH miOCKO- |
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cTI!, |
ao acex cpopMyJiax .ll.O.'DhHa OLITb OT6pomeHa rrpoeKUHH |
Ha ocb z. |
B c.1yqae me npHMO.'IHaelmoroJJ.BIImemm, KoTopoe 3aJJ,aeTc5I O,ll.HHM |
||
ypaaHenueM x=f(t), 6y,.J,eT |
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ch: |
(16) |
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Vx = dt ' |
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PaseHCTBa (16) If orrpe,:J;e,TJHIOT snaiieHHH cKopocTH H ycrwpemm TO'IK!i B 3TO.M CJlyqae.
§ 41. PElliEHHE 3A}],AlJ KHHEMATHKH TOlJKff
3aJJ,atiH, pemae:vib!e MeTO,!l,3\IH KHHeMaTHKH TOtiKH, MoryT COCTOHTb B orrpel('.:~.'IeH!III TpaeKTOp!HI, CKOpOCTH HJIH yc;wpeHHH TOtiKH, B 0Tb!CK3HIIH apeMeHH, B Te'leHHeKOTOporo TO'IKarrpOXO,D,HT TOT HJIH H!;fOH ITYTb, H.llf rryTH, rrpOXO,J,mwro 33 TOT II.'IHHHOH npoV1emyTOK BpeMeHU, II T. IT.
DpemJJ,e qe:,r pemaTL ,TJI06yiO H3 TaKoro poJJ,a 3aJJ.ati, HaJI,o ycTaaoBllTL, rro KaKmty saKoHy JI:BIImeTcH TOtiKa . .3ToT 3aKoH MO>KeT OhlTb HerwcpeJJ.CTBeHHO 33;!,3H B yc.10BI15IX 3a,n;aqn (33)J,3'IH 47, 48) HJIH me H3 ycJioBHH 3aJJ,aliii onpeJI:eJieH (3a,1,aq11 49, 50).
§ 42. OCH ECTECTBEHHOfO TPEXfPAHHHI(A.
LIHCJIOBOE 3HA4EHHE CI(OPOCTU
PaccMorpHM, KaK BbJIIHC.'IHIOTCH cKopocrb 11 ycKopeHHe roqKH npn ecrecrBeHHOM crroco6e 3a,n;aHHH ,n;BHR(eHHH (eM. § 37), T. e. Kor,n;a 3a,n;aEu rpaeKTOpHH TOqJ(!{ 11 3aJ{OH JI;BH}KemiH TOqKH BJI;OJlb ::nofr TpaeKTOp!UI B Bl!,J;e S= f (t).
B 3TOM c.riy'lae3HatJ:eHlU! BeKTOpOB v 11 aonpeJI.eJIHIOT no HX rrpOeKUHHM He Ha OCH CHCTeMbl OTC'leTaQxyz (KaK B § 40), a Ha ITO;tBHlliHble
ocn Mr:nb, HMeiOm;He Haqa.'IoB TOIIKe M H JI:BHMym.HecH BMecre c H::'D (puc. 122). 9TH ocu, Ha3biBaeMhie OCRhttt ecmecmeenno~o mpexcpaHHUKa
(HJIH C!WpOCTHb!MH OCHMH), HarrpaBJieHbl C.'Ie,J:yiOIUHM o6pa30M: OCb
M1:- no KacareJihHOii K rpaeKropHH B cropoHy rroJIOMHTeJibHoro orcqera paccTmiHHH s; ocb Mn- no HopJYra.rm K rpaeKropHH, JieMaiUeii B COilpH!<aCaiOJUeHCH ITJIOCKOCTH 11 HarrpaB,'Iei-IHOl"!B CTOpoHy BOrHyTOCHI rpae:nop11n; ocb Mb- rreprreHJI:HKymipHo K
nepBbiM .zJ.ByM TaK, 1ITOObi oHa |
o6pa30Ba.aa |
c |
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HHMH rrpaByiO CHcTeMy oceii. HopMaJib M n, Jie- |
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:iK3IU3fl B |
COI1pHK3C3IOIUeHC5! |
ITJIOCKOCTli |
(B |
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IlJ10C!<OCTH |
C3MOH |
KpHBOH, |
eCJIH KpHB3H |
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ITJIOCKa>I), H33biBaeTCH C.JWBHOU HOpAWJlbiO, a |
0'}---b~--------~ |
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nepneHnHKYJIHpHaH |
eii |
HopMaJib Mb- 6u- |
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nopMaAbiO. |
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1 |
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CKopocTh TOIIKH, |
narrpaBJieHHaH no Kaca- |
PHc. 122 |
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reJibHofi K rpaeKropHH (pHc. 122), orrpe;r.e.TJHeT- |
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cH B OCHX |
M1:nb TOJibKO |
O.J:HOH |
rrpoeKUJieH Vt |
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Ha OCb M1:. Dp11 3TOM v,=v HJIH v,=-v. C.!!e.l!,oBare.!!bHo, v, IIJIH coB-
na.n.aer C MO.lJ.yJieM CKOpOCTH V, HJIH OTJIIi'IaeTC51 OT V TOJibKO 3H3KO\I. YCJIOBH:>.1CH Il03TOMY B .lJ.3JlbHeiiweM o503H3'!3Tbv, TOMe CHMBOJIOM V,
orrycKaH HH;teKC 'r, |
11 H33biB3Tb V t{UCJl06blM (H.TIH aJ?ce6paUtteCKUM) 3Na- |
tteHUeAt CKOpocmu. |
MO.ll.YJib CKOpOCTH EO BCeX CJiy'laRX, Ror.n.a 3TO He |
MOMeT Bb13B3Tb He,n;opa3yMeHHH, 6y.n.eM TOme 0003HaqaTb CHMBOJIO!Il V, a Kor.n.a Ha.n.o rro.n.qepKHyTh, qro pe% H.lJ.eT o MoAyJie ciwpocni,- npa-
MeHHTh CHMBOJI \V\. |
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Haii,n;eM |
3Ha'!eHue v. |
Ec.rm |
3a rrpoMelliyroK BpeMeHII /).t TO'!Ka |
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coaepwHT |
BAOJib |
nyrH |
rpaeKropnH rrepeMerueHHe MMt=/)..s (cl\r. |
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puc. 115), |
r.n.e o,n;noBpeMeHHO |
/)..s- rrpupameHHe KOOPJI:HHaTbl s, To |
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qncJieHHO cpe.n.HeH |
CKOpOCTb!O |
TO'!KH 33 3TOT npoMe)KyTOK BpeMeHll |
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107
6y,ll.eT vcp= ~s//).t H |
B npeJJ,eJie Haft,rr,eM, |
'ITO |
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. |
!ls |
ds |
• |
(17) |
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V = hm |
M IV! 11 V |
= dt |
= S. |
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llt-+0 |
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TaKH'\1o6pa301\1, |
tttlC/l060e 3ftattmue C!Wpocmu |
mottf(U e oartftbtU MO- |
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.MeHm ape~vteHu pG.BHO nepaou npouseoihwu om paccmoJUlU.fl (KpuaoAuHet1Hou KoopouHambt) s amou molfxu no epeNeHu.
3HatieHHe V MOIKHO TaKJKe HaXOJJ,l1Tb KaK OTHOllleiiHe 3JieMeHTapHOf0 rrepe\1ei.Il;eHYIH ds 10'lKHK COOTBeTCTBYhl~.IJ;eMy rrpoMelliyTKy BpeMemr df.
TaR KaK BcerJJ,a dt>O, TO 3HaK v coarraJJ,aeT co 3HaKoM ds. CJieJJ,oBaTeJIGHo, KOf)J,a v>O, CKOpOCTb HanpaBJieHa B CTOpoHy TIOJIOIKI1TeJibHOf0 OTCtieTa paCCT0f!HI15I S, a KOf)J,a v<0, - B I1pOTI1BOITOJIOIKHYhl CTopoHy,
TaKI1M o6pa3oM, aeJIHtiHHa v OJJ.Hoape,reHHO onpe.u.eJIHeT H MOJJ.YJIL cKo-
pocn!, |
n cTopoHy, KyJJ.a oHa uanpaBJieHa. |
§ 43. |
KACATEJihHOE H HOPMAJiblWE YCKOPEHH51 T01.JKH |
B § 39 6LIJIO ycTaHOBJieHo, 'ITOyciwpeHI1e aTO'lKI1Jie:>KHT B corrpnKa-
Cal{)I.Il;elicH rrJiocKOCTH, T. e. B rrJIOCIWCTI1 M-rn. CJieJJ,oBaTeJILHo, rrpoeKU11H BeKTopa aHa 611HOpMaJib Mb paBHa nymo (ab=O). Hail:,ll.eM npoeK-
UI1H aHa ,ll.Be JJ.pyrne OCH. DpoeKTI1PYH ooe 'l8CTHpaaeHCTBa (10) |
Ha OCH |
||
M-r 11 Mn H 0603H8IJaH CHMBOJiaMH (dv)'t H (dv)n |
rrpoeKUHH BeKTOpa dv |
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Ha ~TI1 |
OCI1, TIOJIY'li1M: |
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(18) |
BeKTop dv rrpeJJ,cTaB,TIHeT co6oil: pa3HOCTb MeJKJJ.Y |
cKopoCTHMII |
B JJ.Byx |
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coceJJ,Hnx |
TO'lKax M n M' (p11c. 123, a), T. e. |
dv=V'-V. OTJIOJKHM |
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BeKTOpbl v=MA 11 v' =MB OT o6mero HatiaJia (p11c. 123, 6); Tor,n;a dv=
a) |
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o) |
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!!' |
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n~,TJ di1 |
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rf, |
v |
4 c r: |
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r' |
- |
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M |
lJ |
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Pflc. |
123 |
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=AB, a qmrypy |
ACED rrp11 6ecKoHetiHO |
M:aJioM yrJie dcp MOJKHO pac- |
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C\1aTpnaaTL KaK |
rrpm1oyro.TibllHI<. |
0TCIOJJ,a |
(dV),=AC=DB=MB- |
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-MA -=v'-v=dv, rJJ,e dv- 3Jie\leHTaprwe |
rrp11pamenHe tii1CJioBoro |
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3HarreHI15I CKOpOCTI1. ,UaJiee, TIOCKOJibKY npeJJ,eJI OTHOllleHRfl JJ.YfH K xop;r:e paaeH eJJ,I1HHue, MmKHO AD paccMaTp11BaTL KaK 3JieMeHTapnyro
.n.yry pa,n;11yca MA, pa3Mep !WTOpoti orrpeJJ,eJJHeTcH npo113BeJJ,eHI1eM
paJJ,Hyca na |
ueHTpaJibHLrti yroJI. |
TorJI:a |
(dv),:=AD=MA ·dcp=vdcp. |
|
DoJI:cTaBJIHH |
Hail:.u.eHHLie 3HatieHHH |
(dv), |
11 (dv)n a paaeHcTaa |
(18), |
rroJiy'l11M: |
a't=dvjdt, an=vdcp;dt |
(19) |
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108
YroJI Mem.n;y KacareJihHbiMH K KpHBoii B .n;Byx ee roqKax Ha3biBaercH yzAoM cMe:>FCHocmu; ror.n;a dcp- 3JieMeHrapHbiH yroJI cMemHoCTH. Ha-
rroMHHM, |
qro onroweHHe dcp K ds = MM', |
onpe.n;emrer |
KpHBH3HY |
KpHBOH |
B TOqKe M, a KpHBH3Ha k HBJIHeTCH |
BeJIHqHHOH, |
oopaTHOH |
pa.n;Hycy |
Kp11B113Hbl p B ::nofi TO'IKe, T. e. |
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dcplds=k= 1/p, |
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(20) |
BBe)J;eM 3TY BeJIHqHHY BO BTOpoe H3 paBeHCTB {19) ero, ytrTH eme paneHCTBO (17), K BH;J,Y
a |
d~ ds |
I |
v2 |
=V -- =V· - ·V= - . |
|||
n |
ds dt |
p |
p |
B pesyJihTare oKoHqareJihHO rroJiyqHM:
d.; d2s |
v2 |
at= at= dt2' |
an- p'ab = 0. |
H rrpeoopasyeM
(21)
TaKHM o6pasoM, liibi JI:OKasaJ1H, qro npoeKl{ttJl ycKopeHuJl mott!W Ha KacameAbHYIO paena nepeou npou3so8Hou om ttuc;weozo 3HattefluJl CKO· pocmu tl/lU emopoii npOU3B08flOU om paccmOJlflUJl (KpUBOJIUHeUHOU !WOp· aunambt) s no epeMeHu, a npoeKlfUJl ycKopeHUJl Ha cAaeHyiO nopMaAb pasHa KBa8pamy CKOpocmu, aeJleflflOJ.ty fla pa8uyc KpU6U3flbl mpaeKmopuu 6 aawwu mo'iKe Kpueoii;
Hymo. 3TO OJI:Ha !13 BamHbiX reope11,1 |
KHHeMaTHKH. |
BeJIHqHHbi at H |
an. H33blB3IOT KaCame/lbl-lblM H HOpMaflbflblM ycKopeHHHMH TQqKH. |
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Opn ll:BIUKeHHH TOqKH M B o,n:noil n.~OCKOCTH KacaTe.'lbHal! M-. noBopaqnBaeTcll |
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BOKpyr 6rmopMaJJH Mb c yrJJOBOil: cKopocTb!O |
w=dcp/dt. Tor,n:a |
BTopoe 113 paBeHCTB |
( 19) .naeT ern.e o,n:ny, qacTo ncnoJJb3yeMyiO B |
HH:lli.eiiepiioil: npaKTHKe ljlopMyJJy ,n:Jill |
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BbJqHcJieHHll an: |
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(21') |
Yh IIee CJie,n:yeT, qTQ HOpMaJibHOe ycKopeHHe paBHO npOH3Be,'l,eHHIOCKOpOCTH TOqKH na yrJIOByiO CKOpOCTb noBopoTa KacaTeJibl!OH K TpaeKTOpHH.
OrJimKHM BJI:OJib KacareJihHoii M-r 11 r JiaBiroii HopMaJIH M n BeKropbi at H Gn 1 T. e. KaCaTeJibHYIO H HOpMaJibHYIO COCT3BJIHIOI.Il;He ycKopeHHH
(pHC. |
124). |
DpH |
3TOM CO· |
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CT3BJIHIOIIJ;3H a~ oyJ(eT BCer- |
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)J;a HanpaBJiena |
B |
cropoHy |
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a |
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BOfl!YTOCTH KpHBOH, TaK KaK |
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BCerJ(a a11>0, |
a COCTaBJIHIO· |
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w.aH a, Momer 6biTb HarrpaB- |
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Jieaa HJIH B IlOJimKHTeJibHOM, |
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HJIII B |
OTpHu.areJibH0\1 |
Ha- |
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rrpaBJieHHH ocH |
M-r B saBH- |
P11c. |
124 |
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CHMOCTH OT 3HaKa rrpoeKU.lUI |
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at (eM. |
pnc. |
124, a, 6). |
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BeK1 op |
ycKopeHHH |
roqK!I a H3o6pamaercH |
J(HaroHaJibiO napaJI- |
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JieJiorpaM~a. rrocrpoeHHoro Ha coc1aBJIHI01.IJ;HX at |
H G,,. TaK KaK 3TH |
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COCTaBJIHIOlll,He |
B3311MHO neprreHJ(HKYJIHpHbl, TO |
MOJI:YJib BeKTOpa a |
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109