(формат)Тарг
.pdfOKPY*HOCTb pa,.uHyca h, ITJIOCKOCTb KOTOpofr ITepiTeH,UHKYJIHpHa o:::H spameHHH, a ueHTp C Jie)KHT Ha ca:,wii: ocn. EcnH 3a speMH dt nponcXOLI.HT 3JieMeHTapHbiH rroaopoT TeJia Ha yro.1 df:P, TO T04Ka M npH 3TOM cosepwaeT B.UOJib csoeii: TpaeKTopnu 3JieMeHTapHoe rrepe:.IerueHHe ds=
Tor.na 4HcJiosoe 3Ha4eHHe cKopoCTu T04KH 6yn.eT pasHo OTHO-
weHH!O ds K |
dt, T. e. |
dm |
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d~ |
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V=dt= h dt |
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HJIII |
v = hffi. |
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(44) |
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CKopOCTb |
V B OT.'IH4HeOT yr.'IOBOHCKOpOCTll TeJia H33b!BaiOT HHOrll.a |
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erue /lUHeUflOU HJ!H OKpy'J/CHOii CKOpOCTb!O TO'IKII M.
TaKHM o6pa30M, ttUCAoaoe 3Hatt.eHue cxopocmu mottKU epaUJf110ll{EcOCR meepaoco me;w paeHo npou3eeaeHUIO yc.!loaoii cKopocmu me!la Ha paccmoRHtte om :;moii mottKll iJo ocu epau;.eHufl.
HaiTpaBJieHa CKOpOCTb ITO KaCaTeJibHOH K OIIHCb!BaeMOi'f T04KOII OKPY*HOCTH rwn ITepiTeH,.uHKy.'IHpHomwc:wcru, ITpoxo,.uHruefr qepe3 ocb sparueHnH u TO'IKY M.
TaK KaK ,U.'IflBCeX TO'-IeKTeJia (f) HMeeT B ,UaHHbl!f MOMeHT BpeMeHH OLI.HO l1 TO JKe 3Ha4eHHe, TO H3 ¢op~1y.'Ibi (44) CJie,.uyeT, 4TO CKOpOCTH
T04eK sparuaiO:uerocH TeJia rrpoiTopuiioHaJibHhi nx paccTOHHIIHM OT ocu
BparuemrH. 00JI2 C!WpoCTefi TO'!eKBpaill,a!OlUCfOCH TBep,.uoro TeJia HMeeT BHLI., ITOKa3~HHblH Ha pHC. 136.
Pv.c. 135
2. |
Y c K o p |
e H H n T o q |
e K T e JI a . |
.Unrr Haxo:JKn.emrH ycrwpc- |
HliH |
TO'-IKH M |
BOCITOJib3yeMCH |
cpopMyJia~.!H |
a,=dv'dt, an=V2 /p. |
B Hawe~.l CJiyqae r=h. no.'l;CTaB.TJH5l 3HatiCHUe v H3 paBeHCTBa (4~)
B BbipaJKeHHH at H an, ITOJIY'IHM:
dw h~w2
ar=hdT' an=h
(45)
KacareJibH?H COCTaBJIHIOIUa51 ycKopemm a, narrpaB.TJeHa ITO KaCaTe.'lbHOli K TpaeKTOpHH (B CTOpOHy ,UBHllieH!Ili IItJH ycKopeHHOM :sparueHHii TCJia H B Oopamy;o CTOpOHy ITpH 3aMeJJ,JieHHm1); HOpMaJibHa51
COCTaB.1HIOIU35I an BcerJ].a HanpaeJieHa no pa,.uuycy LVI c K OC!l sparueHH}1 (pHc. 137).
123
DoJIHoe ycKopeHIIe T04KII M 6y,JJ,eT a = Va~+a; IIJIH |
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a=hVs2 -l-w'. |
(46) |
0TKJIOHeHHe BeKTOpa ITOJIHOrO ycrwpeHHH OT pa,JJ,uyca OITHCbiBaeMOH T04KOH OKpy)KHOCTII OITpeL(eJ!fieTCH yrJIOM J-1, KOTOpb!H Bbi4HCJIHeTCH ITO
¢opMyJie |
tg[1=a,/an [BTopaH |
H3 ¢opMyJI (22)]. DoL(craBJIHR CIO)J,a 3Ha- |
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qeHHH a,; |
ll an H3 paBeHCTB |
(45), IT?JiytraeM |
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tg f1=B/cu 2 • |
(47) |
TaK KaK cu u s HMeiOT B ,JJ,aHHb!H MOMeHT speMeHH L(JIH scex T04eK TeJia 0):\HO ll TO )Ke 3Ha4eHIIe, TO H3 QJOpMyJI (46) II (47) CJieL(yeT, 4TO ycKopeHHH scex TotreK Bpaw;aiOw;erocH TBep,JJ,oro TeJia ITpoiTopuuoHaJibHbl HX pacC-TOHHHHM OT OCII Bpam;eHI15I ll o6pa3yiOT 1;3 ,JJ,aHHb!H MOMe!IT BpeMeHH O,JJ,HH H TOT 1Ke yroJI f1 C paL(HycaMII OITHCb!BaeMbiX Hl\iH OKpy)K- HOCTef.f. TioJie ycKopemn1 TotreK npaw;aiOmerocH TBep,JJ,oro TeJia HMeer BH,JJ,, llOKa3aHHbiH Ha pHC. 138.
PHc. |
138 |
Puc. 139 |
¢opMyJibl (44) - |
(47) TI03l30Jl5IIOT orrpeL(e.II!-!Tb CKOpOCTb H ycKopeHHe |
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JIIOOOH TO'IKHTeJia, eCJII1 li3BeCTeH 33KOH Bpam;eHIIH TeJia H paCCTOHHHe L(aHHOH TOqKH OT OCH Bpaw;eHHH. Do 3TH~ li{e <!JopMyJia~ MO)i{HO, 3HaH
,JJ,BH)KeHHe 0):\HOH TO'IKHTeJia, Ha}iTi! L(B!DKeHIIe JIIOOOH L(pyro:ii ero T04KH,
a |
TaK)Ke xapaKrepHcriiKII ,JJ,BmKemm |
scero |
reJia B |
ueJioM. |
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3. B e K T 0 phi C K 0 p 0 C T 11 |
II y |
C K 0 p |
e H 11 H T 0 q e K |
T e JI a. 4TOObl HaHTH Bbipa)KeHHS'IHerrocpe,JJ,CTBeHHO L(JIH BeKropOB V H |
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a, rrpone,JJ,e~ 113 ITpOI13BOJibHOH roqKH |
0 OCH AB paL(Hyc-BeKrop |
rT04KH |
M (pHc. 139). TorL(a h=r sin a n |
ITO cpop:-.-ryJie (4·1) |
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IvI= I(!) Ih =I(!) Ir sin a |
mm IvI= f'W X rI· |
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T aKHM o6pa30M, - MOL\YJib BeruopHoro ITpOH3BeL(eHHH ffi X r paBeH Mo)i"yJIIO CKOpOCTH TO'IKHM. HanpaBJieHHH BeruopoB wX r II v TO)Ke cosiTa,JJ,aiOT (o6a OHH rrepiTeHL(IIKYJIHpHbi ITJIOCKOCTH OMB) u pa3MepHoCTH ux OL(HHaKOBbi. CJie)J,onareJihHO,
V=wXr, |
(48) |
124
T. e. BeKmop cKopocmu AI06oii mo•u\.u epaU{a!Owp.ocfl meAa paeeH eeKmopHOMY npou3eeaeHUIO YcAoeou CKOpocmu meAa Ha paauyc-eeKmop amoii tnOtiKU. ¢OpMyJiy (48) Ha3b!BaiOT cpOpMyJIOH 3ii:Jiepa.
Bepa or o6eux qacTeii paseHcTBa (48) rrpou3BOJJ.Hbre no speMeHH, llOJIY4HM
HJIH |
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a = (eX r) +(@Xv). |
(49) |
ci>opMyJia (49) oiTpeJ].emrer BeKrop ycKopeHHH moooii T04KH spamaiOmerocH reJia.
BeKrop BX r HarrpaBJieH, KaK II BeKTOp ffi X r, T. e. ITO KaCaTeJibHOH K TpaeKropuu TO'IKHM, a (exrl=er sin a=eh. BeKTop *e wxv HaITpaBJieH BJJ.OJib MC, r. e. ITO HopMaJiu K rpaeKropuu ro4KH M, a lw x Xv\=wvsin 90°=w2h, raK KaK v=wh. Y4HTbiBaH see ::~ru pe3yJihTaThr, a TaK>Ke cpOpMyJibi (45), 3aKJIIO'!aeM, 'ITOeXr ~ H ruXV=a11 •
3a)J.a'la 54. BaJI, ,!(eJ!aJOW:HH n=90 o61MHH, nocJie BbiKJIIO'IeHHH ,!(BHraTeJia na- 'IHHaeTnpaw:aThCR panno3aMe!l,.TieHHO H OCTaHaB.'IHBaeTcRqepe3 t1=40 c. OnpeAeJIHTb, CKOJibKO o6opOTOB Cl\eJiaJ! B8•1 3a 3TO BpCMH.
p e Ill ell He. TaK K3K BaJ1 BpamaeTCH paBH03aMe,!(J!eHHO, TO J],JlH Hero, C'IHTa5!
<i'o=O, 6y,!(eT
(a)
HaiJaJibHOH yr.ITOBOH CKOpOCTblO npH 3aMC,!(J!eHHOM Bpaw:eHHH HBJIS!eTcH Ta, KoTopyro BaJI HMeJI AO BbTKJIJO'!eHHHADHraTeJia. CJie,!(oBaTeJibHO,
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w0=nn130. |
B MOMeHT OCTaHOBKH |
llllH f=f1 |
yrJ!OBa5! CKOpOCTb BaJia, w 1=0. IlO,!(CTaBJillH STR |
8Ha'leHH5! BO BTOpoe H3 |
ypaBHeHHH |
(a), I!OJiyqaeM: |
O=:rtn/30+ct1 H e=-:rtn/30t1.
ECJIH o6o3Ha'IHTbqHcJio c,!(eJiaHHblX BaJJOM 3a apeMSI t1 o6opoToB qepea N (He CMeiiJHBaTb C n; n - yrJIOB35! CKOpOClb), TO yroJI IIOBOpOTa 38 TO JKe BpeM5! 6y,!(eT paBeH <p1 =2:rtN. ITO,!(CTaBJI5!H H3H,!(elliii>le 3Ha'leHH51 e H q1 B nepnoe H3 ypaBHeHHH (a),
TIOJIY'IHM
2nN= (nn/30)t1-(nnj60) t1 =(:rtn/60)ti,
N= nt11120=30 o6.
3a,IJ.a'la55. l'vlaxoBHK paAuycoM R=0,6 M npam,aerca paBHOMepHo, ,IJ.eJiaa n= =90 o6fMHH. Onpe,!(eJIHTb CKOpOCTb H ycKopeHHe TO'IKH,JieiKalll.eH Ha 060,!(e M3XOBHKa.
p e lii e H H e. CKopOCTb TO'!KH o6o,IJ.a V=R W, r,!(e yrJIOBa5! CKOpOCTb W ,ll.OJIJKH3 6h1Tb BblpaiKeHa n paAHaHax B ceKyHAY· TorAa w=nnl30=3:rt a v=R·3:rt~5,7 M/c•
.UaJiee, TaK KaK w=const, To e=O, H, cJie,IJ.oBaTeJihHO,
a=an=Rw2=R·9n2~53,3 Mlc2•
YcKopeHHe TO<[KH HanpaBJieHO B ,!(3HHOM cJiyqae KOCH Bpaw:eHHH. |
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3a,IJ.aqa 56. IloJiaraH, 'ITOnpH pa3rone MaXOBHK npaw:aeTCH no saKoHy |
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cp = c0 +c1f +Cze-kt, |
(a) |
onpe,IJ.eJIHTb 3HaqeHR5! noCTORHHbiX Ko:;,¢¢mJ,HeHTOB c0 , c1 , c2 H k H3 ycJioBHH, 'lTDnpH t=O ,1:\0JIJKHO 6b!Tb t:p0 =0 H W0=0 H 'ITOnpe,!(CJibH35! yrJIOBa5! CKOpOCTb, ,ll;O KOTOpofi
125
pasroHHeTCH MaxoBHK Wnp=50 c- 1 , aero ynwBoe ycKopeHne npa pasro11e ne )l;OJJJKHO npeBbiiiJaTb 3H3'leHUHeH= 10 c- 2. HaihH Ta!oKe, KaKoe ycKopen:ae 6y.r~eT npn 9TOM y TO'leKo6o.r~a \13\0BHKa B MOMeHT spC11enn t1= l c, ecJm pa;~.nyc MaxosnKa R=0,4 M. p e llJ e H He. l13 ypaBHeHHH (a) BH)l;HO, 'ITOllpH t=O <p~O, ecJJH c0+c2=0, T. e.
c0=-c2• |
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,'J.a.1ee!!3 ypaBHeHUH (a) HaxonnM, liTO w=~=c1-kc2e-kt. CJieJJ.oBaTeJ!.bHo, npu |
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t=O lu~O, Cl\~11 c1 -kr2 =0, T. e. c1~kc~. |
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IlpH smx 3Ha'lell!ll!Xc0 H c1 ypannerme (a) npn:,leT Bllil. |
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<p=c2 (kt+~:-"1 -1). |
(6) |
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0Tcro,:z.a naxo;<mi |
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(B) |
I1epnoe !13 paBeHCTB (B) noKa3wBaeT, |
'ITO w co Bpe\Jene\I pacTeT n npu |
f-'>-= |
crpC\1l!TCH K npeJ1eJibllOMY sna'lenmoc~k; |
c.1c;zonareJibHO, W 0 p=c2k. H3 BToporo JKe |
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paBCHCTBa Bll;J.llO, 'ITOC cO BpeMeHe~l YCblB<l~T, CTpCWlCb ({ ny,IIO, a t·JaH60.~bU:ee 3Ha-
'!CHne c2k" |
nMeeT npn t= 0; |
cJJe;wsaTem.uo, |
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f'H"--Czk2. |
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w0 ,=50 |
c- 1 11 |
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Ho no |
ycijOBI!HM sana<m |
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f 11 = 10 c- 2. |
Tor.r~a /I.0.1lh.l!O 6biTb' c"k=50, a |
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c;!~z~ 10, omy.ua k=0,2 H c2=2)0. -I1pn 31HX |
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3lW'leH<!llX k H C2 paBeHCTBO |
(6) /1.8eT OKOH'lll· |
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1e.%HO C.1e.Z::'>Cllilllr3aKOH BpaW,ellHH MaXOBl-IKa: |
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cp=253(0,2t+e-~·2l-l). |
(r) |
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Tor,:;.a, 'l10sn;~.no H aa paBeHcTB (s), 6y- |
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p,er |
" |
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w=58 (l-e-o,2\ e= 10e-o,21. |
(,l\) |
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PHC, 14.0 |
.1_.-q MO\leHra BpeMeHH |
f1= 1 C, |
Y'lHTbl- |
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B3Sl, |
<no e- 0 •2:=o:0,82, no.1y'l!nr ffi ""9,0 c-1, |
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e ""8,2 c- 2. CJie/I.OBaTeJibHO, B 3TOT MOMeHT npeneHH a,=Rc |
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1 |
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1 |
=3,3 M1c2, a =Rw2= |
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1 |
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11 |
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=32,4 Mlc2 n a=Vai+a~=32,6 M,'c.2 •
3aJI.aqa 57. rpys B (pnc. l40l npncopnT ao rparr enne ra,T panrrycm-r r n cn,c~.H· ill)'lOHa O,!(HOH OCH C B3JIOM llJCC1epH!O J pa~h:\CC\1 r1 . Ll,BHJKeHHe rpysa H3'llmaeTCH 1!3 COCTOHHHH IIOKOH II npOHCXO,JHT C !lOl'TOlHIJlbl\1) CKopeHHeM a. 0npe,c~.e.1!1Tb, IIO K3· KOMY saKoHy oyJieT np11 3TOM BpammvcH H3~\0,'!i'IL(a>.cH B 3a:J.enJien,m c mccrepHeii 1
rnecrcpHH 2 pa.J:Hyet M rz·
P em e 11 H e. TaK KaK rpya B H3'1l-lllaeTJJ.Pllr3TbCH 6e3 Ha'la.1bHoi1 cKopocTH, TO era CKOpOCTb B JII06oii MO~IeHT Bper,Je!lll t p:lSHa Cf (vs=af). 3ry CKOpOCTh uy,ayr
l!MeTb H TO'!KH 060J1ll BaJJa. Ho, C !L(lyroii CTO(JOHH, CKOpOCTH 3fHX TO'leKpaBI!bl w1r, r,!(e w1- o6ruaH ;J.JJH BaJia H rnccTepHH 1 yrJJoBall chopocrh. C1ei'\oBaTe.'lbHO,
w1r=at, w1 =ai'r.
Teneph rmi'!.'I,eMUJ 2. TaK KaK |
rKopocTb TO'-lKII C([en.leHI!H C ,c~.oJIJKHa 6L1Th O)l;HOH |
H TOH JKe Jl.JlH o6eux mccTepcH, |
10 z·c=w1r1 "--WJz, orKy,!(a |
W2 = (r1 'r2)w1 =(r1a r"r)t.
HTaK, yrJronaH cKopocTh meCTepHH 2 pacreT nponopnuona.JJhHO BpCMCHH. Yt!llThiBaH, '!TOw2=dlh'dt,r.r~e <p2 - yroJ! nosopora mecrepna 2, no.'ly'IIIM
dcp1= (r 1 a'r2r)ldt.
0TC!0,!(3, 6ep5! OT o6eHX 43CTCH HHTerpaJib! H C'lliT351,'ITOllpH f=Q yro.'!<p2=01 Hall/I.CM OKO!!'laTeJibHOsaKOH paBnoycKope!!HOfO Bp<?.lll,Ct!Hfl lllCCTepHll 2 B BH,!(e
<p2= (r1 a:'2r2r)t2•
* 3Ha'leHHe(I) MOMeT H3MeHHTbCll !10 3aKOHY (JI), KOf.Ila Ha MaXOBIIK .'leilcTBYIOT !10CT05!HHblll Bpalll,a!Olll,HH MOMei!T H MOMeHT CH.'lconpOTHB.lJeHHll, nponOpi.\HOHaJlbHbiH w (3a/l.a'la 150 B § 128).
126
rnasa XI
OJIOCKODAPAJIJlEJlbHOE ~BR>KEHME TBEP~OfO TEJIA
§52. YPABHEHlUI flJIOCKOflAPAJIJIEJihHOfO ABH)I(EHH.SI ~BH)I(EHH.SI llJIOCKOfl «<lHfYPbl).
PA3J10)1(EHHE ABH)I(EHH51 HA flOCTYOATEJlbHOE H BPAJ.UATEJibHOE
llMcKonapaA.Ile.1bHb/.Jft (mm nAoCKU.M) Ha3bwaemcH maKoe OBUJ/Cettue meepaoco meAa, npu Komopo.M ece eco mottKu nepeJdeU{momcH napaAAeAbHo HeKOmopoii cpUKCUpOBaHHOU nAOCKOCtnU ll (piTC. 141). fJJIOCKOe )].B!ltKeHITe
coBepwaiOT MHoriTe '!aCTITMexaHH3MOB IT Marrnm, nanpiTMep KanmJ.eecH KOJieco na npHMOJIITHeiinoM Y':WCTKe nyTI-I, waTyH B KpHBownnHononsyHHOM MexaHIT3Me IT JJ.p. LJ.acTHbiM cnyqaeM rrnocKonapanneJihHoro )].BHJKeHITH HBJIHeTCH BpaiiJ,aTeJibHOe )].BITJKe!IITe TBep)].OrO TeJia BOKpyr HeOO)].B!DKHOif OCIT.
y
PHc. 142
TeJia KaKofr-HHOYJJ.h ITJIOCKOCTbiO Oxy, rrapa.'I- JieJihHOif IT.'IOCKOCTH fl (piTC. 141). f1piT IT.'IOCKOITapaJIJieJibHOM )].Bl!JKeHITH Bee TOl!KIT TeJia, Jie)KaiiJ,He Ha rrpm>1oii MM', rreprreHJJ.HJ{YJIHpnoii ceqeHITIO S, T. e. nnocKOCTIT TI, JJ.Bin!<yTcB TO)KJJ.eCTBeHHO.
OrciO)].a saKJIIOtraeM, liTO )].JIII IT3yl!eHHH JJ.B!ltKeHITH Bcero TeJia JJ.Oc· TaTOl!HO IT3Yl!ITTh, KaK JJ.BH:rKeTcn B ITJIOCKOCTH Oxy ceqenue S 3Toro TeJia ITJIH HeKOTOpaH ITJ10CKMI 4J11rypa S. f103TOMY B )].aJibHei'uneMBl'.JeCTO ITJIOCKOrO )].BIT)!{eHIUI TeJia oy)].eM paccMaTpiTBaTb )].BIT)KeH!Ie ITJIOCKOlf ¢nrypb! s B ee ITJIOCKOCTIT, T. e. B ITJIOCKOCTH Oxy. T1piT 3TOM Bee pe3YJihTaTbi, KOTOpbre OYJJ.YT rronytreHbi B § 53-59 )].JIH TO'!eK rrnocKo!"r ¢ITryphr, cnpaBe.!l.Jll1Bhi, KOHetrno, IT )].JIH Tol!eK cetieHITH S TBep)].oro TeJia, lJ.BHJKyUJ.erDCH ITJIOCKOITapaJIJieJibHO.
T1oJIO)Kemre ¢Hryphr S B rrnocKOCTH Oxy orrpe)].eJIHeTC5I rroJIOJKeHHeM KaKoro-nu6yJJ.b rrpoBe)].ennoro na 3Tofr ¢urype mpe3Ka AB (puc.
B cBOIO otrepeJJ.b rroJio)KeHHe OTpesKa AB MO)KHO onpeJJ.eJIHTb, 3HaH KoopTOtrKH A u yroJI cp, KOTOpbiH OTpe3oK AB o6pagyeT c ocbiO x. Tol!KY A, Bbr6pannyiO )].JIH orrpeJJ.eJieHHH rroJIO)KeHmr q:mryphl S,
6y.n.eM B )].aJibHeiiweM Ha3biBaTb noAIOCOM.
T1pH JJ.BH)KeHHH ¢Hrypbr BeJIH'!HHbixA, YA u cp 6yJJ.yT H3MeHHTbCH. lJ.TOOhl 3HaTb 3aKOH )l.BHtKeHIUI, T. e. DOJ!O)KeHHe <\lHrypbi B DJ!OCKOCTU
127
Oxy B JIIoooil MOMeHr speMemi, Ha,n:o 3Harb saBHCHMOCTH |
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XA=fi(t), YA=fz(t), cp=fa(t). |
(50) |
YpasHeHHH (50), orrpe,n:eJIHIOill.He saKoH rrpoucxo,n:mn;ero L\BH>KeHHH, H33hiB3lOTCH ypaeHeHUH.MU 06Ua!CeHUH nAOCKOU rpucypbl B ee ITJIOCKOCTII. 0HH )Ke HBJIHIOTCH ypa6HeHUHJvtU nAOCKonapaA/leAbH020 06UJ/CeHUfl meepOOcJ meAa.
DepBb!e L(Ba H3 ypaBHeHlJii: (50) orrpe,n:eJIHIOT TO .lJ:Bll)KeH!Ie, KOTOpoe
qmrypa coBepwaJia 6bi |
npn cp=const; 3TO, otJeBH.lJ:HO, 6y,n:er |
rrocryrra- |
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y |
TeJibHOe .lJ:BH)KCHHe, rrpH !WTOpOM |
BCe TOtJ- |
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KH q:mrypb! ,n:BII)Kyrcn raK )Ke, KaK nomoc |
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A. Tperbe ypasnemre |
orrpe,n:eJiner ,n:BII)Ke- |
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Hne, Koropoe qmrypa |
cosepwaJia Obi rrpn |
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xA =canst H YA =const, |
r. e. Kor,n:a rroJIIOC |
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A Heno,n:BIUKeH; aro 6yJI,er BPBili.eHHe 4m- |
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rypbr BoKpyr rroJIIOca |
A. 0Tc10,n:a MO)KHO |
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33KJII01U!Tb, 'ITOB OOill.eM CJiytrae 06U:J/CeHUe |
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nAOCKoti rpucypbt a ee MOCKocmu .MO:J/Cem |
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pacc.Mampueambcfl KaK |
CAaca/OU{eecH us no- |
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0'----------~x cmyname/lbHOco aeu:J/CeHUH, npu |
Komopo.M |
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PHc. 143 |
ace mottKU rpuzypbt aauJJCymcH maK JJCe, KaK |
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noAIOC A, u us apaU{ameAbHOcO |
oau:J/CeHUH |
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aoKpyc. amoc.o noAIOCa *. |
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0CH06Hbl.MU KUHe.MamutteCKU.MU xapaKmepucmUKa.MU paCCMaTpHBaeMOrO .ll:BH)KCHHH HBJIHIOTCH CKOpOCTb H ycKoperme rrocryrrareJibHOrO
.lJ:B~)KeHHH, paBHbie CKOpOCTH H ycrwpeHHIO ITOJIIOCa fvnocr=VA, a:0 CT= =aA), a raKme yrJioBaH cKopocrb ro H yrJioBoe ycKopemre e sparu:.a- reJibHOro .lJ:BH)KCHIIH BOKpyr ITOJIIOCa. 3Ha'!eHI1H3THX xapaKrepHCTHK B JIIOOOH MOMeHT BpeMeHl! t MO)KHO HaHn!, BOCITOJlb30B3BiliHCb ypaBHeHHHMH (50).
DpH usyqeHIIli ,JJ,BlliKeHHH MO>KHO B KatrecrBe ITOJIIOCa BblOHparb JIIOoyiO TOIIKY qmrypbi. PaccMOTpHM, qro ITO.JJyqllTCH, ecJIH BMecro A Bbl6parb B KaqecrBe ITOJIIOCa KaKyiO-HHDY.lJ:b ,n:pyry10 TO'-IKY C H orrpe,n:e- JIS'ITbITOJIO)KeiiHe cpurypbr orpesKOM CD, o6pasyiOIIJ.HM c OCbiO Ox yroJI cp1 (puc. 143). XapaKrepHCTHKII rrocrynareJibHOH qacrH .lJ:BH)KeHHH rrpu 3TOM H3MeHHTCH, TaK KaK B o6ru;eM cJiytrae Vc=FVA II ac=FaA (HHatre .lJ:BH-
)KeHHe |
cpHrypbi |
DbiJIO 6hr |
rrocrynareJibHbiM). X apaKrepHCTHKH .JKe |
Bparn.areJibHOH |
'I3CTH.lJ:BH)KC!lllS'I,T. e. ffi H B, OCT3IOTC5l Helf3MCHHbL\1H, |
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B caMoM ,n:eJie, |
rrpose,n:H H3 |
C rrpHMYIO CBr, rrapaJIJieJibHYIO AB, Mbi |
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qro B JIIOoofi MoMeHr npeMeH!1 yroJI cp1 =cp-a, r,n:e a=const. |
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~1=cp, |
~1=~ IIJIH |
cu 1 =w, £ 1 =e. |
CJie,n:osareJibHO, 6paU{ame,lbHW1 1WCtnb aau:J/CeHUH om 6bt6opa noAIOCa He saaucum.
* CooTBeTCTBeHHo rrJiocKonapaJI,'lemo,Hoe .QBH)KeHHe TBep.Qoro TeJia MO)KHO pacCMarpHBaTb KaK c.ilaraJOm.eecH H3 nocTynareJibHoro ,1\BH)KeHJHI BMecre c noJIJOCOM A H Bpam.aTe.'lbHOfOBOKpyr OCH, nepneH.QHKYJillpHOH llJlOCKOCTH II (eM. pHC. 141) H npoxo- )l;lllll.eH '1epe3nomoc A .
128
§53*. OflPE,nEJIEHHE TPAEKTOPHA TOliEK OJIOCKOA $HfYPbl
DepeHiJ.eM Terrepb K ll3yqeHHIO iJ.BIDKeHHH OTiJ.e.llbHbiX T04eK ITJIOCKOH qmrypbl, T. e. K OTb!CKaHHIO TpaeKTOpHH, CKOpOcTeH H ycKOpeHHH 3THX TotreK. Ha4HeM c orrpeAeJieHHH rpaeKropuii.
PaccMorpHM TOtrKy M rrJiocKoii qmryphr, noJIO)KeHue Koropoii orrpeAe.rHrercH paccTOHHHeM b=AM or noJIIOca A H yrJIOM BAM =ex. (pnc. 144). EcJIH )l.BHllieHIIe 3a,IJ,aHo ypaBHeHHHMH (50), ro KOOp)J.HHaTbi x H y T04KH M B OCHX Oxy 6y,nyr:
x=xA +bcos (cp+cx.), y=yA +bsin (cp+cx.), |
(51) |
riJ.e xA, YA, cp- H3BeCTHbie no ypaBHeHHHM (50) ¢YHKUHH speMeHH t. PaBeHcrBa (51), onpe)l.eJIHIOU~He 3aKoH L\BHllieHHH T04KH M B rrJioc-
Kocru Oxy, )l.3IOT OL\HOBpeMeHHo ypaBHem1e rpaeKrop HH .3TOH TotrKH B rrapaMerputrecKOM BHL(e. Oob!'!Hoe ypaBHeHue rpaeKropmr rroJiytruM, HCKJI!04HB 113 CHCTeMbi (51) BpeMH t.
EcJIH paccMaTpusaercH L\BHllieHue 3BeHa KaKoro-HHDYL\h MexaHH3Ma, TO L(JIH onpeL(eJieHHH rpaeKTOpmr Jl!OOOH T04KH 3TOrO 3BeHa L(OCTaTO'IHO Bbipa3HTb ee KOOpiJ.HHaTbi trepe3 KaKOli-HHDYL\b rraparv!eTp, orrpeiJ.e.IIHIOIII,HH ITOJlO)KeHHe MeXaHH3Ma, a 33TeM HCKJII04HTb 3TOT napaMeTp. YpaBHeHH.H )l.BJ1)KeHu.n (50) nprr 3TOM 3HaTb He o6H3areJibHO.
PRe. 144 PRe. 145
3a}l.a'la 58. TioJJ3YHbl A R B, K KOTOpb!M npHKpenJJneTCR JlRRenKa 9JlJJRncorpa!pa, nepeMelll.a!OTCR no B3aRMHO nepneH,!(HKYJlRpHb!\1 HanpaBJJRIOIIl.RM (pnc. 145). PaccTOHHHe AB=l. Onpe,!(eJJRTb TpaeKTopmo TO'IKR M mmeilKR.
P e III e H 11 e. Bsas sa rromoc TO'IKYA, 6y.QeM onpe.QeJJHTb noJJOMeHRe TO'IKR M Ha mmefiKe oTpesKOM AM= b. OoJ!O}!{eHRe ca~10f1 JlRHeHKH 3all.aeTcH yrJJoM <p. Tor,!l,a
,!(JlH Koop,!(RilaT x R y TO'IKHM noJJy'IRM:x=(b-l) cos <p, y=b sin <p. HcKJJ!OlJaR napaMeTp <p, !IaXO)~RM, 4TO TpaeKTOpReH TO'IKR(H<~3aBRCRMO OT 3aKOHa ,!(BHJKeHR5! JlRHeHKR)
6y,!(eT 3JlJ!RTIC
x2/(b-[J2+y2fb2= 1
C TIOJlyOCHMR a= [b-ll R b R C UeHTpOM B TOlJKe 0.
MeHH5! C TIOMOlil.biO COOTBeTCTBy!Olli.RX BHHTOB paCCTOHHR5! [ R b, MOJKHO Bbi'lepTRTb KapaHJ(aiii0\1 M 3J1JJRnc c Jli06biMR 3aiJ.aHHhiMH noJJyocRMR, He npesocxoJJ.HW.HMR pasMepos JJHHeiiKR. OTC!O,!(a a Ha3BaHRe MexaHH3Ma - 9/l/luncocparp.
5 Jlf• 2173 |
129 |
§54. ORPE,IJ,EJIEHHE CJ(OPOCTEfl TOtJEI( RJIOCI(Ofl .HTYPbl
B § 52 obiJIO OTMe4eHo, 4TO )J:BRmeHHe rrJiocKoii qmryphr MO}KHO paccMaTpHBaTh KaK cJiararorn.eecH H3 noCTyrraTeJibHoro )J:BH.memm, npH
KOTopoM see T04KH qmryphr .U.BH}KyTca co cKopocTbiO vA nomoca A, 11 us Bparn.aTeJibHOrO )J:BH}KeHY!H BOKpyr 3TOf0 ITOJIIOca. f10Kax-<eM, 4TO CKO-
pOCTb JIIOooii T04KI1 M qmrypbi CKJia)J:biBaeTCH |
reoMeTpH4ecKH |
H3 |
CKOpOCTeH, KOTOpbre T04Ka noJiyqaey B Ka}KiJ.OM |
!13 3THX ]J.BM}KeHr!H. |
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B caMOM .n:eJie, noJio}KeHHe JII06oii T04KH M qmryphl onpe.n:eJIHeTC.H no OTHOIIIeHHIO K OCHM 0xy pa.n:HyCOM-BeKTOpOM r t:_ +f' (puc. 146),
fAe rA - pa.n:uyc-BeKTOp flOJIIOca A' ? =AM - BeKTOp' orrpe.n:eJI.HIOlli.HM noJio}KeHHe TOtrKH M OTHOCHTeJibHO oceil Ax'y', nepeMemaromMxCH BMeCTe c rroJIIOCOM A nocTyrraTeJibHO (L\BHx-<eHue cp11ryphl no OTHoweHHIO K 3THM OC.HM rrpe)J:CTaBJI5!eT COOOH Bparn.eHHe BOKpyr flOJIIOca A). TorAa
-ctr drA ct?
VM=dt=dt+ctt•
B noJiyqeHHOM paseHCTBe seJIH4IIHa d~/dt vA ecTb cKopocTb BeJIH4HHa JKe pasHa Clmpocm v.MA• KOTOpyiO TOtiKa M
rroJiyqaeT npu rA =canst, T. e. OTHOCI!Te.'lbHOocei1 Ax'y', HJIH, HHatre rosopn, npu epawptuu tjJucypbt eoKpyc noA/Oca A. TarmM o6pa3oM, H3 npeiJ.biL\yruero paseHcTBa AeiicTBHTeJibHO c.rre)J:yeT, 4TO
(52)
T1pn 3TOM cKopocTb vMA, KOTopyro T04Ka M rroJiyqaeT npu sparueHHii cpurypbl BOKpyr ITOJIIOCa A, oy)J:eT (eM. §51):
VMA=W·MA (vMAj_MA), |
(53) |
r.n:e (!) - yrJIOBa.H CKOpOCTb qmrypbl.
TaKHM o6pa30M, c1wpocmb mo6ou mot~xu M nAocKou tjJuaypbt ceo-
.MemputieCKU CK/lGObt6aemCH U3 C1Wpocmu I'WIWU-Ht.t6yob apyzou nl04XU A'
Pnc. 146 Pnc. 147
npUHHmou sa noAIOC, u c1wpocmu, KOmopyw mottKa M noAytf.aem npa epaw,eHttU_ tjJucypbt BOKpyz amoco no.~wca.
CKOpOCTH VM HaXOA.HTCH flOCTpOeHHeM. COOTBeTCTBYIOill.ero napaJIJieJIOrpaMMa (puc. 147).
130
3a,n.a'la 59. Hal'ITHcKopocTh TOl!KH M o6o.r~a KOJieca, KaTllw:erocH no npliMOJIHHeiiHmly peJibCY 6e3 cK0.1h:lKeH!Ill (pHC, 148), eCJII{ CKOpOCTb UeHTpa C KOJieca paBRa vc. a yroJI DKM=ct.
P e III e H He. DpHHliB TO'lKYC, cKopoCTh KoTopoii H3BeCTHa, sa nomoc, Haii.r~e~l,
'IToVA1=Vc:+v,uc. r.r~e Vft1cj_CH H no ~Io)l;y.110 v111c=w·MC=wR (R- pa)l;Hyc Ko-
Jieca). 3nal!eHHe yrJioBoii cKopocTU w nafr)l;e\1 ns ycJIOBHll roro, lJTO TO'IKaK KoJieca He CKOJ!b3HT no peJihCy H, CJ!e)l;OBaTeJ!b!IO, B )l;3HIH>IH MO\IeHT BpeMel!H VK=O. C ,u;pyfOH cropoHhl, raK JKe KaK n l!Jill TOlJKH M, t7K=~+Vk-c. r.r~e vKc= w·KG= wR. TaK
KaK )];Jill TOlJKH K cKopocrH vKc n vc HanpaB.1eHhi B)l;OJ!h Ol\HOH npliMOi'!,To npH Vl(=O vKc=Vc, OTKy,u;a w=t'c R. B pe3)Jlhrare Haxo.u;HM, 'ITOvl\Ic=wR=vc.
6y.r~er npa 3TOM poM6o\l.
YroJI MeJK.!IY vc Hv,uc paBeH ~. TaK KaK cTopoHh!, o6pasyJOW:He aroT yroJI H yroJI B, ESaHMHO nepeneH,u;HKy.1HpHhl. B CBOJO o'lepe)l;hyroJI t\=2ct, KaK u.earpaJibHhlH yroJI, onnpaiOmal'lcllHa TY JKe ;ayry, 'ITOH BnJfcaHHhiH yroJI ct. Tor.r~a no CBOHCTBa~l poM6a
) r!lhl MeJK)l;y vc H 1!111 11 Me.-K::\Y VJ;c n VJI! To:lKe paBHhl a. OKOil'mTeJihHO, TaK KaK ,!(HaroHaJlH po"16a Bsa:nwo ne;,er.<eH;:(HKYJillpm>~, rroJJy'IHM
PacqeT, KaK BH,UH\1, oKa3hJBaercH .QocrarolJHO rpoM03.QKHM. B ,!(aJibHei:iiiieM 11hi TI03H3KOWI\ICH C ~1eTO,'J3~1H, ll03BOv1liJOW,m1ll peiiiaTb aH3.1J0f!{lJHh!e 33)1;3'111 ropa3,!(0 npow.e (eM. aa.r~a'ly 61 B §57).
§ 55. TEOPEMA 0 flPOEK~H51X CKOPOCTEfl |
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;:J.BYX TOliEK |
TE.IJA |
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Onpe,n:eJieHHe |
CKopoCTeii |
TOtJeK |
nJIOCKOH |
4JHryphi (HJIH TeJia, ,ll,Bll- |
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)K)'Uieroc}J !1J10CKOITC!p<1lJJieJibllO) |
C |
ITOMO!IIbiO |
cpOpMyJihi |
(52) |
CBH3aHO |
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OObi'!HO c ,n:oBOJlhHO c.'Ionmbi~m |
pactreTaMH (eM. 3a,n:aqy |
59). |
O.n.HaKo |
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I!CXO,I\H H3 3TOf0 OCHOBHOfO De3y.'ILT3Ta, |
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MOlliHO ITOJiyqHTb pn,n: ,n:pymx, npaKTII- |
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'!eCKH6oJiee y,n:o6HL!X II rrpOCTb!X |
MeTO- |
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.UOB orrpe,n:eJiemw |
CKopocTeii: TotreK cpH- |
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ryphi (HJIH TeJia). |
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O,n:IIH H3 Tamrx MeTO.:!.OB |
,n:aeT |
Teo- |
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pow: npoeK"fUU CIWpocme(i OBljX fr10'1C:;; |
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mcep8oco me.tza I-ta ocb, npoxooHU{YtO <te- |
PHc. |
149 |
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pe3 amu mo4Xll, pa'3rtbt apyc opy2y. |
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PaccMOTPII\! KaKlle-Hli6);IL ,lBe TO'lKHA H |
B rr.'IOCKOilqmryphr (liJm |
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TCJra). DpHHHMaB ToqKy A sa no.1!DC (puc. 149), rro.'IytraeMnocpopMyJie (52), trTO vn=vA+vnA· 0Tcrc;:,a, npoeraHpyH o6e '!UCTH paBeHCTBa Ha ocb, Harrp2s.1emryro no AB, a Y'IliThiBMI,qTo BeKTop VnA neprreH,D.HKyJinpeH AB, Haxo,JJ..H:\1
(54)
H TCOpe11!3 ,I\CKD3a!la. 3a:'IIeTJ::\I, IJTO 3TOT pe3yJihT3T HCeH H Jl3 qHCTO Q":3ll'ICCKTIXCOOupm;<eHl!Y!: eCJIH paEe:'CTBO(54) He 6y,leT BbliTOm!SlTbCH, TO rrpn ,n:BmEeH!Il! poccronHHe Men..:!.y TOtrKaMH A I! B ,n:ommro II3l.reHnTLC5l, 'ITOHC'I:03\10iKHO,TDK K3K 1 CJlO C'liiTUCTC5Ia6COJII01 :m TBCp,n:bil\1.
Oo:YTOMY panencmo (54) BbiiTOJIJT<TClCH ne TOJihKO rrpa DJIOCKOrrapaJIJIC,lLHO~r, HO II rrpll JII060M ,ll.BlDi<CHIIH TBep,D.OfO TeJia,
S' |
131 |
,UOK333HH35I Teope:vta IT03BOJHieT JierKO ll3XO,JJ,HTb CKOpOCTb ,JJ,aHHOH T04KH reJia, eCJIH l!SBCCTHbl HarrpaBJICHIIe CKOpOCTH 3TOH T04KH II CKO· pOCTb K3KOH·HHDYL\b ,JJ,pyrorr TOqKH TOrO )Ke TeJia.
3aJI.a'la60. HaiiTH 3aBHCIIMOCTb MeM.QY CK opoeTl!MII TolJeK A 11 B mmei1KII 3Jl-
JJHneorpa¢a (eM. p11e. 145) np11 )l;aiiHOM yn1e <p.
P e III e H H e. Hanpas"~eHHll eKopoereft TolJeK A 11 B 113BeeTHhl. Tor.Qa, npoeKTHPYll BeKTOphl VA 11 VB Ha oeb, HanpaBJieHHyiO no AB, noJJy'liiMcorJiaeHo ):\OKa3aHHOH
reope~1e
VA COS <p=VBeOS (90°-<p),
§ 56. OUPEAEJIEH HE CI(OPOCTEfl TOtfEK UJIOCKOfl $ HfYPhl C UOMOJ.UbiO MrHOBEHHOrO ~EHTPA CI(OPOCTEfl.
UOH51THE 0 ~EHTPOHAAX
.Upyroii rrpocroil II HarJI5I,JJ,llhiH MeroL( orrpe,JJ,eJieHH5I CKopocreii ToqeK l!JIOCKOH qmrypbl (HJIH reJia rrpH ITJIOCKOM ,JJ,Bll)KCHHH) OCHOBaH Ha ITOH5I· TllH 0 MrHOBeHHOM Ue!irpe CKOpOCTCH.
M mooeHHbt.M upmzpoM c1wpocmeil Ha3bt6aemcH mottiW nAocKoii if;ucy- pbt, c1wpocmb Komopoli 6 oaHHbtil MOMeHm 6peMeHU pMHa HYAIO.
JlerKO y6e,JJ,HTbC5l, qyo CCJIH 4Jnrypa ,JJ,BH)KeTC5I HCITOCTyrraTeJibHO, TO
T3K35I T04K3 |
B K3)K,Jl,blH MOMeHT EpeMCHH t CYlli.CCTBYeT H rrpHTOM CL(HH· |
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c rBeHHMI. flycTh B MoMeHr speMeHn t T04KH A |
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II B nJiocrwii: (jmrypLI |
HMeiOT |
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He rrapaJIJieJibHh!e |
L(pyr |
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150). |
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py vB, H 6yL(eT MrHOBeHHh!M ueHrpoM crwpocreii, |
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P11c. 150 |
TaK KaK Vp =0*. B caMoM ,JJ,eJie, ecJIH L(orrycTHTb, |
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4TO Vp=#=O, TO ITO reopeMe |
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neprreH,JJ,HKYJI5IpeH H AP, (raK KaK vA..LAP) H BP (raK KaK VB iBP), 4TO HeB03MO.IKHO. J.-h TOH .IKC reopeMbl BIIL(HO, 4TO HHK3K35I L(pyra5I T04-
Ka ¢urypbl B 3TOT MOMeHT BpeMCHH He MO.tKeT HMeTb CKOpOCTh, paBHyiO HyJIIO (HarrpHMep, )J.JI5I TotrKH a rrpoeKurm vB Ha JIHHHIO Ba He pasHa HyJIIO H, CJieL(OBaTeJibHO, Va=fo0 II T. ,JJ,.).
EcJIH rerrepb B MOME'HTBpeMCHH t B35ITb T04KY p 33 ITOJIIOC, TO flO cpopMyJie (52) cKopocrL T04KH A 6yL\eT
raK KaK Vp=O. AHaJiorHtrHbiH pe3yJ1LTar rroJiytraercH )J.JIH JII06ofi L\PY· roi1 rotrKH ¢HrypLr. CJieL(oBareJibHO, cKopocmu motteK nAocKoii tjJuzypbt
onpeoeAHIOmCH 6 OaHHbtii MOMeHm epeJ>~eH.u maK, Kat\.. ecAu 6bt 06UaJCeHue ipucypbt 6bt/lo 6paU{eHueM 60Kpyc J.tcHoBeHHoco ~?eHmpa cKOpocmeil. flpu
* Pa3MCpbl qmryphl Beer,!(a MOMHO npe,!(eTaBHTb ee6e TaKIIMII, lJTO TO'lKap Oy)l,eT npHHa,!(.~e.lKaTb <>TOH qmrype (eM. HII)KC npiiMep II p11e. 151).
132