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§ 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. |
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 |
|
coaepwHT |
BAOJib |
nyrH |
rpaeKropnH rrepeMerueHHe MMt=/)..s (cl\r. |
|
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 |
||
107
6y,ll.eT vcp= ~s//).t H |
B npeJJ,eJie Haft,rr,eM, |
'ITO |
|||
. |
!ls |
ds |
• |
(17) |
|
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 |
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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) |
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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) |
o) |
!!' |
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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- |
|
-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. |
||
dcplds=k= 1/p, |
(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= - . |
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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') |
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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 |
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
n yroJI I+ ero OTK.TioHeHIIH oT HopMaJIH Mn onpe;a:eJIHTC.H ¢opMyJiaMHl
(22) |
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rn:e- :n/2~!-L~:n/2; npH |
~t>O BeKTOp aOTK,1IOIIeH |
OT HopMa.'Ill Mn |
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n CTopoHy OCI! /\111:: (p!ic. |
124, a), a rrpn J1.<0- B rrpoTIIBOIT0,101KHYtJ |
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CTopony (puc. |
124, 6). |
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Ta101~1 o6pasmz, ec:m .ll.Bri£KeHHe TOtrKII 3c~aHo ccreCTBeHHbB1 cnoco- |
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Cmf, TO, 3!!3H |
TpaeKTOpiUO (a CJle.J,OilaTeJibHO, H ee |
pa;a:nyc KpHB!!3Hbi |
|
p B .11060U TOllKe) H 33KOH JJ.BmKeHI-IH, T. e. 2aBHCID10CTb S=- f (t), MO)KEO no ¢opr.ryJia:-1 ( 17) II (21), (22) onpeJ.emnh MOL\YJib 11 nanpaBJieHHc BeKTOpOB CKOpOCTH ll ycr<apeHHH TO'lKH B Jl!060i{ MOMCHT BpeMeHII.
§ 44. HEKOTOPbi.E lJACTHbiE CJIYlJAH ,l{BH)l(EHIUI TO'-II(H
TioJib3)'HCb noJiytieHI!bf',lH pe3yJibTaTaMH, paCCMOTpHM HeKOTOpbie qacTHbie C.'Iyllal! ):{BllMeHIIH TO'JKH.
1. n p l1 M 0 JI u H e H H 0 e ):{ B H )K e H H e. EcJIH TpaeKTOpHeii TOllKH HB.1HeTCH rrprrMas mmrm, TO p=oo. Tor;a:a an=V2 ip=O H Bee
)'CKOpeHHe TO'I!UI parmo |
O;J,H0\1Y TO,'lbKO KacaTeJibHOMY |
ycKopeHIIIO: |
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a= at= dv;dt. |
(23) |
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TaK KaK B .aanHoM c.l!piae cKopocr& li3MeHReTcR TOJJ&Ro tiJICJJenHo, |
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TO oTciO;a:a 3aKJifOllae~r. |
'iTO !Wcarne;ZbHOe yctwpeNue xapatcmepusyem |
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U3-t!BHCHUC ttUC.10!3020 3fW1fCflUH. CKOpOCtnU. |
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2. p a B H 0 M e p H 0 e K p II B 0 ll 11 H e ii H 0 e I\ B H )K e If H e. |
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PaBHOMcpHbi\I Ha3biBaeTcH Tmwe KpHBOJIIIHei:JHoe .ll.BlDKeHHe TO'iKII, B |
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hOTOp0\1 'IHC,llOBOe 3HalleHne CKOpOCTH |
Bee |
BpeMH OCTaeTCH IlOCTOHH- |
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HbiM: o=const. TorJ.Xa |
a,=dv,'dt=O |
H see |
ycrwpenue |
TO'IKH paBHo |
O):{HD:I'IY TO,'IbKO HOp!AaJ1bHO'v!Y ycKopeHIIIO: |
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(24) |
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BeKTOp ycxopemm aHarrpaB.'IeHrrpu 3TOM BCe Bpe.MH no HOpMaJIH !{
tpaeKTOp !HI TO'HOI.
TaK KaK B naHHmr c.11yqae ycKopeHIIe no>IBJI>IeTCH TOJibiW sa cqeT H3MeHeHHH HanpaB.1eHHH CKOpOeT!I, TO OTCIOJ:a 3aKJIIO'Iae;o,.l, 'ITO flOp- J.taAbflOC ycKopeflue xapaKmepusyem U3Jrtefleflue cKopocmu no ttanpaal!e- flUJO.
Hafue.1 saKoH paBH0'<1epHoroKpnBommeimoro ;I.BHLKeHHH. H3 cpop- 1\IYJibr (17) HMeeM ds=vdf. 0yCTb B Ha'l2..1bHblilMOMeHT BpeMeHH (t=Q)
TO'IKaHaXOJJ,HTCH OT Ha'Ia.'!aOTC1leTa Ha paCCTOHU!IH So. Tor21:a, 6epH OT
,'IeBoli H rrpasoii qacre!! paneHCTBa orrpe;a:e.'IeHHbie HHTerpaJlhl B cooTBeTCTBYIOW.HX npe;I,eJiaX, IIOu'IYlllnl
s |
t |
~ |
ds= ~ vdt IJ.TIII s-s0 =vt, |
S0 |
0 |
Jl<)