(формат)Тарг
.pdfn yroJI I+ ero OTK.TioHeHIIH oT HopMaJIH Mn onpe;a:eJIHTC.H ¢opMyJiaMHl
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(22) |
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 |
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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. |
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(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) |
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<)
TaK KaK V=COTISt. 0KOHt:raTeJibHO HaXOD.HM 33KOH paBH0r\IepHOfO KpHBOJII!HeiiHOfO lJ.BIDKeHIUl TOL!Kll B BI1,le
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s=s0+vt. |
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(25) |
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EcJIH B paaeHcTBe (25) rroJimKHTh s0 =0, TO |
s ll.aCT nyTb, rrpoflneHHhiil |
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TOLIKol't 3a apeMH |
t. C.TienoaaTeJihHO, |
rrpu |
paaum.1epEO\! JI.Bml<eHuu |
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nyTb, |
rrpoiineHHhili |
TO'IKoii, pacTeT |
rrponopunoHaJibHO |
apeMeHH, |
a |
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CKOpOCTb TO'm!l paBHa OTilOilleHHIO |
rryni KO |
Bpe:VleHH: |
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(23') |
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3. |
p |
a B H 0 M e p H 0 e IT p 5! M 0 .TI H H e H H 0 e 11. |
B H )K e H II C. |
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B :Ho:\1 CJiyt.tae an=a,=O, a 3H3'1HT,H a=O. 3a'.1eTII\[,'ITOeuwtcmeefl- |
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Hbl.H 88UaiCCHUe.lt, B KOtnOpOM ljC!WpeHue |
tnO'IKU BCe 6peJl!Jl paBHO flljll!O, |
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fl8AJl£tnCfl |
paBHOJ,tepnoe npJZMO.IltlfleUHOe |
aeUJ!Cefltte. |
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4. |
p |
a B H 0 IT e p e M e H H 0 e |
K p H B 0 .'IH H C li |
H 0 e 11. |
B II· |
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}1\ e II u e. Pamtonepe'.reHHbi\rI-T83biBaeTcH TaKoe Kprmo.1rwclinoe nBH· )KeHHe TO'lKH, rrpH !<OTOp0\1 KacaTeJibHOe ycKopeH!Ie OCT3eTCH BCe Bpe~IH nocTol1HHLlM: a,=const. Hai'!neM3aEoa 3Toro namKeHHH, C'lHTaH,
t.tTO rrp11 |
t=O s=s0 , a v=v0 , rn:e v0 - HJ.'la.JihHaH C1<0pOCTb TO'lKI!. Co- |
rJiacno |
rrepaoi1 H3 cpopMyJ1 (21) dv=a,dt. TaK KaK a,=comt, TO, 6epH |
OT 06eHX 'laCTe}IIlOCJieD:Hero paBeHCTBa HHTerpaJibl B COOTBeTCTBYIOIUHX
rrpe~eJiaX, ITOJIY'!HM
v=v0+a,t. |
(26) |
<J>opMy.Tiy (26) rrpe;J.CT3BHM B BIIJ(e
dsidt=v0 +a,t IIJIII ds=v0 df-Latf.if.
BTopnqHo HHTcrpHpyH, uai\nec.r saKoH pamwnepc:>.ieHIIoro KpHao.rmueiiuoro namKeHH5J ToqKH
(21)
ITprr 3TOM crwpocTr:. TO<JKrr orrpe.T(eJirrcrcH cpop'.ryJroir(26).
Ec.'IIIrrpn KpnnommeiiiiOM nanllieHHH TO'IKII'.1ony.'IhcKopoCTll Bo3pacTaeT, TO nBmKeHHe H33biBaeTc5l yCiwpeHHbW, a ecJIH y6biBaeT,- ::;aAte8AeHHbtM. TaK KaK H31\Ierrerme Mo,n:y.TIH CI<Opocnr xapaKTepH3yeTcH
Y')~ 11r |
0 |
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A |
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A • _.1 X |
!1 v M |
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Pn~. 123 |
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KacaTe.ribHhiM ycKopeuHe~1, TO narr:>Keune 6yneT ycKopeHHbi\I, ec.rrH ae-
miqHHbi V H Gt Hii!CIOT OJ:HHaKOBb!e 3HaKH |
(yro.TJ MeHqy |
BeKTOpaMH V |
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a~CTpb!H, |
pHC. 125, a), H 3aMe,U.~eHHbiM, |
eCJIH pa3Hble (yro.rr Me:>Kny |
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CJ |
If a Tyrroif, |
puc. |
125, 6). |
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B qacTHOCTH, rrpu |
paBHorrepe:-,reHHm.r ll.BH:iEeHHH, ec.rm |
B paseHCTBe |
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III
HM:elOT O,llHHaKOBbie 3HaKH, ,llBH)KeHHe oy,n:eT pM1-lOYC1WpeH- flbt.M, a ecJIH pasHhre sHaKH,- paBN03a.MeoJteflflbl.M.
<PopMyJihi (25)- (27) onpe,n:emiiOT TaK)Ke saKOHhi paBHOMepHoro
IIJIH paBHOrrepeMeHHOfO rrpHMOJIHHeHHOrO )J.BH)KeHIIH TO'!KH, ecJill
C'!HTaTh s=x. npn 3TOM B paBeHcTsax (26) H (27) |
a,=a, r.n:e a - qnc- |
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JIOBoe |
3Ha'!eHHe ycKopeHHH )J.aHHOH |
TO'!KH (CM. <flopMyJiy (23)]. |
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5. |
fa p M 0 H H q e C K He K 0 |
JI e 6 a H H H. |
Pacc:-.wTpHM rrpH- |
MOJIHHeihroe )J.BH)KeHHe TO'!KH,I!pH KOTOpOM ee paCCTOHHHe X OT Ha'!aJia
KOOp)J.HHaT 0 |
H3MeHHeTCH CO BpeMeHeM ITO SaKOHy |
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x=A cos kt, |
(28) |
r.n:e A H k - |
nocToHHHbre BeJIH'!HHhl. |
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To'!Ka M (pnc. 126) cosepwaeT np11 .9TOM JJ.BH>KeHHH |
KoJie6amm |
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Me*.n:Y noJio)KeHHHMH M 0 (+A) H M 1 (-A). KoJie6arniH, rrpoHCXO)J.HJ.IJ;I!~ no saKoHy (28), HrpaiOT 6oJibWYIO poJib B TeXHHKe. 0HH HaSbiBaiOTCH npocTbiMH zap.MOHu<tecKuMu KOAe6aHUJlAtu. BeJinqmm A, paBHaH HaH6o.nhrneMy OTKJIOHeHHIO TO'!KH OT UeHTpa KOJieoaHHH 0, Ha3biBaeTCH a.Mn;wmyoou Ko.ne6aHnti.
JlerKO Bll)J.eTb, 'ITO,HaqnHaH )J.BH)KeHHe B MOMeHT f=Q H3 nOJIO)KeHHH
M 0 , TO'!Ka BHOBb rrpH)J.eT B 3TO nOJIO)KeHHe B MOMeHT B~MeHH fl> )J.JIH
KOTOporo COS kf1 = 1, T. e. |
kt1 =2.n. |
npoMe)KyTOK BpeMeHH |
T=t1 =2n/k, B Te'leHne KOToporo TO'!Ka |
COBeprnaeT O)J.HO I!OJIHOe KOJieoaHHe, Ha3b!BaeTC5! nepUOOO.M KOJieoaHHH. EepH rrpOH3BO)J.Hbie OT X ITO f, HaH)J.eM 3Ha'!CHHHCKOpOCTH H ycKO·
peHIUI TO'!KH:
(29)
C.ne)J.oBaTeJibHO, B 3TOM JJ.BH)KeHHH n cKopoCTb, n ycKopeHne TO'!KH
H3MeHH!OTC5! C Te'!eHHeM BpeMCHH no rapMOHH'leCKOMY 3aKoHy. no 3HaKaM V H a JierKO rrpoBepHTb, IJ:TO KOr)J.a TO'!Ka)J.BH)KeTCH K ueHTpy
KOJie6aHHH, ee )J.BmKeHHe HBJIHeTCH ycKopeHHbiM, a KOf)].a OT UeHTpa KOJie6aHHH,- 3aMe)J.JieHHbiM.
AHaJIOrH'!Hbie KOJie6aHHH rrpoHCXO)J.HT H rrpH saKoHe x=Asinkt,
TOJibKO )J.BH)KeHHe B 3TOM CJiyqae Ha'!HHaeTCH H3 ueHTpa 0. fapMOHH'!eCKHe KOJie6aHHH no 3aKOHy S=Acoskt (HJIH s=A sinkf)
TO'!KaMO)KeT COBeprnaTb, )J.BHraHCb B)J.OJib JI!OOOH Kp HBOH ( CM., Harrp I!-
Mep, |
B § 46 sa.n:aqy |
51). |
Bee cKasaHHoe o |
xapaKTepe JJ.BH)KeHHH rrp11 |
3TOM |
coxpaHHTCH c |
Toil |
JIHrnh pasHnueti, |
'ITO nocJie.UHHH H3 ¢opMy.n |
(29) 6y.n:eT orrpe)J.eJIHTb KacaTeJibHoe ycKopeHHe TO'!KH;KpoMe Hero TO'!Ka
6y)J.eT eJ.IJ;e HMeTb HOpMaJibHOe |
ycKopeHHe an=V2 /p. |
§ 45. fPM>HI(H }].BH)I(EHH51, |
CI(OPOCTH |
H YCI(OPEHH}I. TOI.JI(H |
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EcJIH B COOTBeTCTBY!OIUHX MaCWTaoax OTKJia)J.biBaTb B)J.OJib OCH a6cUHCC BpeMH f, a B)J.O,Tib OCH Op)J.HIIaTpaCCTOHHHe S, TO TIOCTpOeHHaH B
3THX OCHX KpHBaH 6y)J.eT H3o6pa)KaTb cpat/JUK paccmOJlflUU,
HJIH cpacpUK 06U:JJCeflUfl TO'!KH.no 3TOMy,rpa<flHKy HarJIH)J.HO BH)J.HO, KaK H3MeH5leTCH rroJio)KeHHe TO'IKH(ee Koop.n:nH~I.Ta s) c Te'!eHneMnpeMeHn.
112
AHaJwrnqHo, B cooTBeTCTBYIOIII.HX MacwTa6ax MoryT 6hiTb rrocTpoe-
Hbi Kpi1Bb!e, |
):{8!0UJ.I1e |
38BHCI1MOCTb V (t) - cparj:JUK CKOpocmu |
U a, (f), |
an (f), a (f) - |
cparpUKU |
KllCame.7bflOZO, rlOpAtaAbH020 U nOAflOZO |
ljCKDpe- |
liUU.
Ha p11c. 127, a, 6, a cBepxy rroKa3aHhi rpacp11KH Jl:BI1JKeH11H, orrpe- ;a:eJIHeMbiX cooTBeTCTBeHHo ypaBHeHH5nm (25), (27) 11 (28). H111Ke Ha TeX )Ke p11CYHKaX 11306pa1KeHLI ):{.TI5I 3THX ):(BHJKeHI1H rpacp11KIJ, CKOpOCTeH H rpacp11KI1 KacaTeJibHbiX ycKopeHHH.
Pnc. 127
fpacfmK paBHOMepHOfO ):{BI11KeHI15I H3o6paJKaeTCH, K8K Mbi BH):{HM, npHMOH JII1HI!eii, HarrpaBJieHIIOH IlOJ( yrJimi KOCH 30C1!HCC, rpacp11K CKOp0CTI1 B 3TOM CJiyqaerrpHII!OH, rrapa.'!JieJibHOllOCH 80C1!11CC (v=const),
a rpacpHK K3C3TeJibHOfO ycKopeHH51npHMoi:'r, COBil3JI:8lOUJ.eH C OCb!O 30C1!11CC (a,=O). Ll.JIH paBHOnepeMeHHOfO ):{BI1JKeHI151 (B 11306pa1KeHHOM Ha p11c. 127, 6 cJiyqaeycKopeHHoro) rpaqmK ;a:BHJKeHI1H 113o6pa- :>KaeTCH BeTBbiO rrapa60Jibl, rpacpHK CKOpOCTI1npHMOH, HanpaBJieHHOH ITOJI: yrJioM K oc11 a6cuHcc, a rpaqmK KacaTeJihHoro ycKopeH11H - rrpHMOH, napaJIJieJibHoli oc11 a6cu1111C (a,=const). HaKoHeii, ;a:mr rapMOHII'ICCKHX KOJie6aHHH (p11c. 127, B) COOTBeTCTBY!OUJ.I1e rpacpmm 11300pa:x<aiOTCH KocrmyconJ(aMH 11JII1 CI1HycoH)J.3MI1.
fpaqJHK ):{BI11KeHI15I He CJie;J,yeT CMeiiii1B3Tb C TpaeKTOp11eH, KOTOpaH Bo Bcex paccMmpeHHbiX cJiyqaHx ;a:oJIJKHa 6hiTb sa;1_aHa ;a:orroJIHHTeJibHO. fpacp11KH HOpM3JibHOfO 11 IlOJIHOfO ycKopeH11H Ha p11C. 127 He IlOKa33Hbi, TaK KaK an 11 a KpOMe 38KOII8 J(BHllieHHH 38BI1CHT ern.e OT p, T. e. OT BH):{3 TpaeKTOp11H, 11 npH OJ(HOM II TOM JKe 33KOHe s=f (t) OYJl:YT )T.JIH
pa3Hb!X TpaeKTOp11H pa3HbiMfl.
113
fJiaoa X
llOCTYflATE.llhHOE H BPA~ATEJihHOE ,l{BH)I(EHH.SI TBEP)J,OfO TEJ1A
§ 48. UOCTYUATEJlbHOE ,!J,BH)l(EHHE
B KHHeMaTHKe, K3K H B cTaTIIKe, 6y)J.e\1 p3ccMaTpHB3Th Bee TBepL\hie TeJI3 KaK 'a6COJIIOTHOTBep)J.ble. 33)J.3LJH KHHeMaTHKH TBep)l,OfO TeJia p3C113)1.3lOTCH Ha )].Be LJ3CTH:
1) 33,[\31-IHe )J.BHllieHIIH H onpe;:I;eJieHHe KHHeM3THLJeCKHX X3p3KTepHCTHK )J.BHllieHHH TeJia B ueJIOM; 2) orrpe)J.eJiei1!1e KHHeM3THLJeCKHX xapaKTepiiCTHK )J.BmKeHHH OT)J.eJihi-Ib!X TO'!eK TeJI3.
H3LIHeM c p3cCMOTpeHH5I rrocTyrr3TeJihHoro JJ.BHi!(eHHH TBep)J.oro TeJI3.
Jlocmyname!lbflblM fla3bteaemcfl maKOe 8eu:JJCeHue meep8020 me!la,
117
npu twmopoM A1o6aJl npR.!rtaR, npoeeaeflflaRe BmoM meAe, nepeMetupemc,q, ocmatJaRCb napaAJtellbflOU ceoeMy Hatta!lbflOMY HanpatJileflum.
DocTyrraTeJibHoe JJ.BI1JKemre He cJienyeT cMewi1BaTh c rrpHMOJII1HefiHhiM. Dp11 nocTyrrareJibHm.1 JJ.BI1JKeHI111 TeJia TpaeKTOp1111 ero ToqeK Moryr 6biTh JII06hiM11 Kp11BhiMI1 JII1H11HMI1. Dp11neneM rrp11Mepbi.
l. Kyson aBTOMo6HJIH Ha rrpHMOM rop1130HTaJibHOM yqacTKe noporn
.ll;BI1iKeTc5I nocTyrraTeJihHO. Dp11 3TOM TpaeKTOPHH ero ToqeK 6ynyr npHMbiMI1 Jl11H115IMH.
2. CrrapHHK AB (p11c. 131) rrp11 npameHim Kp11now11rroB 0 1A n 0 2 8 (QtA =02B) TaKH\e )J.BHJKeTCH IlOCTyrraTeJibHO (JII06aH rrpoBeneHHaH B HeM rrpHMa5I OCT3CTCH rrapaJIJieJibHOH ee HaqaJibHOMY HarrpaBJieHH!O), ToqKH cnapH11Ka JJ.BHJKYTCH np11 .9TOM no oKpymHocTHM.
z
Puc. 131 |
Puc. 132 |
CnoiicTna rrocTyrraTeJibHoro .ll;BHJKeHHH onpeneJIHIOTCH cJienyiOrueii reopeMoii: npu nocmynamellbflOM aeuatcmuu see motaw me11a onucbteawm 08Ufla!W6ble (npu 1-laAO:HCeflUU coena8aww,ue) mpaeKmOpUU U UMeiOtn 8 Ka:J/Cawi MoMeflm epeMeflu o8uHaKoeue no ~rway1210 u HanpatJ,1eHum cKopocmu u ycKopeHUR.
ll,JIH noKasaTeJihCTBa paccMOTpHM TBepnoe TeJio, conepwaiOmee rrocrynaTeJihHoe .n,nnmeHHe oTHOCIITeJibHO cucTeMbi orcqeTa Oxyz. BoshMeM B TeJie nBe rrpoHSBOJibHbie ToqKI1 A 11 B, rroJioJKeHI1H KoTopbiX B MOMenT
BpeMeHI1 |
f orrpe)J.eJIH!OTCH pa)J.I1yCa:\1!1-BeKTOpaM11 ~ 11 fB (p11C. 132); |
np.one.n.eM |
BeKTop AB, coe.n.uHHIOIU:Hii 3TH TO'IIUI. Tor.n.a |
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(35) |
Dpn 3TOM .ll;JIHHa AB nocTOHHHa, KaK paccTOHHHe Mem.n.y ToqKaMu
TBepnoro TeJia, |
a HanpaBJieHI1e |
AB ocTaeTcH HeusMeHHhiM, TaK |
KaK |
reJio .ll.BI1JKeTc5I |
nocTyrraTeJihHO. |
TaKHM o6pa3oM, BeKTop AB BO |
nee |
BpeMH )J.BI1llieHH5I TeJia OCTaeTCH ITOCTOHHIIb!M (AB=const). BcJie,ll.CTBHe 3TOfO, KaK Bl11l.HO 113 paBeHCTBa (35) (II Henocpe)J.CTBeHHO H3 qepTeiKa), TpaeKTOp11H TO'IKI1B rroJiyqaeTcH 113 TpaeKroprm ToqKI1 A napaJIJieJihHbiM cMemeHI1eM ncex ee ToqeK Ha nocToHHHhlii BeKrop AB. C.nenonaTeJihHO, rpaeKTop1111 roiieK A 11 B 6ynyT neiicTBI1TeJihHO O.li.I1HaKOBhiM11 (rrpH H8JIOJKeH1111 COBITa)J.a!OI.Il;11M11) Kp11BbiM11.
1lm1 HaxomneHI1H cKopocTeii ToqeK A 11 B rrpo.n.11qxpepeHu11pyeM o6e qacrH paBeHcTBa (35) no BpeMeHI1. Do.nyq11M
drB/di = drA/dt +d (AB)jdt.
118
Ho rrponsaonHaH or noc~miHI~oro BeKTopa AB paana ny.mo. nsBOllHhle me OT BE'KTOpOB rA H r 8 ITO BpeMeHH ,!l,a!OT CKOpOCTH TOI.J.eK
A u B. B pesyJibTaTe HaxoAHM, I.J.TO
T. e. I.J.TO cKopocni rolJ.eK A n B TeJia B mo6ofi MOMeHT BpeMemr onnHaiWBbi H ITO MOJJ.y.'liO,!! ITO HarrpaBJieHHIO. oepH OT o6eHX Y.JCTeif UOJIY· 'lemroropaBeHCTBa rrpOII3B01lHhie no Bpe:,remr, nail1leM:
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dvA/di=dvBjdt |
HJIH |
aA=aB. |
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CJieJT,oBaTeJibHO, ycKopeHHH TolJ.eK A H B TeJia B mo6oir MoMeiiT |
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spcMeHH TmKe onnnaKOBhl rro MOllYJIIO |
n |
Hanpanm~HFiiO. |
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Tal< KaK TOI.J.KH |
A H |
B 6biJIH Bbi6paHbl rrpOIBBOJibHO, ro H3 nai11leH- |
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Hb!X pe3yJibTaTOB |
CJie,'l.yeT, 'lTO y |
BCCX |
TOI.J.eK TeJia |
liX Tp3.C!{T0pHH, a |
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T3K/Ke CKOpOCTH |
II ycKope- |
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HIIH B JII06oii r.10MeiiT BpeMe- |
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HH 6ynyr Oli.I1HaKOBbi. TaKH1\t |
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o6pa3oM, reopeMa llOKasana. |
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CKopocnr H ycKopeHIIH |
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TO'!eK J(BH)KyUJ.efOCH |
Te.TJa |
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o6pasyiOT BeKTOpHb!e ITOJIH- |
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JlO.'le cKopocTeir |
H |
no.'le |
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PHc. |
133 |
ycrwpeHHH TOI.J.eK |
re.Tia. |
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113 1lOKa3aHHOrO CJie;r.yeT, I.J.TO ITOJfH |
C!WpOCTeH |
H ycKopeHHII TOI.J.eK |
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TeJia, ABHIKYUJ.erocH TIOCTyrraTe,T!bHO, 6yAyT OiJ.HOpOfl.Hb!Ml! (pnc. 133),
HO B006rn.e He CTaUHOHapHb!MH, T. e. H3MeHHIOUJ.HMHC5I BO BpeMeHU (CM.
§ 32).
113 reopeMbi cJienyer TaKme, I.J.To rrocryrrareJibHoe ABHmeHne TBep-
.n.oro TeJia BTIOJIHe oupe,ZJ;e.'lHeTCH.UBIDKeHHeM KaKoii-HH6Y1lb OAHOH ero TOI.J.KH. C.'Ie,UOBaTeJibHO, H3y'-leHHe IIOCTyrrareJibHOfO ABII/KeHHH TeJia
CBOL\HTCH K 331l3'leKHHe:-.1aTIIKH TO"'KH, HaMil yme paCCMOTpeHHOft.
Dpn nocTyrrareJibHo:\1 JI:BHmemm o6rn.y10 .UJIH Bcex TO'leKTeJia CKO·
poCTb V H33b!BaiOT CKOpOt::tnb/0 nocmljnatneJlbfl020 06UJICeHUJl TeJia, a |
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ycKopeHHe |
a- ljCKOpeNUeJ1t nocmyname/lbfl020 06UJICCflUJl TeJia. BeK- |
TOpbl V If |
aMOIKHO !1306pamaTb I1pHJIOLKeHHbl1\HI 1\ Jll06oil TOI.J.Ke TeJra. |
3aMeTHM, I.J.TO ITOHHTHH 0 CCWpOCTH H ycKopeHHII TeJia H\1eiOT CMbiCJf |
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mOJlbKO npu nocmynameilbflOM oeuJtceNuu. Bo Bcex ocr<WbHbiX cJiyqaHx ToqKH TeJia, KaK Mbl YBllAHM, 1lBHJKYTCH C pa3HblMii CI<OpOCTHMH II
ycKopeHHHMH, H TepMHHbl «CKOpOCTb TeJia>> HJI!i «yCKOpeHue Te.aa» |
.JVIH |
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3THX ABH)KeHHH TepHIOT |
C:\fb!C.'l. |
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§ 49. BPA~ATEJibHOE |
)lBH)I(EHHE TBEP)lOfO TEJIA BOI(PYf |
OCH. |
YfJIOBA}l CI(OPOCTh H YfJIOBOE YCI(OPEHHE
BpCUJ.{ameJlvflbtM iJeuJFCenue,lt meepiJoco meJla eoKpljc nenooeumcHou ocu flG3bl6aemcJl maKoe eco OBUJFCeHue, npu KOmopoM KaKUe-nu6yiJb oee mottrw, npuflaOAeJFCaU{ue meAy ( UJlU neus.MeHHO c HUAt C6fl3aflflble), ocmarotnefl eo
119
see epeJ.Ul aeua~eeflWl HenoiJBUa!CflblMU (pHc. 134). IIpoxo;:vrmaH qepes
HeiTOJJ.BHmHhre TO'lKHA H B rrpHMaH AB Ha3hlBaeTeH OCbiO epaU{eflUJl.
TaK KaK paeeTOHHIHI MemJJ,y TO'lKaMHTBepJJ,oro TeJia JJ.OJI)KHbi oeTaBaTbeH HeH3MeHHbiMH, TO OIJeBHJJ.HO, 1ITO rrpH BpamaTeJibHOM JJ.BH)KeHHH
z1 |
Bee TO'lKH, npHHaJJ.JiemamHe oeH BpameHHH, |
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~ r2B |
6yJJ,yT HerroJ(BHmHbi, |
a Bee |
oeTaJihHbie |
TOIJKH |
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TeJia 6yJI:yT OITHebiBaTb OKpymHOCTH, |
ITJIOCKO- |
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eTH KOTOpb!X rreprreHJJ,HKYJIHpHb! OeH |
Bpame· |
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a IJ,eHTpbl JiemaT Ha :noli OeH. |
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llJIH orrpeJJ,eJieHHH ITOJIO)KeHHH BpamaiO- |
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meroeH TeJia rrpoBeJJ,eM qepes oeb BpameHHH, |
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BJJ.OJib KOTopoi1 HarrpaBHM oeb Az, no.rryrrJioe- |
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KOCTb |
I - HeiTOJJ,BHmHyiO |
H TIOJiyrrJIOeKoeTb |
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I I, BpesaHHYIO B eaMo TeJio H BpamaiOIUYIO- |
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eH B\1eeTe e HHM (eM. pHe. |
134). TorJJ,a |
noJio- |
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meHHe TeJia B JII06oti |
MOMeHT BpeMeHH |
OJJ:HO· |
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3H31JHO orrpeJJ,eJIHTeH |
B3HTb!M C COOTBeTeTByiO- |
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IUHM 3H3KOM yrJIOM cp |
Me/KJ(y 3THMH riOJIYfi.TIOeKOCTHMH, KOTOpb!H liaJO· |
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BeM lj2JWM noeopoma meAa. oyJJ,eM eiJHTaTb yroJI cp ITO.'IO/KHTeJibHbii\l, eeJIH OH OTJIO)KeH OT HeiTOJJ.BH)KHOH ITJIOCI<OeTH B HanpaBJieHHH rrporHB XOJJ,a qaeOBOH CTpeJIKH (JI:JIH Ha6JIIOJJ,aTeJIH, eMOTpHmero e ITOJIO)KHTellh· HOfO KOHU.a oeH Az), H OTpHIJ.3TeJibllbiM, eeJIH ITO XOJJ.Y qaeOBOH eTpeJIKH. HsMepHTb yroJI cp 6yJJ,eM BcerJJ,a e paouaflax. 4To6hi 3HaTh noJioA<eHHe TeJia B JII060ti MOMeHT BpeMeHH, Ha;J;O 3H3Tb 3aBHeHMOeTb yrJia cp OT
BpeMeHH f, T. e. |
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cp=f (t). |
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YpaBHeHHe (36) Bh!paJKaeT 3aKOfl epaU{atneAbfl020 aeua~eeflUJl meep- |
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0020 meAa eoKpyz HenoiJeua~eHou ocu. |
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OeHOBHbiMII KHHeMaTwreeKIIMH xapaKTepueTHKaMn |
BpamaTeJibHoro |
JJ,BH)KeHHH TBepJJ,OrO TeJia HBJIHIOTeH ero yrJIOBaH eKopoeTb (l) H yrJIOBOe
yeKopeHHe e*.
EcJIH sa rrpoMemyToK BpeMeHH 1'1t=t1- t TeJio eoBeprnaeT noBopoT Ha yroJI 1'1cp=cp1-cp, TO IJHeJieHHO epeJJ,HeH yrJIOBOH eKopoeTbiO TeJia 3a
3TOT rrpoMemyToK npeMeHH 6yJJ,eT fficp= 1'1cp/M. B |
npeJJ,eJie npu M-O |
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HaH,ll.eM, I.J.TO |
. |
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dqJ |
(37) |
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ffi=crr |
HJIH (l) = cp. |
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TaKHM o6pasoM, ttuc!losoe |
sflatteflue ljc!lOeou |
cKopocmu me11a e |
(}aflflblU MOMeflm epeMeflU paeflO nepeou npOU3600flOU om lj2Aa nosopoma
no epeMeflu. PaBeHeTBo (37) rroKa3biBaeT TaKme, IJTO BeJIH'lHHaffi paBHa OTIIOWeHHIO 3JieMeHTapHoro yrJia riOBOpOTa dcp K eOOTBeTCTByiOIIJ,e\1y npoMemyTKY BpeMeHH dt. 3HaK ffi orrpe.u.eJIHeT HarrpaBJieHHe Bpamemm
* 00}l,06HO TOMy, KaK B § 42 Mbl ycJIOBIIJU!Cb o603Ha1IaTb CHMBOJIOM V 'lliCJIOBOe SHalJeHl!e CKOpOCTll II Oll:HOBpeMeHHO ee MO!l:YJlb, 3,n:ecb CO II 8 6y,n:yT 0603H;.t'I3Tb'lHCl!D- BbJe (aJire6paii1JeCKlle) 3Ha1JeHIIH yrJIOBOll CKOpOCTll II yrJIOBOfO ycKopeHHH If O,n:HoBpeMeHHO !IX MO)l,yJIH, KOr}l,a 3TO He MOJKeT Bh!3B3Tb He,n:opa3yMeHJ!lL
126
TeJia. JierKo BU)J.eTb, 4TO KOr)J.a Bpaur..eHue npoucxo)J.HT npoTHB xo,n;a qacOBOH CTpeJIKU, w>O, a KOr,n;a no XOJ.!.Y 4aCOBOH CTpeJIKU, TO w<O.
Pa3MepHOCTb yrJioBoii cKopocTu liT (T. e. 1/BpeMH); B Ka4eCTBe e)J.HHUll.bl H3MepeHUH 06hPIHO npHMeHHlOT pa)J./C UJIU, 4f0 TO Me, 1/c (c- 1),
TaK KaK pa)J.HaHBeJIIIquHa 6espa3Mep- |
~71 |
'2; |
ol |
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HaH |
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ll~ ~ |
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YrJioByiO cKopocTh TeJia MOR<HO |
Hso6- |
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pa3UTh B BU,ll;e BeKTOpa,w, MOJ.!.YJih KOTOpO- |
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fO paBeH IwI u KOTOphm HarrpaBJieH BAOJih |
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ocu Bparn.enmt TeJia B TY cropony, orKy)J.a |
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BpaiiJ,eHHe BH)J.HO |
npOHCXO,ll;5IIIJ,H'A: npOTl1B |
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xo.u.a qacoBoii crpeJIKH (puc. 135). TaKoii |
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BeKTOp onpe)J.eJlfieT cpa3y ll MO.U.YJib |
Yf JIO- |
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PHc. lSfi |
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BOll CKOpOCTH, H OCh Bpamemm, U HarrpaB- |
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JieHHe BpaiiJ,eHHH BOKpyr 3TOH OCII. |
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Y r JI o B o e |
y c K o p e H u e |
xapaKrepu3yeT |
U3MeHe~me c Te- |
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qeHHeM BpeMeHH yrJioBoii cKopocTH TeJia. EcJIH 3a npoMeMyToK BpeMeHH yrJioBaH CKOpOCTh TeJia H3MeHHeTCH Ha BeJIU4HHY /1(!)= =w1-w, TO qucJioBoe 3HaqeHue cpe)J,Hero yrJIOBoro ycKopeHHH TeJia 3a 3TOT npoMeMyroK BpeMeHu 6y)J.eT ecp= 11w/11t. B npe,n;eJie npu M--+0
Haii.u,eM, y4TH OJ.!.HOBpeMeHHO |
paBeHCTBO |
(37), 4TO |
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d(l) |
d2cp |
8 = W = <p. |
(38) |
8 = df = |
dt2 UJIH |
TaKuM o6pa3ovr, tiUC!lOBOe3HatteHue yc!loaozo yc1wpeHUfl me!la B iJaHHbtCt MO.MeHm BpeJtteHu paaHo nepBou npou38oiJHou om ycAoeou c1wpocmu UAU emopoii npoU3BDiJHou om ycAa noBopoma me;w no epe.MeHu.
Pa3MepHOCTh yrJIOBoro ycKopeHIIH 1/P (l/BpeMH2); B Ka4eCTBe
e,n;UHIII.I,hl U3MepeHUH 06bi4HO npHMeHSieTCH pa.u./c2 HJIU, 4TO TO Me, l/c2
(c-2).
ECJIH MO)J.YJib yrJIOBOH CKOpOCTU CO BpeMeHeM B03paCTaeT, BpameHue TeJia Ha3biBaeTCH yctwpeHHbtM, a ecJIH y6biBaeT,- 3aMeiJAeHHbtM. JierKo Bll,ll;eTb, 4TO BpameHHe 6y,n;eT ycKOp€HHbiM, KOf)l,a BeJIU4UHhl W U 8 UMeiOT O,ll;HHaKOBbie 3HaKH, II 3aMe,ll;JieHHbiM,- KOf)J.a pa3Hble.
YrJioBoe ycKopeHue TeJia (no aHaJioruu c yrJIOBOH cKopocrbiO) MO)I{HO TaK)Ke H306pa3UTb B BU)J.e BeKTOpa B, HanpaBJieHHOfO B,lJ;OJib OCH BpameHUH. flpH 3TOM
(38')
HanpaBJieHue 8 COBna,n;aeT c HarrpaBJieHHeM @, K~r)J.a TeJio BpamaeTcH ycKOpeHHO (puc. 135, a), II npOTHBOITO.'IOMHO (!) IIpU 3aMe)J.JieHHOM BpameHuH (puc. 135, 6).
§ 50. PABHOMEPHOE H PABHOOEPEMEHHOE BPAll.J.EHH51
EcJiu yrJioBaH CKOpOCTb TeJia ocTaeTcH BO Bee BpeMH .z:t.BHMeHuH rroCTOHHHOH (w=const), TO Bpameime Te,7Ja Ha3biBaeTCH pa6HO.MepHblJlt. Haii)J.eM 3aKoH paBHOMepHoro Bpai.I.J..eHuH. 113 ¢opMyJihl (37) uMeeM
121
df:P=rudt. OrciO)J.a, CI.JIITaH, qyo B naqaJibHbiH MOMeHT speMeHII yroJI f:P=f:Po, H 6epH IiHTerpaJibi cJieBa OT f:Po )J.O f:P, a cnpaaa or TIOJiyquM OKOHqaTeJibHO
Yl3 paBeHCTBa (39) f:Po=O,
B rexmme cKopocTb pasnoMepHoro sparn.eHnH qacro onpeL\eJIHIOT quCJIOl\1 ooopOTOB B MIIHYTY'o603HaqaH 3TY BE'JIJlquHyqepe3 n o6/MuH*. Hafr)J.eM 3aBHCHMOCTb Me*L\Y n o61MHH u ru 1/c. flpn O)J.HOM o6opore reJio nosepnercH na yroJI 2n, a npu n oooporax Ha 2nn; 3TOT nosopor
.D,eJiaeTCH 3a BpeMH t= 1 MHH = 60 C. 113 paBeHCTBa (40) CJie)J.yeT TOf)J.a, qro
ru=nn/38 ~ O,ln. |
(41) |
Ecmr yrJiosoe ycKopeHIIe TeJia so see spe:vrH L\BH)KeHHH ocTaeTc51 TIOCTOHHHbi:\1 (e=COTISl), TO Bparn.eHHe H33hlBaeTCH pa6HOnepe.Me/iflbl.M. Haii)!;eM 3aKOH pasnonepeMeHHoro sparn,eHHH, cqiiTaH, qro B Ha'IaJih-
Hbiii |
1\IOMeHT BpeMeHII t=0 yrO.lJ f:P=f:P0, a yrJIOBaR CKOpOCTb (J)=Wo |
(ru 0 - |
HaqaJibH3H yrJIOBaH CKOpOCTb). |
lb ¢opMy.'Ihi(38) HMeeM dru=edt. HnrerpnpyH JieByiO qacTb B npe-
)J.eJiax or ru 0 .uo ru, a npasy10 - B npe)J.e.'Iax or 0 )J.O t, |
nai!)J.eM |
ru=ruo+et. |
(42) |
'
flpe)J.CTaBH.M Bb!paiKeHHe (42) B BH)J.e
d'fl/dt=ru0 +et HJIH df:P=ru 0 dt+etdt.
BropuqHo unrerpHpyH, um1.ueM orciO)J.a 3aKOH paBHonepeMeHHoro spa-
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(43) |
YrJioBaH cKopocTb |
ru |
3Toro spar.u;ennH onpe,c.eJIHeTCH |
¢op:.ryJioii |
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(42). Ecmr BeJIHI.JliHbi |
ru |
n e |
m1e1or |
O)J.HHaKOBbie 3HaKH, |
spar.u;e1me |
6y)J.eT paBHoycKopeHHbiM, |
a |
ecm1 |
pa3IIbie - paBH03aMe)J.Jiel!HbiM. |
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§51. CKOPOCTH H YCKOPEHH5I T04EK
BPAll.J.AIOJ.UEfOC51 TEJJA
Ycrad"osus B rrpe,:!,hiL\YID;HX naparpa{pax xapaKrepncrnKH )J.BHX{eHHH BCero TeJia B UeJIOM, nepeii;J.eM K H3yqeHHIO )J.BIDKeHHH OT)l.eJ1bHh!X ero roqeK.
1. |
C K 0 p 0 C T H T 0 q e K T e JI a. |
P'aCC"r.fOTpHM KaKyiO-HH6y)J.b |
TOI.JKY |
M TBep)J.OfO Teo1C:l, HaXO)J.HID;Y!OCH |
H<l pacCTORHHH h OT OCH Bpa- |
m,eHHH (c:vr. pn::. 134). flpn span:,emm reJia roqKa M 6y)J..eT omiCbiBaTb
* CJI€.UyeT ocooo no):(tJepKHYTb, 'ITO n no paaMepHOCTH He yroJI, a yniOBaa CKOpOCTb.
122