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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- |
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YrJioByiO cKopocTh TeJia MOR<HO |
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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- |
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 |
|
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) |
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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-
(43) |
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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
OKPY*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 |
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- |
|
qeHHH a,; |
ll an H3 paBeHCTB |
(45), IT?JiytraeM |
|
tg f1=B/cu 2 • |
(47) |
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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 |
|
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. |
. |
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) |
|
IvI= I(!) Ih =I(!) Ir sin a |
mm IvI= f'W X rI· |
|
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