DD' <BE', TO ,n:oJDKHO 6biTb F2>F1• CJie,n:oBaTeJibHO, CJUCileHHO CU/la e nomenl{Ua/lbHOM
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no11e 60ilbtue ma,u, zae nooepx11ocmu |
ypOBHfl |
npoxoiJflm eyUJ,e. |
OrMe'leHHhie csoil:CTsa |
ll03BOJilll0T |
HarJillll;HO npell;CTaBI!Tb |
KapTHHy pacnpe.n:eJieHHll |
CHJI B !!OTeHIJ.HaJibliOM |
t:HJIOBOM noJie |
c noMOIIJ.hiO noBepxHocTei'! ypoBHH. KpoMe Toro, KaK BH.D:HO H3 paBeH- |
CTBa (57), |
pa6oTa'llOTe!IUH3JibHOH |
CHJ!bl 3aBHCHT B KOHe'!HOMC'leTeTOJ!bKO OT TOfO, |
C K3KOii llOBepxHOCTH ypOBHll H H3 |
KaKyiO |
npOHCX0)1HT nepcMCllJ.CHHC TO'!KH. |
floliCHHM |
cKa3aHHoe npHMepaMH. |
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1. ,Il,JIH O)lHOpO,n:HOfO TIOJ!ll CHJI TSDKCCTH (eM. pHC. 231), KaK BH;J.HO 1!3 <jJOpMyJihl
(59), U=const, Kor.n:a z=const. CJie,n:oBaTeJibHO, noBepxnocTl!MH ypoBHll liBJIHIOTCll ropH30HraJibHhre nJioCKCCTH. CHJia Tll:ll{eCTil P HanpaBJiena no HopMaJIH K 3THM nJio-
CKOCTliM B CTOpOHY B03paCTaHHll U H BO Bcex TO'!KaX I!OJill UOCTOliHHa.
2. ,UJIH IIOJill CHJI TSrroTei!Hll, corJiacHO ¢opMy.1e (59"), U=conot, Kor.n:a r=const.
CJie.n:oBaTeJibHO, noBepxHOCTFMH ypoBH5! liB.'liiJOTCll KOHuenTpwrecKHe c<jlephl, uellTp KOTOpbiX COBUa.n:aeT C npHTHrHBaiOIIJ.HM ueHTpOM. CHJia B KaJK.QOH TO'!Ke !!OJ!ll HanpaBJieHa no HOpMaJIH K COOTBCTCTBYIOIIJ.CH ccpepe B CTOpOHY B03paCT3Hllll U (y6biBa-
HHll r), T. e. |
K |
ueurpy c<jlepb!. |
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EcJm |
B |
noTeHu,HaJibHOM CHJIOBOM |
noJie HaXO)l;HTCH |
c H c T e M a |
M a T e p |
H a JI b H hi X T 0 q e K, TO |
CHJIOBOH cpyHKJJ,HeH |
6y)l;eT |
Ta- |
KaH cpyHKilHH |
KoOp,!J;HHaT TO'-leKCHCTeMbl U (x1, Yt. Zt. |
., |
Xn, Yn• |
Zn), |
)J.JIH KOTOpOH |
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(62) |
T. e. ,!I;HqxpepeHu,HaJI KOTOpOH paBeH cyMMe 3JieMeHTapHb!X pa60T BCeX )J.eHCTBY!:Oill,HX Ha CHCTe:\-ty CHJI ITOJIH.
§ 127. OOTEHl.J,HAJibHMI 3HEPfH~.
3AI(OH COXPAHEHH~ MEXAHH'-IECI(Ofl 3HEPfHH
llJIH ITOTeHUHaJibHOfO CHJIOBOfO ITOJIH MOlliHO BBeCTH ITOHHTHe 0
noTeHu,HaJibHOH 3HeprnH KaK o BeJIH'lHHe, xaparnepn3yJ:Omefi «3anac pa60Tbl», KOTOpblM o6Jia,[J;aeT MaTepHaJibHa5I TO'-!Ka B JJ.aHHOM rryHKTe CHJIOBOrO ITOJIH. 4T06b! cpaBHliBaTb Melli,!J;Y COOOH 3TH «3anaCbi pa60Tbi», HYlliHO ycJIOBHTbCH 0 BbJ6ope HyJieBOH TO'IKll0, B KOTOpOH 6y)J;eM YCJIOBHO C'lHTaTb«3arrac pa60Tbl» paBHbiM HYJI!:O (Bbi6op HyJieBOH TO'-!KH,KaK H BCHKOfO Ha'laJia OTC'leTa, I1pOH3BO)J;HTC5I I1pOH3BOJibHO). flomerlll,UallbHOii 3Hepr!Uet1 .Mamepua!lbHOU mottKU (J aarmO.IIt noAOJ/CCHUU M Ha3bteaemcfl cKaA.RpHaJl 8CIIUttUNa IT, paeHa.R motl pa6ome, KomopyiO npous8eoym CUAbl nO/Ifl npu nepe.AteUtptuu mottKU U3 nOAO:JfCCHUfl M 8 HyAeeoe
il=A(MO)·
J.13 onpe,!l;eJieHHJI cJie)J;yeT, '-ITO rroTenuHaJihHaH |
3HeprHH II 3aBHCHT |
OT KOOp,!I;HHaT X, y, Z TOIIKH M, T. e. '-ITOll=ll (x, |
y, z). |
Ey)J;eM B )J;aJibHeiirneM C'IHTaTb HyJieBhre TO'IKH AJIH cpyHKUHH II (x, |
y, z) H U (x, y, z) cosrra)J;aiOill,HMH. Tor11.a U0 =-0 |
H no cpopMyJie (57) |
A,10 =U 0-U=-U, r)J;e U- 3Ha'INIHecHJiosoii cpyHKUHHBTO'-!KeM noJIH. TaKHM o6pa3oM,
IT (x, y, z)=-U (x, y, z),
T. e. nomeHt(UaAbHa.R :mepcu.R 8 AI06ot1 mottKe CUA080eo noA.R paeHa 3HatteHUIO CU/108011 c/JyHKf1UU 8 3mot1 mOttKC, 83.RnlOMY C o6pamliblM 3/iaKOM.,
320
0TCIO)l,a BH)l,HO, 'ITOnpH paCCMOTpeHHH BceX CBOiicTB TIOTeHI.J.HaJibHOfO CHJIOBOfO TIOJIH BMeCTO CHJIOBOH ~yHKI.J.HH MOJKHO TIOJib30BaTbCH TIOHH·
THeM |
noTeHuHaJibHOii ::~HeprHH. B qacTHOCTH, |
pa6oTy noTeHuHaJibHoii |
CHJibi |
BMeCTO paBeHCTBa (57) MOJKHO Bbl'IHCJIHTb ITO ~OpMyJie |
|
A(M,M.) = II1-II2. |
(63) |
CJie;a:osareJibHO, pa6gma nomeliVyUa.llbliOCi cu.llbt pasHa pa:mocmu sliatteliuii nomeliVyUG.IlbliOU Emepc.uu iJsu:JJCyU{eiicH mo'tKU s liG'lG.IlbliO.M u KOHe'tliO.M ee nO.IlO:J/Cet-LUHX.
BbrpaJKeHHH noTenuuaJibHoii 3Hepriiii )l,JIH H3BecTHbiX HaM noTeH-
urraJibHbiX CHJIOBblX TIOJieH MOJKHO HaHTH |
H3 paBeHCTB (59) - (59"), |
y'IHTbiBaH, 'ITO II=-U. TaKHM o6pa3oM, |
6y;a:eT: |
1) ;ri:JIH TIOJIH C H JI bi T H JK e C T H (OCb Z BepTIIKaJibHO BBepx) |
II=Pz; |
(64) |
2) ;rJ:JIH TIOJIH C H JI bl y IT p y r 0 C T H (JIHHefmoro)
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II =cx2 /2; |
(64') |
3) )l,JIH TIOJIH |
C H JI bi T |
H r 0 T e H |
H H |
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(64") |
floTeHI.J.HaJibHaH |
3HepnHI |
CHCTeMbi onpe,n:eJIHeTCH TaK JKe, KaK H |
)l,JIH o,n:Hoii ToqKrr, |
a |
HMeHHo: |
nomeliVyUG.IlbliaR Emepc.uR II .MexaliU'teCKOii |
cucme.Mbt 8 iJaliliOM ee no.llo:JJCeliuu pasHa pa6ome, KOmopy10 npous8eiJym CU.Ilbl nOAH npu nepeMeU{eliUU cucme.Mbl U3 iJaliliOcO nO.IlO:JICCiiUR 8 liY.IleBOe,
T. e.
flpH HaJIHqHH HeCKOJibKHX TIOJieH (HanpHMep, ITOJieH CHJI THJKeCTH
H CHJI yrrpyrOCTH) ,U:JIH KaJK,U:OfO ITOJIH MOJKHO 6paTb CBOe HyJieBoe ITO· JIOJKeHHe.
3aBIICHMOCTb MeJK,n:y ITOTeHUil3JlbHOH 3HeprHeH H CHJIOBOH ~YHK·
I.J.Heii 6y,n:eT TaKOH JKe, KaK II |
;ri:JlH TOqKH, |
T. e. |
II (xh Yl, zl> ••• , Xn, Yn· |
Zn)=-U(xlo |
Yt. zl ,... , Xn, Yn· Zn)• |
3aKOH coxpaHeHHH MexaHII'IeCKOH3HeprHH•
.lJ.orryCTHM, qTO Bee ,n:eHCTBYIOII.J,IIe Ha CIICTeMy BHelliHHe H BHyTpeHHHe CHJibi rroTeHuHaJibHbi. T or;a:a
2:Ak=II0 -II1 •
flo,n:CTaBJIHH 3TO BbipaJKeHHe pa60Tbl B ypaBHemre (50), ITOJiyqHM )l,JIH
JIJ06oro noJioJKeHHH |
CHCTe:vrbr: T- T0 =Il 0-II HJIH |
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T+II=T0 +II 0 =const. |
(65) |
CJie,n:osareJibHO, |
npu iJ8U:JJCeliuu noiJ iJeiicmsueM |
nomeliVyUG.IlbliblX |
cu.ll cyMMa KUiiemutt,ecKoii u nomelil.t,Ua.llbliOU :mepc.uu cucme.Mbl e Ka:JJCiJO.M
ee noAo:JJCeliuu ocmaemcR 8e.llU'lUiiOU nocmoRiiliOU. B 3TOM |
H COCTOHT |
3GK01i |
COXpalieliUR .MeXGiiU'teCKOU c:!liepcUU, HBJIHIOII.J.IIHCH 'IaCTHbiM CJIY· |
qaeM |
o6II.J.ero ~H3HqecKoro 3aKoHa coxpa:neHHH ::~:neprrrrr. |
BeJirrqrr:na |
11 N' |
2173 |
321 |
f,llC ffix, Wy, Wz - TipOCKIJ.I!I! Ha OCI! Oxyz MrHOBeHHOH yrJIOBOH CKOpOCTI! TCJia; Xk, Yk• Zk - KOOp,lli!HaTbl TO'IeK TeJia.
Tio,llCTaBI!M 3TI! 3Ha1!e!II!51 Vky I! Vkz B npe,llbl.llYIIJ.eC paBeHCTBO; npl! 3TOM 3aMeTIIM, '!TO '!JieHbl C Tip0113Be,lleHI!51MI! KOOp,lli!HaT MOlKHO He TIO,IlC'II!Tb!BaTb, TaK KaK OCI!
Oxyz 51BJI5110TC51 fJ13DHblMI! OCHMII IIHepiJ.IIll ll ,llJIH HIIX Dee IJ.eHTpo6elKHble MOMeHTbl
HHepD.III! |
paBHbi HYJIIO, T. e. |
'2. mkXkYk="L Fnf,xkzk=O. B pesyJibTaTe, Bb!HOC51 |
06IIJ.I!H |
MHOlKl!TeJib Wx sa CK06KII, |
H3H,IlCM |
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Kx = |
L.mx (m,;v,,) ~ [~mk (yk +z.k)] Wx, |
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r,lle BeJIH•mHa B KBa.llpaTIIblX cKo6Kax |
npe.!l,cTaB,151eT co6oii, cor.~acHO ¢op\ryJiaM (3) |
H3 § 102, |
fJJaBHb!H MOMeHT |
II!IepiJ.IIII |
Te,1a OTHOCHTe.1bHO |
OCII |
Ox. AHaJIOfl!'IHbie Bbi- |
palKeHI!51 |
ROJIY'IHM .llJIH K,1 , Kz |
n |
OhOH'IaTe,JbHO 6y,n:eT· |
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Kx=lxWx, |
Ky=lyffiy, |
Kz=lzWz• |
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(78) |
<DopMyJibi |
(78) ,llaiOT |
BhrpalKeHHH |
npoeKu.nii: |
BeKTopa K0 fla |
eAaeHble ocu |
uHeptjuu |
n;e.11a ,llJIH TO'IKH 0. |
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EcJin ocH Oxyz He 6y.llyT rJiaDHbiMH, To, KaK neTPY.IlHO TIO.IlC'lHTaTb,¢opMyJihi (78) |
np!UviYT |
CJie,llyiOIIJ.IiH |
6oJiee |
CJJOlKHhlH |
Bli)l: |
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Kx=lxWx-lxyWy-lxzWz, |
) |
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Ky=-lxyffix+lyWy-l 11 zffiz, |
~ |
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(78') |
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Kz = - J xzWx- J yzffiy+ lzffiz· |
J |
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2. |
K II n e T II q |
e c K a |
51 |
::~ |
H e p r H H |
T e JI a, |
.ll B II lK y Ill. e r o c |
5I |
n o- |
K p y r |
|
H en o .ll B H lK 11 o if |
To q K H. |
TaK |
KaK Jllo6oe |
::~JieMeHTapHoe |
nepeMe- |
IIJ.enHe TBep,lloro TeJia, HMeiOIIJ.ero Heno,llBH:lKHYIO TOlJKY 0, npe,llcTaBJI51eT co6oft 3JieMeHTapHbiii llOBOpOT C yrJIOBOH CKOpOCTbiO (;} BOKpyr MfHOBeHHOH OC!i BpaiiJ.eHH51 0[, npoXO,Il511IJ.eH qepe3 3TY TO'lKY{eM. § 60), TO KIUieTH'leCKYIO3HeprHIO TeJia MOlKHO OTI-
pe,lleJIHTb |
no ¢opMyJie |
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T= J w 2j2. |
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1 |
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Tio,llCT3BiiM CIO,Ila 31!3'leHHeJz li3 <\JOPMYJibl (12) (eM. ~ 105, p!iC. 280) H O,llHOBpe- |
MeHHO y'!TeM, 'ITOw cos a=wx, w cos~= wy, w cos y=wz, TaK KaK neKTOp |
w Ha- |
npaBJien |
no ocH Ol. Tor,lla noJiy'IHM |
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2T= J xffi;+ J yw~+Jzw;-2lxyffixWy-2lyzWyffi::-2lzxWzWx• |
(79) |
EcJIII B Ka'leCTBeKoop,llHHaTHblX oceii: B351Tb eAaeHbte ocu uHepu,uu meAa .llJIH To'I- |
KI! 0, |
TO |
BCe 1J.eHTpo6elKHb!e |
MOMeHTbl |
HHepiJ.!iH |
06paT51TC51 B |
HYJII! H |
TOf,lla |
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2 |
2 |
J 2 |
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~) |
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~=4~+~~+z~ |
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3. |
,ll |
1i H a M 1i 'Ie C K 1i e |
y p a B H e H li 5I |
8 H JI e p |
a. TiyCTh |
Ha TBep,llOe |
TeJio, |
HMeiOIIJ.ee neTIO,IlBI!lKHYIO TO'lKY 0, )l..eHCTByiOT 3a)laHHble |
CliJibl ~. f';,..., Fn |
(pnc. 341). O.llHOBpeMeHHO Ha TeJio 6y,lleT ,lleiicTBOBaTh peaKIJ.HH Ro CBH3H (Ha pi!-
cyrme He |
nOKa3aHa). l.J:T06bl liCKJIIO'IIiTb li3 ypaBHeHHH ,llBiilKeHHH 3TY |
Heli3BeCTHYIO |
peaKIJ.HIO, |
BOCnOJib3YeMC51 |
TeopeMOll MOMeHTOB OTHOCHTeJibHO IJ.el!Tpa 0 (§ 116), npe,ll- |
CTaBIIB ee |
B Bl!,lle (74), T. e. B BH,Ile |
TCOpCMhl PeaaJIH. TOr,'J,a nOCKOJibKY |
mo(Ro)=O, |
ypaBHCr!IIC (74) ,ll3CT |
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vn=Mo, |
(80) |
r,lle Mo='2. mo(F~). a vn - |
CKOpOCTb no OTHOllieHHIO K IIHCPIJ.Ii3JibHOH CliCTeMe OTC'IeTa |
Ox1y1z1 TO'IKH B, coana,llaiOIIJ.eif c |
KOHIJ.0\1 aeKTopa R0 . |
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.ll,BHlKeHHe TeJia li3Y'IaeTC51TOlKC no OTHOlliCHiiiO K li!!CpiJ.HaJibHOH CliCTeMe OTC'leTa |
Ox1y1z1• Ho 'ITOOblnoJiy'IHTbypanHeHnH 3Toro ,llBHlKeHHH a naH6oJiee npocroif ¢opMe, CllpOeKT!ipyeM o6e '13CTHHpC,Ilbli'\YIIJ.CfOpaBeHCTBa Ha lKeCTKO CB5133HHble C TeJIOM 1i ,llBI!- JKYIIJ.HeC51 BMeCTe C HliM OC!i Oxyz, 51BJIH!OIJJ.HCC51 <!AaBflbl.MU OCJl.MU UHepU,UU meAa .1\Jilf
TO'IKH0. Tor,lla BblpaJKeHHH npoeKIJ.HH neKTopa R0 6y.nyT HMeTb npocTOif BH,Il, .nanaeMbiii <\JopMyJiaMll (78), a BXO,Il51IIJ.l!e B HliX MOMCHThiHHeplJ.I!II Jx, Jy, Jz 6y,llyT BCJIH- 'Il!HaMH nOCTOHHHb!Mii.
fJiaua XXVII
UPHHUHU ~AnAMBEPA
§ 133. nPHHU.Hn }J,AJIAMBEPA }J,JI,Sl TO'IKH
H MEXAHH4ECKOii CHCTEMbl
MeTO.Il.bi peweHHH sa,naiJ: MexaHIIKH, KOTOpbie ,no cux nop paccMaTpH· BaJIHCb, OCHOBbiB810TC.H Ha ypaBHeHHHX, BbiTeKaJOmHX I:IJII:l Henocpe,ll.CT• BeHHO ns 38KOHOB HhiOTOHa, IMH :m:e H3 o6m.ux TeopeM, flBJifllOlll,HXCR CJie,ncTBHeM 3THX saKOHOB. O,nHaKo 3TOT nyTh He HBJIHeTCfl e,nHHCTBeH· HbiM. 0Ka3biBaeTCH, IJ:TO ypaBHeHHH ,ll.BH:m:eHHH_ HJIH ycJIOBHR paBnoBe. Cll.H MexaHuqecKQH CI:ICTeMbi MO)f(HO IIOJIYIJ:HTb, IIOJIO)f(I:IB B OCHOBY BMeCTO saKoHOB HbiOTOHa ,npyrue o6mue noJio:m:eHHH, H83biBaCMbie npufttJUnaMU MexaHww. B pH,ne cJiyiJ:aeB npuMeHeHHe 3THX npHHUHIIOB no3BOJIHeT,
K8K Mbl yBH,ll.HM, H8HTI:l 6oJiee 3qxpeKTHBHbie MeTO)].bl pemeHI:lfl COOTBeTCT•
BYJOmux sanaiJ:. B 3TOfi rJiaBe 6y,neT paccMoTpeH O.Il.HH H3 o6I.UHX npm-r· UHIIOB MeXaHHKH, H83biBaeMbiH npUfll{UnOM J1,a.tWM6epa.
Haii:,neM CHaTJaJia Bbipa:m:eHHe npHHluma )J.Jifl o,nHofi MaTepHaJibHOH TOTIKH. f1yCTb Ha MaTepuaJibHYIO TOIJ:KY C Maccofi m ,neficTByeT CHCTeMa aKTI:IBHbiX CHJI, paBHO)J.efiCTByJOmyJO KOTOpbiX o603H8TJI:IM pa, H peaKUHfl Cl3H3H N (ecJIH TOIJ:Ka flBJIHeTc.H HecBo6o,nHofi). ITo,n ,neficTBHeM Bcex 3THX CI:IJI TQqKa 6y,neT ,ll.BHraTbC.H IIO OTHOJEeHHIO K HHepUH8JibHOH CHCTe•
' Me OTCTJeTa C HeKOTOpbiM yCKOpeHHeM a.
BBe,neM B paccMOTpeHHe BeJIHIJ:HHY
(84)
HMeiOI.I.J.YIO pa3MepHoCTh CHJibl. BeKTOPHYJO BeJIHTIHHY, paBHYJO no Mo-
.U.YJIJO npOH3Be,neHHIO MaCChi TOTIKH Ha ee ycKopeHHe H HanpaBJieHHYIO npOTHBOIIOJIO)f(HQ 3110My ycKopeHHIO, H83b!B810T CUAOU UHeptJUU TOIJKH.
§ 106.
Tor.n.a oKa3biBaeTc.H, liTO .D.BIDKeHHe TOliKH o6Jia,n.aeT cJie,n.yiOI.l.J.HM CBOHCTBOM: eCAU 8 Jl106ou MO.Meflm 8pe.MeflU K aeUcmBljiOU{UM Ha mDttKY aKmU8flbtM CUJla.M U peDKlfUU C8Jl3U npucoeaUflUmb CUJllj UHepl{UU, mo noAylieHflafl cucmeMa cuA 6yaem ypaeHoBeULeHHou, T. e.
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(85) |
3To llOJIOJKeHHe BblpaJKaeT |
ll p |
I:l H U I:l ll n a JI a M 6 e p a ,lJ. JI li |
M aTe p I:l a JI b H 0 ll T 0 ll K |
H. |
HeTpy.D.HO y6e,n.HTbC.H, liTO OHO 3KBI:l· |
BaJieHTHO BTopoMy 3aKoHy HhJOToHa n Hao6opoT. B caMOM ,n.eJie, BTO· poii 3aKOH Hh!OTOHa ,ll.JIH paccMaTpHBaeMOH TOliKI:l .n.aeT ma=Fa+iJ.
TiepeHOCH 3)J;ech BeJIH'IHHYmii B npaByJO llaCTb paBeHCTBa H YliHThiBaH o6o3HalleHHe (84), npu,n.eM K cooTHoiiieHHJO (85). Hao6opoT, nepeHocs:r B ypaBHeHHH (85) BeJIHTIHHy F" B .n.pyry10 qacTb paBeHcTBa H ytJHThlBaH o6osHalleHue (84), noJiyliHM BbipaJKeHHe BToporo saKoHa HhJOTOHa.
PaccMoTpHM Tenepb MexaHwiecKyJO cHcTeMy, cocTOHJUYIO us n
MaTepuaJibHbiX TOlleK. Bhi,n.eJIHM KaKyJO-HH6y,n.h H3 TOTieK cucTeMbl c Maccoii mk. ITo.n. .n.eiicTBHeM npHJIOJKeHHbiX K Heii BHeiiiHHX H BHyTpeHHHX cuJI Fk H F~ (B KOTOpbie BXO)J;.HT n aKTHBHhie cHJibl, H peaKUHH cBH· 3eii) TOliKa 6y,n.eT )J;BHraTbC.H no OTHOIIIeHHIO K HHepuHaJibHOH CI:ICTeMe OTClleTa C HeKOTOpb!M ycKopeHHeM ak.. BBe)J;H )J;JI.H 3TOH TOliKI:l CHJIY HHep· uuu F~=-mk.a1., noJiyliHM corJiacHo paBeHcTBY (85), liTO
(85')
T. e. liTO F:, Fk H F~ o6pasyJOT ypaBHOBerneHHyJO cncTeMy CHJI. DoBTOpHH TaKHe paccyJK,n.eHH.H )J;JIH K3JK)J;Oll I:l3 TOlleK CI:ICTeMbl, npH,ll.eM K
CJie,n.yJOIUeMy |
pesyJihTaTy, ~bipaJKaJOlll,eMy ll p I:l H U I:l ll n a JI aM- |
6 e p a .n. JI H |
c u c T e M hi: ecAu e A106ou MoMmm epe.Mmu K KaJ~Caoa |
U3 motteK CUCme.Mbl KpOMe aeuCmBljiOUfUX fla flee BfleUlflUX U BflljmpeHHUX cu11. npucoeauHumb coomeemcmeyiOU{Ue CU/I.bt uHepttuu, mo noll.ytteflflaJt cucmeMa cuA 6yaem ypaeHoeeULeHHou u K Heu MOJ/Cflo npu.MeHRmb ece ypaeHeflufl cmamuKu.
MaTeMaTHlleCKI:l npmnum naJiaM6epa )J;JIH CI:ICTeMbl BbipaJKaeTC.H n
BeKTOpHb!MH paBeHCTBaMH BH,ll.a (85'),KOTOpble, OlleBH)J;HO, 3KBHBaJieHT· Hbl ,n.mtxpepeHUHaJibHb!M ypaBHeHHHM ,ll.BHJKeHHH CHCTeMbl (13), llOJIY· lleHHb!M B CJie)J;OBaTeJibHO, H3 npHHUHna naJiaM6epa, KaK H H3 ypaBHeHHH (13), MOJKHO llOJIYliHTb BCe 06IUHe TeopeMbi ,ll.HHaMHKH.
3HalleHHe npHHUHna naJiaM6epa COCTOHT B TOM, liTO npH Henocpe)J;- CTBeHHOM ero npHMeHeHHI:l K 3a,ll.allaM ,ll.HHaMHKI:l ypaBHeHH.H ,ll.BHJKeHHH CHCTeMbl COCTaBJIHIOTCH B cpopMe XOpOIIIO H3BeCTHbiX ypaBHeHHH paBHOBeCH.H; 3TO ,n.eJiaeT e)J;HH006pa3Hb!M llO)J;XO,ll. K perneHHIO 3a,n.aq H tJaCTO ynpolll,aeT cooTBeTCTBYJOIUHe paclleTbi. KpoMe TOro, B coe,n.HHeHHH c llp1iHUI:lllOM B03MOJKHb!X nepeMeiUeHHH, KOTOpblll 6y,n.eT paCCMOTpeH B CJie)J;yJOIUeii rJiaBe, npHHUI:lll naJiaM6epa ll03BOJI5IeT llOJiyTIHTb HOBblll 06lli,Hll MeTO,ll. perneHHH 3a,ll.all ,ll.HHaMHKI:l (eM. § 141).
Hs cTaTHKH H3BecTHO, liTO reoMeTpnllecKaH cyMMa cHJI, Haxo,n.SIUJ;HXC.H B paBifOBeCHH, H CYMMa I:IX MOMeHTOB OTHOCHTeJibHO JII060rO ueHTpa 0 paBHbi HYJIJO, npnlleM, KaK noKa3aHo B §'120, 3TO cnpaBe,n.JIH·
BO ){JIH CHJI, .ueiicTBYIOI.I.J.HX He TOJibKO Ha TBep,n:oe TeJIO HO II Ha JIJ06yJO HSMeMneMyJO MexaHII'IecKyJOcHcTeMy. Tor.n:a Ha ocHoBaHHII npHHU.Hna ,[laJiaM6epa .UOJI)KHO ObiTb:
~ (F1+Fk +F~) = o; |
|
~ [m0 (F1) +m0 (Fk) +mo (F~)] = o. |
(86) |
BBe,n:eM o6osHaqeHHH: |
|
RH = ~ F~, Ml'J = ~ m0 (F~). |
(87) |
BemrqHHhi RH, Ml'J npe.n:cTaBJIHJOT co6o10 eACU3Hbtt1BeKmop u eAaBHbttl MOMeHm omHocumeAbHo uptmpa 0 cucmeMbt cuA uHepttuu. B pesyJibTaTe,
yqHTbiBaH, qTQ reOMeTpHqecKaH cyMMa BHyTpeHHHX CHJI H cyMMa HX MOMeHTOB paBHbl HyJIJO, noJiyTIHM H3 pa13eHCTB (86):
~Fk+RH=O, ~m0(FV+M~=O. |
(88) |
TipHMeHeHHe ypaBHeHHH (88), Bb!TeKaiOI.I.J.HX H3 npHHIJ,Hna naJiaM6epa, ynpoi.I.I,aeT npouecc perneHHH sa,n:aq, TaK KaK 3TH ypaaHeHHH He co-
.uep)KaT BHyTpeHHHX CHJI. Tio cymeCTBY ypaBHeHHH (88) 3KBHBaJieHTHbl ypaBHeHHHM, Bblp8)K8JOI.I.J.HM TeopeMbi 06 H3MeHeHHH KOJIHTieCTBa )J.Bll•
)KeHHH H rJiaBHOfO MOMeHTa KOJIH'-!eCTB.!J.BH)KeHHH CHCTeMbl, H OTJIH'-18· IOTCH OT HHX TOJibKO no cpopMe.
YpaBHeHHHMH (88) oco6eHHO y.uo6Ho noJib30BaTbCH npH Hsy'-leHHII
){BH)KeHH.H TBep,n:oro TeJia HJIH CHCTeMbl TBep.UbiX TeJI. nJIH nOJIHOfO H3Y· '-!eHHH ,l{BH)KeHHH JIJ06oH H3MeHHeMOH CHCTeMbi 3THX ypaBHeHHH 6y,ll,eT He,ll.OCT8TO'-IHO,TaK )Ke KaK He.!J.OCT8TO'-IHOypaBHeHHH CTairHKH ,l{JIH H3y•
'-leHHH paBHOBeCHH JIJ060H MeXaHH'-!eCI{QH CHC'De:Mbl (eM. § 120). |
B |
npoeKuHHX Ha Koop,n:nHaTHhie ocH paBeHCTBa (88) )l,aJOT ypaBHe- |
HHH, |
aHaJIOfHTJHbie COOTBeTCTBYIOI.I.J.HM ypaBHeHHHM CT8THKH (eM. § 16, |
30). |
1ho6hi nOJib30BaThCH 3THMH ypaBHeHHHMH npH pemeHHH 3a,l{aiJ, |
Ha,l{O |
3H8Tb Bbipa)KeHHH rJiaBHOfO BeKTOpa H rJiaBHOrO MmleHTa CHJI |
HHepii,HH.
B saKJIJOqeHHe cJie.uyeT no.uiJepKRYTh, IJTO npH HsyTiemm .UBH)KeHHH no OTHOIUeHHIO K UHept{UaJlbflOU CUCmeMe omcttema, KOTopoe 3)J,eCb H paccMaTpHBaeTCH, CHJibl HHepUHH BBO,l{HTCH mOJlbKO m02iJa, KOr)J.a ,li;JIH peiiieHI!HI sa,n:aq npHMemieTCH npHHIJ,Hil naJiaM6epa * .
§ 134. fJIABHhlii BEI(TOP U fJIABHbiA. MOMEHT CUJl UHEPU:HH
CpaBHHBaH nepBoe H3 paBeHCTB (88) C ypaBHeHHeM mac=J:.Fg,
Bbipa)K810I.I.J.HM )J.OK838HHYIO B § 107 TeopeMy 0 ){BH)KeHIUl IJ,eHTpa Mace
* 8 § 91 Mbl paCCM3Tp!iB3JIIi CI!Jibl I!HepiJ.!ili (rrepeHOCHYIO li Ki)p!iOJIHCOBy), KOTO· pb!e BBO,!U!TCll )lJIH TOfO, 'IT06blITOJIY'II!TbB03MOMHOCTb COCT3BJI5!Tb ypaBHeiiiili )lBH)Ke• Hill! B HeHHepiJ.!i3JibHOH CI!CTeMe OTC'IeTa B TOM BH)le, KOTOpblfi Olili HMelOT B CHCTeMe OTC'IeTaHHepiJ.H3JihHOi1. 3.necb CHJibl HHepiJ.!ili BBO)lliTCli )lJill TOrD, 'IT06b!B HHepU.!i3Jib· HOH CHCTeMe OTC'IeTa ITOJIY'IIiTb B03MQ}KH0CTb COCT8BJI5!Tb ypaBHeHHli )lBH}KeHHli B ou.ne ypaBHeHHil paBHooecHH. Bee 3TH CI!Jibl Hllepumt K Karetopunt41K3H'!ecKHXCHJI, rrpHMepbl KOTOpblX 6bl.'llipaCCMOTpeHbl B § 76, He rrpRHa)lJleJKaT,
(B 3TOH rJiaBe Macey CHCTeMbl 0003HaqaeM 6yKBOH m), Haif,neM, t.ITO
(89)
T. e. C!/Ja8flblU8eKmop CUll UHepl{UU .MeXaflUtteCKOU CUCme.Mbl (B t.IaCTHOCTII, tn8epaoco me;w) pasen npouseeaeflluo htaCCbt cucme.Mbt (me;w) Ha ycKOpeHue l{eHmpa .Mace u H'!_npas!lefl npomueonollo:J/Cflo amoMy ycKopeHU/0.
EcJIII ycKopeHIIe ac pasJIOJKIITb ua KacaTeJibHOe H uopMaJihHoe, TO BeKTop RH pasJioJKIITC.H Ha cocTaBJIHIOIUHe *
|
|
(89') |
CpaBHHB Tenepb BTopoc II3 paBeHCTB (88) c ypaBHeHIIeM |
dK0 /dt= |
=~mo (ft), BblpaJKaiOill,HM TeopeMy MOMeHTOB (eM. § 116), |
H YTIT5!, |
'ITO aHaJIOfii'-IHbiM 6y.u.eT COOTHOIIIeHIIe )l:JI.H |
MOMeHTOB OTHOCIITeJibHO |
OCH, nOJIY'-IIIM: |
|
|
MH= _dKz |
(90) |
z |
dt ' |
|
T. e. z;zaeflblu MOMeHm cull uHepl{uu MexaHuttecKou cucmeJvtbt (meepaozo me11a) omHOcumellbflO HeKOmopozo l{eHmpa 0 u/lu ocu z paeefl e:mmou co 3flaKO.M MUHyc npouseoaHotl no epe.Meflu om KuHemuttecKOzo MoMeHma cucme.Mbt ( me!la) omrwcumellbflO mozo ;)ICe tteHmpa u11u mou :J/Ce ocu.
fi p II B e ){ e H II e C II JI |
II H e p U II II T |
B e p )1. 0 |
r 0 |
T e Jl a. |
CorJiaCHO pesyJihTaTaM § 12, |
cnpaBe)l.JIIIBhiM )l.JIH |
JII06biX |
cHJI, |
ciicTeMy |
ciiJI IIHepumi meepaoco me.aa MOJKHO saMeHIITh o.u.Hoii ciiJioii, paBHoii RH
H npHJIOJKeHHOif B npOH3BOJibHO Bb16paHHOM II,eHTpe 0, H napoil C MO-
MeHTOM, paBHbiM Ml!J. PaccMoTpHM HeCKOJibKO IJ:acTHbiX cJiyi.IaeB.
1. Il o c T y n a T e JI b H o e .u. B H JK e H II e. B 3TOM ~yqae yc-
KOpeHHH BCeX T01IeK TeJia O)l.IIHaKOBbl II paBHbr ycKopeHIIIO ac ueHTpa Mace C TeJia (ah=ac). Tor.na Bee ciiJihi IIHepuim F~=-mkac~6pasy~T
CHCTeMy napaJIJieJibHbiX CIIJI, aHaJIO!IIlJHbiX CIIJiaM T.HJKeCTII Pk =mkg,
H 1103TOMY, K8K H CIIJibl T5!JKeCTH, IIMeiOT paBHO)l:eHCTBYIOIUYIO, npOXO·
.namy10 qepe3 ToqKy C.
CJie)l.OBaTeJibHO, npu nocmynamellbflOM aeu:JICeflUU CU/lbl UHepl{UU
meepaozo me!la npueoaflmCfl K paeHooet1cmeyJOU{etl, paeHou R!'. u npoxo-
aflttteu ttepes l{eHmp .Mace mella.
2. B p a m aT e JI h H o e .u. B II JK e H II e. flycTh TBep.noe TeJio IIMeeT ITJIOCKOCTb MaTepiiaJibHOH CIIMMeTpiiii 0xy H BpamaeTC.H BOKpyr OCII Oz, nepneH)l:IIKYJI.HpHoii 3TOH nJioCKOCTII ** (piic. 343, r,n:e noKasaHo ceTieHIIe TeJia nJiocKOCThiO Oxy). EcJIII npiiBeCTII ciiJibl IIHepuHH K ueHTPY
0, TO BCJie)l.CTBIIe CIIMMeTpiiii pe3yJibTIIpyiOill,a.H CIIJia II napa 6y,nyT JieJKaTh B nJIOCKOCTH Oxy II MOMeHT napbi 6y,neT paBeH Mgz. Tor.na, TaK KaK
*HopM3JibHYIO COCT3BJI51101IlYIO CHJ!bl HHepU.HH H33b!Ba!Ot eme U.eHTp06eJKHoii COCTaB.~HIOIIleH HJIH U.eHTp06eJKHoH CHJIOH HHepU.!iH.
**.ll,JIH TeJia npo!iSBOJ!bHOH <!JopMhl, 13pamaiOmerOCH BOKpyr HenO)(BHlKHOH OCH, npuMep np!iBe)(eHHH cun unepmm )(aeTcH B § 136.
Kz=J ozW, TO no BTopoii H3 cpopMyJI (90) |
|
M~z= - J0 z·W= -loz•8, |
(91) |
r)J.e e - yrJiosoe ycKopeHHe TeJia.
CJie,ll.oBaTeJihHO, cucTeMa cHJI uHepuuu TaKoro Bparu.aiOruerocH TeJia npHBO,ll.HTCH K CHJie R't, onpe,ll.eJIHeMoii ¢opMyJiofr (89) H npHJIOMeHHoii
|
B TO<IKe 0 (puc. 343), |
H K nape c M011eHTOM M~z, onpe)J.eJIHeMbiM ¢op- |
|
MyJioil: (91), JieiKaiUeR B IIJIOCKOCTH |
CHMMeTpHH |
|
TeJia. |
|
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|
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|
|
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|
3. |
B p a ru. e H H e |
B o |
K p y r |
o |
c H, |
n p o- |
|
x |
o .Il. 51 ru. e fr |
q e p e 3 |
u |
e H T p |
M a c c |
T e JI a. |
|
EcJIH Te.Tio, pacc:viOTpeHHoe B n. |
2, |
sparu.aeTcH |
|
soKpyr ocu |
Cz, npoxo,ll.Hill.eii qepes |
ueHTP Mace C |
|
TeJia, |
TO R"=O, TaK KaK Gc=O. CJie,n:oBaTeJibHO, B |
|
3TOM CJiyqae CllCTeMa CHJI HHepUHH TeJia npHBOJ{HTC51 |
|
K o,n:Hoii TOJibKO nape c MOMeHTOM |
M~z, JieMaru.eii |
|
B |
IIJIOCKOCTH |
CIIM~!eTpHH |
TeJia. |
|
|
|
|
Puc. 343 |
4. |
fi JI o |
|
c K o |
n a p |
a |
JI JI e JI b |
H o |
e |
)]. B H- |
|
e H H e. |
EcJIH TeJio |
HMeeT nJIOCKOCTb |
CHMMeT- |
|
M |
|
pHH |
H )J.BHMeTC5! |
napaJIJieJibHO 3TOR |
nJIOCKOCTH, |
TO, OTJeBH)J.HO, CHCTeMa CH.TI HHepUHH TeJia npHBe,ll.eTC.H K JieMalli.HM B nJiocKOCTH cHMMeTpHH cHJie, paBHOH R.u H npuJioMeHHoii B ueHTpe Mace
C TeJia, H nape c MOMeHTOM M~z=-Jcz ·e.
flpH peWeHHH sa,n:aq no ijlopMyJiaM BH,Ua (91) BbllJHCJI5!eTC51 MO.Il.YJlb MOMeHTa Mi', a ero HanpaBJieHHe, npoTHBonoJioMHoe e, yKa3hiBaeTCH Ha qepTeMe.
§ 138. 803MO)I(HbiE flEPEMEU(EHH~ CHCTEMbl. 'IHCJIO CTEflEHEft CBOISO~bl
3$lJeKT MeXaHHqecKHX CB513efi MO>KHO. ytiitTb!BaTb, He TOJibKO BBO~ll
HX peaKU.HII, KaK 3TO ll.O CIIX nop lJ..eJiaJIOCb, HO H paccMaTpiiBa51 Te nepe• Mern,eHHH, KOTOpb!e TQqKH MeXaHWiecKOH tHCTeMbf MOryT HMeTb npH HaJIO}KeHHbiX Ha nee CBSI3HX. TaKOH n.yrh no3BOJUieT cpa3y noJIYqaTb ypasHeHIIR paBHOBecHH HJIH lJ..BIUKeHH51 CHCTeMbl, He COll.epJKali(He HanepeA
HeH3BeC'I'HblXpeaKUHH CBJl3eH, tJTO Cyti{eCTBeHHO o6JiertJaeT peilleHHe MHOrHX 33lJ..3ti MeXaHIIKII.
flepeMeli(eHHH, 0 KOTOpbiX CKa3aHO Bbiiiie, Ha3blBa!OT 603Jri()JICflbl.MU
(»JIH supmyaAbftbt.MU)· nepeMetl.{euufM'!u. 0HH lJ..OJI}KHhr Yll.OBJieTBOps.ITb lJ..BYM ycJIOBIIHM: 1) 6b!Tb 3JieMeHT3pHbiMH, T3K K3K IIpH KOHe'IHOMnepeMern,emm CHCTeMa MOJKeT npHHTII B llOJIO)KeHHe, flJ..e 3¢lqleK1 H3JIOJKeHHbiX CBR3eH 6y~e-r ll.PYriiM; 2) 6orF& TaKIIMH, trr06bl BCe HaJIO)f{eHHbie R lJ..3HHbiH MOMeHT BpeMeHII Ha CIICTeMy CB.H3II coxpaHHJIHCb, H1'JaqeMOlKe'V
H3MeHHTbCli BHlJ.. pacCM3TpiiBaeMOH MeXaHI-f'IeCKOH CHCT€Mbl.
HanpnMep, B KpnoomiiiiHo-noJI3YHHOM MexaHR3Me, nao6paJKeHHOM HIIJKe na pHc. 356 (eM. § 140), nepeMemenne na noKa3aHnoro noJiolKeHnH B noJioJKerme, npa KOTOpOM cp=(}, HeJib3H p3CCM3T!J'IiBa'!'bK3K B03MOl*HOe, T3K KaK npn cp=Q ~KT HaJIOJKeHHb!X CBH3eii 6y.n:eT .llPYrY!M, qTO, B qaCTHOCTii, li3MeHiiT yc.10Biie paBHOBeCHH MexaHH3Ma no.u. ,!l;eikTBHeM cnJlbl. P u napb! c MoMeHTOM M. Toqno TaK lKe HeJih3H cquTaTb B03MOlKiib!M }.l.alKe 3JleMeHT3j;)HOe nepeMeiii,eHiie TOqKR B maTyHa B,llO.llb JIH• HUH AB; ono 6b!JJO 6bi B03MOJKHb!M, ecJIU B TOqKe B BMecTo noJI3yHa 6biJia 6bl Kaqa!d· maHcH MyqlTa (pnc. 161 B §57, MycpTa C}, T. e. xor.n:a MexanusM 6b!JI 6bl ll.PYrHM.
TaKIIM o6pa3oM, BOOMOJICHblllt nepeMeU{f!HUeM Mexanu!leCiwii cucmeMbl 6ydeM Ha3bt8aJ1lb JlJ06yHJ COBoKynHOCtnb fJ/leJ-teHmapHblX nepe.MeU{eHUU mo· £f£K amou cucmeMbt U3 3aHuMaeMozo e daHHbtt1 MOMeHm epeMenu TWAOJ!Ce· HUH, KOmopbte donycKaHJmCH BCeMU Ha/lOJ/CeHflbl.MU Ha Cucme.My C8Jl3Jl.MU. flpii 3TOM TIOll.lJ..OTIYCKaeMb!Mli B CJiyqae HeylJ..epJKHB3IOJUHX CBH3eii 6yll.eM
IIOHHM3'fbTe B03MQ)KHble nepeMern,eHHH, npii KOTOpbiX CBH3H COXpaHJI• IOTCR (TOtJKH CHCTeMbl OT CBSl3eH He «OCB06om~a!OTCH»).
