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iloGJie.n;Hee paBeHCTBO cJie.n;yeT H3 cpopMyJibl (15'). TaKHM o6pa3oM, TeopeMa .n;oKa3aHa. Pe3yJibTaT, .n;aBaeMbiii TeopeMoii, MOMHO ern.e H306pa3HTb TaK, KaK 3TO noKa3aHo Ha pnc. 37, 6 (cHJIY F Ha 3TOM pncyHKe Ha,Zto Cl!HTaTb oT6poweHHoii). PaccMoTpHM npHMepbi, HJIJIIOCTpHpyiOrn.He TeopeMy.
npHMep 1. ~ho6bi y,n:ep»<aTb B paBHOBeCHH o,n:nopo,n:HblH 6pyc AB AJIHHOH 2a H B~COM p, na,n:o npHJIOJKHTb B ero cepeiJ.HHe C HanpaBJieHHyiO BBepx CHJIY Q, ITO MO,n:yJIIO panny10 p (pHC. 38, a). CorJiaCHO ,n:oKa3aHHOH TeopeMe CHJiy Q MOJKHO 3aMeHHTb CHJIOH Q', npHJIOJKeHHOH K KOHUy A 6pyca, II napoil: C MOMeHTOM, MO,n:yJib KOTOporo m=Qa,
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Pnc. 39 |
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Pnc. 38 |
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EcJIH nJieqo aroil: rrapbi |
yMeHbiiiHTb |
iJ.O BeJIH'IHHblh (puc. 38, 6), TO o6pa3YIOIUHe ee |
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CHJibl F, F, HaiJ.O yaeJIH'IHTbTaK, 'IT06hl6brJio Fh=Qa. CJie.n;oaareJibHO, 'IT06bly,n:ep-
lKIITIY 6pyc 3a ero Koneu A, na.n:o KpoMe cn,'lblQ' rrpHJIOJKHTb e!Ue rrapy F, F'. 3ror pe3yJibTaT, BblTeKaiOIUllif H3 ,!IOKa33HHOil: TeopeMbi, Henocpe,llCTBeHHO «OIUyiUaeT» pyKa qeJioBeKa, y.n:epJKnaaiOIUaH 6pyc 3a ero cepeiJ.HHY (pnc. 38, a) HJIH sa Koneu (pHc. 38, 6).
npHMep 2. Ha 6apa6an 1 pa.n:ayca r naMoranbi B npornaono.'lOJKHbiXnanpaBJie-
HHHX ,!llle HHTH, K KOHUaM KoropbiX npaKJia,n:biBaiOT cHJibl F n F'=-F (pnc. 39); Ha 6apa6aH 2 TOrD JK.e p3,ll.Hyca H3MOTaHa O,II;H3 HHTb, K KOTOpOil: IIpHKJia)lbiBaiOT CHJiy,
paanyro 2F. Paccr.IOTpHM, '!eM6y.n:yr OTJIH'I3TbCH,n:eiicTBHH 9THX CHJI.
Ha 6apa6an 1 .n:eil:crayer TOJibKO napa CHJI F, F' c |
MoMenroM, qncJienno paBHbiM |
2Fr, apaiUaiOIUaH t5apa6an. CnJiy, .n:eiicrayiOIUYIO na |
6apa6an 2, MOJKHO saMeHHTb |
cnnoii 2F'= 2F, npHJioJKennoil: K ocn 6apa6ana, n napoil: 2F; 2F". B pesyJihTaTe naxo,n.nM, 'ITOua 9TOT 6apa6an .n:eiicrByiOT: l) napa c 'IHCJiennoTaKHM JKe, KaK n B nep-
BOM CJiy'!ae,MOMeHTOM 2Fr, BpaiUaiOIUaH 6apa6aH, H 2) CHJia 2F', OK33biB310IU3H ,ll.aBJieHHe na ocb 6apa6ana.
HTaK, o6a 6apa6ana 6yil.YT apa!UaTbCl! Oil.HHaKoao. Ho rrpu 9TOM ocb aroporo 6apa6ana HcnbiTbiBaer .n:aaJienne, paanoe 2F, a OCb nepaoro 6apa6ana HHKaKoro .n:aaJieHHH He HCllb!TbiBaeT,
§ 12. nPHBE)J.EHHE CHCTEMbt CHJ1 K J(AHHOMY l(EHTPY
PeiiiHM Tenepb 3a,n;aqy o npHBe,n;eHHH npoH3BOJibHGH cHcTeMhi clfJI K ,n;aHHOMY ueHTpy, T. e. 0 33MeHe ,n;aHHOH CHCTeMbl CHJI ,n;pyroii, eii 3KBHBaJieHTHOH, HO 3H3l!HTeJibHO OOJiee npocTOH, a HMeHHO COCTOll!UeH, KaK
Mbi YBH)J;HM, TOJibKO H3 O)J;HOll CHJibl |
H napb!. |
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nycTb |
Ha TBep.n;oe TeJio .n;eiiCTBYeT npOH3BOJibH3ll CHCTeMa CHJI Flo |
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F 2, •• '1 |
Fn (pHc. 40, a). Bbi6epeM |
KaKyiO-HH6y.n;b TO'IKY0 3a ueHTp |
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npHBe.1J.eHHH H, nOJib3YHCb TeopeMOH, .1I.OKa3aHHOll B nepeHeCeM BCe CHJihl B UeHTp 0, npHCOe)J:IIHHSI npH 3TOM COOTBeTCTBYIOIIJ,He naphl (eM. pHC. TOr)la Ha TeJIO 6y)leT )leHCTBOBaTb CHCTeMa CHJI
(19}
npHJIOMeHHhiX B UeHTpe 0, II CHCTeMa nap, MOMeHThl KOTOpbiX COfJiacHe ¢QpMyJie (18) paBHbi:
m1=m0(F1), m2=m0 (F2), ... , mn=m0 (Fn)· |
(20) |
v) z
PHc. 40
Cxo)lHUJ,HecH cnJI&I, npuJioMeHHhie B ToqKe 0, saMeHHIOTCH o.rwoil cnJioii: R, npnJioMeHHOH B ToqKe 0. Tipn 3TOM R ='2:.F~ n.nu, corJiaeoo paBeHCTBaM (19),
(21)
qT06bi CJIO:lKHTb Bee nOJiy'llelllibie napbl, Ha)lO CJIOMHTb BeKTOpbl MOMeHTOB 3THX nap. B pesyJI&TaTe cncTeMa nap saMeHnrcH O)lHOfi napofi, MOMeHT KOTopotl: M0 =L.mk HJIH, corJiacHo paBeHcTBaM (20),
(22)
l\aK nsBeCTHO, BeJinqnHa R, paBHaH reoMeTpHIIecKoii: cyMMe Bcex cnJI, Has&maeTCH ZAaBHbtM eeKmopoM cucmeMbt cwz; seJiwnma M 0,
paBHaH reOMeTpll'IeCKOHCYMMe MOMeHTOB Bcex CUJI OTHOCHTeJibHO ueHTpa 0, Ha3&1BaeTCH Z/la81ibtM MOMeHrllOM CUCme.lffbt CU/l OTHOCHTeJibHO
SToro ueHTpa.
TaKHM o6pasoM, Mbi )lOKasaJIH cJie)lyiOIII.YIO T eo p e M y o n p H- B e .11. e H II H C H C T e M bl C H JI: /IJ06a5l CUCmeMa CU.Il, OeUCmBYIOIJ.{UX Ha a6coA10trlftO meepooe meAo, npu npueeoeHuu K npouseo.llbHo ebt6paH-
H.oMy l{eHmpy 0 saMeH.flemcfl oOH.ou cuAou R, paeHou ZAaeH.oMy eeKmopy cucmeMbt cuA u npuAoaJCeH.HOU e ~feHmpe npueeoenufl 0, u ooHou napou
c MoMenmoM M 0 , paeHbtM Z.llaBHOMY MoMeHmy cucmeMbt cuA omHocu- /ne.!lbH.O l{eH.mpa 0 (puc. 40, 6).
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3aMeTHM, liTo cHJia R He HBJHieTCH g,u:ecb paBHo,u:eiicTByrow.eH: ,u:aH-
HOii CllCTe'Vlhi CHJI, TaK KaK 3aMeHHeT CHCTeMy CHJI He O)J.Ha, a BMeCTe
c napoi%.
l13 )J.OKa3aHHOH TeopeMbl CJie,u:yeT, 'ITOaee cucmeMbl CUA, UMetoUipe oi)urtaK06bte ZAU6flble 6eKmOpbl U Zfla6flbte MOMeflmbl omrtocume.llbflO oartozo U mozo alee t{ertmpa, 3K6U6UAeflmflbt
CHJI).
0TMeTHM ew.e, tl:TO 3Hal!eHHe R OT Bbr6opa ueHTpa 0, oqesH,u:Ho, He 3aBHCHT. 3Hal!eHHe Me M 0 npn H3MeHeHHH noJIOJKeHIHI ueHTpa 0 MOMeT
B o6I.UeM CJiyllae H3MeHllThCll BCJie)J.CTBHe ll3MeHeHHll 3HaqeHHH MOMeHTOB OT)J.eJibHhiX CHJI. DmnOMY BCer)J.a He06XO)J.HMO yKa3biB3Tb, OTHOCH-
TeJibHO KaKOrO ueHTpa onpe,u:eJIHeTCll rJiaBHhiH MOMeHT.
PaccMoTpHM B 3aKJIIOlleHne ,u:Ba qacrHhiX cJiyqa5c 1) ecJIH )J.JIH ,u:aHHOO cncTeMbi CHJI R =0, a M 0 :¥=0, TO oHa npHBO)J.HTCH K o,u:Hofi nape CHJI c MoMeHTOM M 0 . B 3TOM cJiyqae 3HalleHne M 0 He 3aBHCHT OT Bbl6opa ueHTpa 0, TaK KaK HHaqe llOJIYllllJIOCb Obi, 'ITOO)J.Ha II TaMe CHCTeMa CHJI 3aMeHlleTCll pa3HbiMH, He 3KBHBaJieHTHbiMH ,u:pyr ,u:pyry napaMH, liTO HeB03MOMHo; 2) ecJIH )J.JIH ,u:aHHOii cHcTeMhr CHJI R=:foO, a M 0 =0, TO oHa npnBO)J.HTCll K o,u:Hofi CHJie, T. e. K paBHo,u:eiicTBYIOI.Ueii, paBHOH R H npHJIOMeHHOH B IJ,eHTpe 0.
§ 13. YCJIOBH51 PABHOBECH51 CHCTEMbl CHJ1.
TEOPEMA 0 MOMEHTE PABHOJ(EACTBYIOII(Efl
DOiwMeM, liTO aml paertoeeCUfl Jlto6or1 cucmeMbl CUA rteo6xooUMO u aocmamot{HO, t{m06bt z.llaertbtrl eeKmop amor1 cucmeMbt cuA u ee ZAU6Hbtr1 MOMeHm omrtOCUme/lbflO Ato6ozo l{ertmpa 6btAU pa6Hbl HYAIO, T. e. liTOObl BbinOJIHHJIHCb ycJIOBHll
R=O, M 0 =0, |
(23) |
r,u:e 0 - JII06oii ueHTp, TaK KaK B KoHue § 12 noKa3aHo, liTO npH R=0
31raqeHHe M 0 OT Bbi6opa ueHTpa 0 He 3aBHCHT.
YCJIOB!ill (23) llBJlHIOTCH He06XO)J.HMhlMH, TaK KaK eCJill KaKoe-HH- oy)J.b H3 HHX He BblllO.lJ:HHeTCll, TO CHCTeMa ,u:eiiCTByiOW.HX Ha TeJIO CHJI
npHBO)J.HTCll HJUi K paBHo,u:eiicTBYIOI.Uefi (Kor,u:a R=:foO), HJIH K nape CHJI
H, cJie)J.oBaTeJihHo, He HBJIHeTCH ypaBHOBeiiieHHOH. uospeMeHHO ycJIOBHH (23) HBJIHIOTCll ll ,U:OCTaTOliHbiMH, nOTOMY liTO npH
R =0 cncTeMa CHJI MOMeT np HBO)J.HTbCH TOJibKO K nape c MOMeHTOM M0 ,
a TaK KaK M 0 =0, To HMeeT MecTo paBHOBecne. |
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DoJI&3yHch noJiylleHHbiM pe3yJihTaTOM, ,u:oKaMeM cJie,u:yiOI.UYIO |
T eo- |
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p e M y |
BapHHbOHa * o M o M e H T e p a B H o ,11. e il c T B y IO- |
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w. e il: |
ec.11u aartrtafl cucmeMa cuA UMeem paertoaerlcmeytoU{yto, mo MoMertm |
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paertoiJer1cmeytoU{er1 omrtocume.llbrto .11106ozo t{ertmpa 0 pasert |
cyMMe |
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MoMertmoe cu.11 cucmeMbt omrtocume.llbrto mo<?-o a~ee t{ertmpa. |
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* n. BapHHhOH (1654-1722)- Bbl,ll;310UlllllCH ¢paHUY3CKHH yqeHb!H, MaTeMaTHK H MeXaHHK. 113JIO)KHJI OCHOBhl CTaTHKH B KHHfe «npoeKT HOBOH MeXaHHKH» (1687).
40
nycTb CHCTeMa CllJI Ft. ~. 0 |
•• , Fn npHBO,ll.llTCH K paBHO,ll.ellCTBYIO- |
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Ill,ell R, JIHHHH |
,ll.ellCTBHH |
KoTopoll npoxo,li.HT qepe3 HeKoTopyiO TO'IKY |
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C (pHc. 41). llpHJIO}KIIM |
B 3TOll |
ToqKe CHJIY R' =-R. Tor)l.a cucTeMa |
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CUJI Ft. F 2 , ••• , |
Fn, |
R' |
6y,ll.eT |
HaXO,li.UTbCSI |
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B paBHOBeCHll |
H |
)l.JISI |
Hee |
,li.OJI}KHO |
BblllOJI- |
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HSITbCSI ycJIOBHe |
A1 0 =0, T. e. COrJiaCHO |
¢op- |
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MyJie (22) ,li.JISI ,ll.aHHblX CUJI (BKJIIOqaSI |
CHJIY |
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R') ,li.OJI}KHO 6biTh 'i.m 0 |
(Fk)+mo (R')=O. |
Ho -, |
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CHJihl |
R |
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T~K KaK R' =-R u o6e |
HanpaBJieHbl |
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B,li.OJib O,li.HOll U TOll }Ke npSIMOll, |
TO m0 (R')= |
!J |
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=-m0 (R). |
Oo,ll.cTaBJISISI |
3TO |
3Haqeaue |
X |
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m0 (R')B npe)l.bi,ll.ylll,ee |
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paBeHcTBO, |
uaii)l.eM |
Pnc. 41 |
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H3 uero, 'ITO
(24)
TeM caMbiM TeopeMa ,li.OKa3aHa. EIO qacTO 6biBaeT y)l.o6uo noJib3QBaTbCSI npu BbliJ:UCJieHUU MOMeHTOB CUJI.
§ 16. PABHOBECHE OJJOCI(Ofl CHCTEMbl CHJJ.
CJIYLJAfl nAPAJJJJEJJbHbiX CHJJ
Heo6xo,ll.I1Mhie 11 .rr.ocTaToqHbie ycJIOBI1H pasHoBeCHH mo6oii CI1CTeMbl
CI1JI |
.rr.aiOTCH paBeHCTBaMH R=0 11 M 0 =0, BbrpamaeMbiMI1 ¢opMyJiaMI1 |
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(23). |
Hai'I.rr.eM BhiTeKarorn.l1e OTCJO.ll.a |
aHaJII1T11LieCKI1e |
ycJioBHH paBHOBe- |
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CHH |
ITJIOCKOH Cl1CTeMhl |
CI1JI. l1x MOJKHO noJiy'IHTb B |
TpeX pa3JIH'IHhlX |
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<}lopMax. |
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l. 0 C H 0 B H a 5I |
<tJ 0 p M a |
y C JI 0 B 11 H |
p |
a B H 0 B e C II H. |
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TaK KaK BeKTOp R paseH HYJIIO, Kor.n.a paBHbl HyJiro ero npoeKU1111 R x
H R y, TO ,!l,JIH |
paBHOBeCHH .rr.omKHhl BbiiTOJIHHTbCH paseHcTBa |
Rx =0, |
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Ry=O H M 0 =0, r.rr.e B .rr.aHHOM cJiyqae M 0 - |
aJire6paHqecKl1H MOMeHT, |
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a 0 - Jiro6aH |
TO'!Ka B nJiocKOCTII .rr.eil:cTBHH |
CHJI. Ho H3 ¢oPMYJI (27) |
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cJie.rr.yeT, qyo rrpe.rr.hl.ll.YIIJ.He paser-rctBa 6y.n.yT BhlrroJir-reHbl, Kor.n.a |
.rr.eil:cT- |
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syrorn.He CI1Jihl |
y.rr.osJieTBop HIOt ycJIOBHHM: |
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'2.Fkx---: 0, 2:.Fky = 0, '2:.m 0 (Pk) = 0. |
(29) |
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<PopMyJihi (29) BblpamaiOT cJie.rr.yrorn.vie ar-raJIHTHIJ:eCKHe ycJioBHH pasHoBeCI1H: Ollfl pa6fto8eCUH npoU380!!1JfWU nJlOCIWU CUCtneMbl CUll Neo6xoOUMO u oocmamo£tno, £ttno6bt cy.MMbt npoeKl{uii ecex CU/l na KaJICOYIO us iJeyx KOopounatnHbtx oceii u cyMMa ux MOMenmoe omnocumellbHO lll06oco u,enmpa, .!leJ!Catu,eeo 6 MOCKOCtnU oeiicmeUfl CUll, 6bt.!lU paeHbt HY.!liO. O.rr.HoBpeMeHHO paBeHCTBa (29) BbipaJKaiOT ycJIOBl15I paBHOBeCHH TBep.rr.oro
TeJia, HaXO,!l,HIUerOCH ITO,ll. ,ll.eHCTBI1eM ITJIOCKOH CI1CTeMbl Cl1JI.
<P o p M a y c JI o B 11 i'I p a B H o B e c 11 H: d/lfl paenoaecuH npou3eollbHor1 nllocKou cucmeMbt cull Heo6xoouMo u oocmamo£t- no, £ttno6bt cyMMbt MOMelimoe ecex :7tnu.x CUll omnocumellbHO KaKuX-Hu- 6yob oeyx l{eHtnpoe A u B u cyMMa ux npoeKl{Uii Ha ocb Ox, ne nepneHiJuKyllflPNY/0 npHMoii AB, 6btllu paeHbt HYil/0.'
1:.mA(F J=0, I.mn(l~k)=O, '2.Fkx=O. |
(30) |
1 |
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