(формат)Тарг

37, 6).

§ 11,

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).

39

0,11.-

(Kor,u:a M 0 =:fo0)

(ycJIOBHll 3KBHBaJieHTHOCTll CHCTeM

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.
DoJI&3yHch noJiylleHHbiM pe3yJihTaTOM, ,u:oKaMeM cJie,u:yiOI.UYIO		T eo-
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-
w. e il:	ec.11u aartrtafl cucmeMa cuA UMeem paertoaerlcmeytoU{yto, mo MoMertm
paertoiJer1cmeytoU{er1 omrtocume.llbrto .11106ozo t{ertmpa 0 pasert		cyMMe
MoMertmoe cu.11 cucmeMbt omrtocume.llbrto mo<?-o a~ee t{ertmpa.

* 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-
Ill,ell R, JIHHHH	,ll.ellCTBHH			KoTopoll npoxo,li.HT qepe3 HeKoTopyiO TO'IKY
C (pHc. 41). llpHJIO}KIIM				B 3TOll		ToqKe CHJIY R' =-R. Tor)l.a cucTeMa
CUJI Ft. F 2 , ••• ,		Fn,	R'	6y,ll.eT		HaXO,li.UTbCSI			z
B paBHOBeCHll	H	)l.JISI	Hee		,li.OJI}KHO		BblllOJI-
HSITbCSI ycJIOBHe		A1 0 =0, T. e. COrJiaCHO						¢op-
MyJie (22) ,li.JISI ,ll.aHHblX CUJI (BKJIIOqaSI								CHJIY
R') ,li.OJI}KHO 6biTh 'i.m 0			(Fk)+mo (R')=O.					Ho -,
-	-			CHJihl				R
T~K KaK R' =-R u o6e						HanpaBJieHbl
B,li.OJib O,li.HOll U TOll }Ke npSIMOll,						TO m0 (R')=			!J
=-m0 (R).	Oo,ll.cTaBJISISI				3TO		3Haqeaue		X
m0 (R')B npe)l.bi,ll.ylll,ee
				paBeHcTBO,			uaii)l.eM		Pnc. 41

H3 uero, 'ITO

(24)

TeM caMbiM TeopeMa ,li.OKa3aHa. EIO qacTO 6biBaeT y)l.o6uo noJib3QBaTbCSI npu BbliJ:UCJieHUU MOMeHTOB CUJI.

2*. B T o p a H

§ 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
(23).	Hai'I.rr.eM BhiTeKarorn.l1e OTCJO.ll.a		aHaJII1T11LieCKI1e	ycJioBHH paBHOBe-
CHH	ITJIOCKOH Cl1CTeMhl	CI1JI. l1x MOJKHO noJiy'IHTb B			TpeX pa3JIH'IHhlX
<}lopMax.
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.

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,
Ry=O H M 0 =0, r.rr.e B .rr.aHHOM cJiyqae M 0 -		aJire6paHqecKl1H MOMeHT,
a 0 - Jiro6aH	TO'!Ka B nJiocKOCTII .rr.eil:cTBHH	CHJI. Ho H3 ¢oPMYJI (27)
cJie.rr.yeT, qyo rrpe.rr.hl.ll.YIIJ.He paser-rctBa 6y.n.yT BhlrroJir-reHbl, Kor.n.a			.rr.eil:cT-
syrorn.He CI1Jihl	y.rr.osJieTBop HIOt ycJIOBHHM:
	'2.Fkx---: 0, 2:.Fky = 0, '2:.m 0 (Pk) = 0.		(29)

<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

46

52).

M MB=O.

MA =0

Heo6xoJ(HMOCTh 3THX ycJioBHii oqeBH)l.Ha, TaK KaK ecmt mo6oe H3 HIIX He BbinOJIHHeTCH, TO IIJIII }~=j=-0, IIJIII MA=i=-0 (MB=/=-0) H paBHOBeCHH

He 6y)l.eT . .Ll.oKaJKeM HX J(OCTaToqnocTb. EcJIH J(JIH )l.aHnoii cHcTeMbi CHJI BbinOJIHHIOTCH TOJibKO rrepBhre )l.Ba H3 ycJIOBHii (30), TO )l.JIH Hee

TaKaH CHCTeMa C11JI, corJiacHo pe3yJibTaTaM, rroJiyqeHHbiM B

§ 15, MOJKeT He HaXO)l.11TbCH B paBHOBeC11H, a HMeTb paBHO)l.eHCTBYIOlll.YIO

R, O)l.HOBpeMeHHO npoxo)l.Hlll.YIO qepe3 ToqKH A H B * (p11c. Ho no TpeTbeMy ycJIOBHIO J(OJIJKHO 6hiTh Rx=~Fkx=O. TaK KaK OCb Ox

rrpoBe)l.eHa He nepneH)l.HKYJIHpHo K AB, TO rrocJie)l.Hee ycJioBHe MOJKeT 6biTh BhinOJIHeHo, TOJibKO KOr)l.a R=O, T. e. KOr)l.a 11MeeT MeCTO paBHo-

secHe.	<P o p M a
3*. T p e T h H		y c JI o B H :ii	p a B H o B e c 11 H

(ypG8HeHUJl mpex .MO.MeHmoo): OllJl pG8H08eCUJl npOU380ilbHOU nllOCKoii cucme,wbt cull neo6xoou.Mo u oocmamottno, ttmo-

6bl cy.M.Mbt .MoMenmoo ecex amux CUll omnocumeAbHO lll06btX mpex qenmpoo A, B H C, He lleJICawpx na oonou npfl.M,OU, 6btllU pG8Hbt nylllo:

~mA(Fk)=O, ~mB (Fk) = 0, ~me (Fk) = 0. (31)

Heo6xo)l.HMOCTb 3THX ycJIOBHH, KaK H B rrpe-
J(bi)l.yUJ.eM	cJiyqae, oqeBH)l.Ha • .Ll.ocTaToqHOCTb	Pnc. 52
ycJIOB11H	(31) CJie)l.yeT 113 TOfO, qTo eCJIII rrpH

O)l.HOBpeMeHHOM BhlllOJIHeHHII 3T11X ycJIOBIIH )l.aHHaH CIICTeMa CIIJI He HaXO)l.HJiaCb 6hi B paBHOBeC11H, TO OHa )l.OJIJKHa 6biJia 6hi rrp11BO)l.IITbCH K

paBHO)l.e:iicTByrom.e:ii, O)l.HoBpeMeHHO rrpoxo)l.HIIJ.eii qepe3 ToqKH	A, B H
C, qTo HeB03MOJKHO, TaK KaK 3TH ToqKH He JieJKaT Ha O)l.Hoii	rrpHMoii.

CJie)l.OBaTeJibHO, npH BbiiTOJIHeH11H YCJIOBHH (31) HMeeT MeCTO paBHOBeC11e.

Bo BCeX paCCMOTpeHHbiX CJiyqaHX ,H;JIH ITJIOCKOii CHCTeMbl CIIJI nc<lJiyqaiOTCH TpH ycJIOBHH paBHOBeCHH. Y CJIOBIIH (29) cqHTaiOTCH OCHOBHMMH, TaK KaK npH IIOJib30BaHHH IIMII H11KaKHX orpannqeHHH Ha Bhl60p KOOp-

.l{lfHaTHblX OCeH H l{eHTpa MOMeHTOB He HaJiaraeTCH.

EcJIH na TeJIO HapH.D.Y c nJiocxoii: cncTeMoii CHJI F1, F 2, ••• , Fn .n.eiicTsyeT cncreMa JielKaJil,HX B TOM JKe llJIOCKOCTH nap C MOMeHTaMH m 1, m 2, ... , mn, TO npH COCTaBJleHHII ycJiosnii pasHosecnH B ypasHeHHH npoeKu,nii napbl ne soii.n.yT, TaK KaK cyMMa npoeKunil: CHJI napbl Ha JII06yiO OCb pasHa HYJIIO. 8 ypaBHeHHl!X JKe MOMeHTOB K MOMeHTaM CHJI aJire6paH'iCCKH npn6aB5!TC5! MOMeHTbl nap, TaK KaK CYMMa MOMCHTOB CHJI napbl OTHOCHTeJibHO JII060f0 U,eHTpa p.aBHa MOMeHTY napbl [§ 9, cpopMyJia (15)]. TaKHM o6- pa30M, HanpHMep, ycJIOBHlil paBHOBeCI15! (29) npH i{eHCTBHH Ha TCJIO CHCTeMbl CHJI H nap npHMYT BHJ{:

~Fkx=O, ~Fky=O, ~mo (Fk) +~mi=O.

(32)

AnaJiornqno npeo6paayroTCH s 9TOM cJiyqae yCJioBHH (30) n (31),

PaBHoBecHe rrJIOCKofl cHcTeMbi napaJIJieJib- H hi X C II JI. B CJiyqae, KOr)l.a BCe )l.eHCTBYIOIIJ.He Ha TeJio CHJ!bl rrapaJIJieJihHbl )l.pyr )l.pyry, MOJKHo HarrpaB11Tb ocb Ox rreprreHJ(HKy.rmpno CH-

* npn MA=O CHCTeMa CHJI MOJKeT llMeTb paBHO~HCTBYIOlUYIO, npoXO)l;lllUYIO qe- 1'e3'lO'iKYA, a npn MB=0- npoxo;u.HlUYIO >repe3 TO'iKYB.

47

§30. PABHOBECHE fiPOH3BOnbHOA fiPOCTPAHCTBEHHOA CHCTEMbl CHn. tnYl.JAA fiAPAnJIEJibHbiX CHJI

Heooxo)I,I1Mhre 11 ,!I,OCTaTO'lHhieyc.JIOBI1H paBHosecHH .JII06oii crrcTeMbl

CHJI	Bbrpa)KaiOTCH	paseHCTBaMI1 R=O, M 0 =0 (eM. § 13). Ho seKTOphl
R 11	M0 paBHhl	Hy.JI!o TOJibKO TOr)I,a, KOr)I,a Rx=Ry=Rz=O 11 Mx=

=My=Mz=O, T. e. KOr,!l,a )I,eikTByiOIU!fe CI1Jlhl, COr.JiaCHO cpopMy.JiaM

(49) 11 (50), oy,!l,yT y,!I,OBJleTBOpHTb YCJIOBIIHM:

~FI<x=O, ~ ~ky=O, ~~kz=O;			}
-	-	-	(51)
~ mx (Fk) = 0,	~my (Fk) = 0,	~mz (Fk) = 0.

TaK!IM oopa30M, 0/lJl pa8H08eCUJl npou380/lbHOii npocmpaHcmeeHHOii cucmeMbt cull Heo6xoouMo u oocmamoliHo, qmo6bt cyMMbt npoeKI{Uii ecex

79

CU.~ Ha Ka:JICOYIO U3 mpex KOOpOUHamHblX oceii U cy.MMbl UX MOMeHmOB omHocume!lbHO amux oceil 6bt/lU paeHbt HY!liO *.

PaseHCTsa (51) BbrpmKaiOT O,!I,HospeMeHHO yc.JioBHH paBHOBeCIIH Tsep- ,!I,Oro Te.Jia, HaXO)J,RIIJ.erocH no)I, )I,eiicTBIIeM mo6otr npocTpancTBemrofr

CHCTeMbl CIIJI.

Ec.JIII Ha TeJIO KpoMe cii.JI )I,ei\crsyeT erng napa, 3a)I,aHHaH ee MOMeH-

TOM m, TO npu 3TOM BIJ)I, nepBbiX Tpex H3 YCJIOBHH (51) He li3MeHHTC5I (cyMMa npoeKuuii: CHJI rrapbi na .JII06y!O och pasna HYJI!O), a noc.Jie)J,l!He

TPH yc.JIOBHH rrpHMYT BU)I,:

~mx (Fk) +mx= 0, ~my (Pk) +my= 0, ~ mz (Fk)+mz=O. (52)

Случай параллельных сил

В качестве произвольной пространственной системы сил рассмотрим си-

стему параллельных сил.

В случае, когда все действующие на тело силы параллельны друг другу и одной из координатных осей (например – оси Z), то условие равновесия, в частности, будет иметь вид: ΣFkz = 0; Σmy (Fk) = 0; Σmx (Fk) = 0.

Система имеет равнодействующую, приложенную в т. С – центр II-ных сил.

Найдем координаты этого центра.

Пусть в произвольной системе координат в точках с координатами A1 (X1, Y1, Z1), A2 (X2, Y2, Z2), … , C (Xc, Yc, Zc) приложены силы F1 , F2 , … , Fk , …, Fn и

R .

Произвольная система параллельных сил

Применив к ним теорему Вариньона, запишем my ( R ) = Σmy ( Fk );

но my ( R ) = R ·Xc; my ( F1 ) = F1 ·X1, my ( F2 ) = F2 ·X2 , … , my ( Fk ) = Fk ·Xk;

тогда:

RXc = F1X1 + F2 X2 + … + Fn Xn = Σ Fk Xk Xc = R1 Fk X k ;

Аналогично:

RYc = F1Y1 + F2 Y2 + … + Fn Yn = Σ Fk Yk Yc = R1 Fk Yk ; RZc = F1Z1 + F2 Z2 + … + Fn Zn = Σ Fk Zk Zc = R1 Fk Z k .

На этих формулах основано определение координат ц.т. тела если принять приложенные к телу параллельные силы за силы тяжести частей тела – как част-

ный случай произвольной пространственной системы II сил, заменив систему сил

Fk на силы pk .

Равнодействующая сил тяжести частиц тела p1 , p2 , … pn равна P . Модуль этой силы – вес тела, т.е. Rp = P = Σpk. Точка приложения силы P (равнодейству-

ющая сил pk ), точка Ср – центр параллельных сил тяжести частей тела, или ц.т.

тела.

Система параллельных сил тяжести

								n
	Поэтому P · Xc = p1X1 + p2X2 + … pnXn = pk X k , т.е. координаты ц.т. тела
								k 1
	определяются по формулам:
		1	n	1	n		1	n
	Xc =		pk X k , Yc =		pk Yk ,	Zc =		pk Z k
		P		P			P
			1		1			1

96).

C Jl y q a :H n a p a JI JI e JI h H bi x c rr JI. B c.Jiyqae, KOr,!l,a Bee

)I,ei.icTBYIOIUHe Ha Te.JIO CH.Jlbl napa.JI.JieJibHbl ,!l,pyr ,!l,pyry, MOJKHO Bhl6parh KOOp,!I,HHaTHbie OCII TaK, qTO OCb Z 6y,!l,eT napa.JI.JieJibHa CHJiaM (puc. Tor}l,a npoeKUHll Ka)K)I,Ollll3 CHJI Ha OCH X ll y H HX MOMeHTbl OTHOCHTeJib-

	z	HO OCH Z 6y,!l,yT paBHbl Hy.'IIOH CHCTeMa		(51)
		)I,aCT Tpli yc.JIOBHH paBHOBeCUH:
		~Fkz=O, ~mx(Fk)=O, ~my(Fk)=O. (53)
	!J	0cTa.1JbHbie paBeHCTBa	OOpaTHTCH	npH
		3TOM B TOM)I,eCTBa BH)I,a 0=0.
		C.Jie,li.OBaTeJihHO, O/lJl paeHoeecuJL npocmpa.'-1.-
		cmeeHHoil cucmeMbt napa!l/leAbHbtX cull Heo6-
		xoouMo U OOCmamQiiHO, lim06bt cyMMa npoeK-
	PHC. 96	[\UU ecex CU/l Ha OCb, napa!l/le/lbHYIO CU!la.M, U
		CYMMbl UX MOMeHmOB OmHocume/lbHO OBYX ()py-
	p e ill e H H e 3 a }1, a	eUX KOopouHamHblX oceu 6bt/lU paeHbl HY/l/0.
		'I. DopH,ll;OK pemeHH5! 3a)I,aq	3)I,eCb OCTaeTCR

TeM iKe, 'ITOu s c.Jiyqae IIJIOCKoil CHCTC:\lhi'CHJI.YcTaHOBHB, paBHOBecue KaKoro TeJia (o6beKTa) paccMaTpusaeTcH, Ha)I,o u3o6pa3HTh Bee )I,eikrsyiOill.He Ha Hero BHeUIHHe CIIJlbl (H 3a,ll.aHHbie, H peaKUHH CBH3e:fi) li COCTaBHTb YCJIOBHH paBHOBeCHH 3THX CHJI. l13 IIO.JiyqeHHbiX ypaBHeHHll li onpe,ll.eJI.HIOTCH HCKOMhie BeJIH'IHHbi.

,U.JIH no.JiyqenrrH 6o.Jiee rrpocTbiX cucTeM ypaBHeHuil peKoMeH,!I,yeTcH OCli npOBO)I,HTb TaK, qTOObl OHH rrepeceKaJIH OOJlbllle HeH3BeCTHbiX CHJI HJIH 6hl.JIH HM nepneHJI.HKY.JIRpHhl (ecJirr 3TO TOJihKO H3.Jiumne He yc.Jio?K- HHeT Bb!llliCJieHHH rrpoeKUHH H MOMeHTOB JI.pyrux CHJI).

HoshlM 3JieMeHTOM B cocraBJieHHH ypasHeHu:H RBJIHeTCH Bhl'lHCJieHlie MOMeHTOB CHJI OTHOCHTeJibHO KOOpJI.HHaTHbiX OCeH.

B CJiyqaHX, KOrJI.a us oollJ,ero qepTeiKa TPYJI.HO ycMoTpeTb, 'IeMypaBeH MOMeHT )I,aHHOll CHJibi OTHOCHTeJibHO KaKOll-HHOYJI.b OCH, peKOMCH- ,!I,yeTCH H300pa3HTb Ha BCllOMOraTeJibHOM qepTeiKe IIpOeKUHIO paCCMaTpHBaeMOfO Te.Jia (BMeCTe C CHJIOH) Ha II.'IOCKOCTb, rreprreH)I,llKYJIHpHy!O

.9TOH OCH.

* Ilpa COCTaBJICHHK ycJIOBHH (51) MOJKHO, CCJIH 3TO ll.CJICC006pa3HO, 6paTb )!Jill Bbi'IHCJICHHHnpoCKUHH O,ZJ.HY CHCTeMy KOOp,!!_HHaTHb!X Ocei1, a Jf.JIH Bbl'IHCJICHHHMOMCII·

TOB - .ll.p yryJO.

so