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CTATHKA TBEPAOfO TEJIA

rnasa I

OCHOBHbiE nOHSITIUI H HCXO)J,HbiE nOJlO)I(EHIHI

CTATHKH

§ I. A6COJUOTHO TBEP,ll,OE TEJIO; CHJIA. 3A,lJ,Al.fH CTATHl(H

Cmamwwt1 Ha3hiBaeTcH pa3,n:eJI MexaHnKII, B KOTopo.\1 II3J1araeTCH o6mee yiieHHe 0 CllJ!aX II H3yqa!OTCH ycJIOBllH paBIIOBeCHH MaTepiiaJibHbiX TeJI, HaXO,ll,HUJ.llXeH IIO,ll, ,n:encTBHeM CHJI.

f10,ll, paBHOBeCHeM 6y,n:eM IIOHHMaTb eOCTOHHlle IIOKOH Te.'laIIO OTHOllieHH!O K ,n:pyrrrM TeJiaM, HarrpiiMep ITO OTHOllieHHIO K 3eMJie. YcJIOBHH paBHOBeeHH TeJia cyrn.eCTBeHHO 3aBHeHT OT TOrO, HBJIHeTCH Jill 3TO TeJIO TBep,ll,biM, JKH,ll,KHM HJIII ra3oo6pa3HbiM. PaBHOBeciie JKH,li,KHX H ra3oo6pa3HbiX TeJI n3yqaeTcH B Kypea,x rn,n:pocTaTHKH HJIH a3pOCTaTHKH. B o6-

meM Kypee MeXaHHKH paceMaTpHBaiOTCH 06b!IJHO TOJ!bKO 3a,n:aqrr 0 paBHOBeCHH TBep,ll,biX Te.'l.Твердые-условно. Поскольку имеют некоторую конечную тв-ть.

Bee BeTpeiiaiOUJ.HeCH B rrprrpo.:I,e TBep,ll,bie TeJia IIO,ll, BJ!H51HlleM BHeillHllX B03,n:eikTBHH B TOll HJ!H HHOH Mepe ll3MeHHIOT CBOIO cpopMy (,n:ecpopMrrpyiOTeH). BeJIHIIHHhi 3THX .n:ecpopMau.rrii 3aBHC5IT OT MaTepHaJia TeJI, HX reoMeTpHIIeCKOH cpopMbl II pa3MepOB ll OT ,ll,eHeTBYIOUJ.llX Harpy3oK. )lJIH o6eerreiieHHH npOIIHOeTH pa3JIHqHbiX IIHJKeHepHbiX eoopyJKeHHH II KOHCTPYKU.HH MaTepnaJI H pa3Mepbi IIX IIacTeii rro,n:onpaiOT TaK, IITo6hi ,ll,ecpOpMaU.HH IIpll ,ll,eiicTBYIOUJ.HX Harpy3KaX 6b!Jlll ,li,OCTaTOIIHO MaJ!bl *.

BeJie,li,CTBne 3Toro rrpn n3yiieHrrrr ycJionuH: paBHOBeCHfl BIIOJIHe .n:orryeTHMO rrpeHe6peraTb MaJib!MH ,AecpopMaU.HHMll COOTBeTCTBYIOUJ.HX TBep,ll,biX TeJI ll paeeMaTpHBaTh llX KaK He,ll,ecpOpMHpyeMbie llJ!ll a6eOJIIOTHO TBep- ,l(bie. A6co;uomrw tn8epabtM me.lloM Ha3biBaiOT TaKoe TeJio, paccTOflHHe MeJK,n:y KaJK,ll,h!Mll ,li,BYM51 TOqKaMH KOTOporo Bcer,l(a OeTaeTCH noeT05IHHbiM. B .n:aJibHeiiweM IIf>H peweHHH 3a.n:aq CTanmn nee Te.'lapaeeMaTpHBaiOTCfl KaK a6COJI!OTHO TBep,ll,bie, XOTH qaeTO ,li,J!H KpaTKOCTH llX Ha3bl· BaiOT IIpOCTO TBep,ll,biMH Te.'laMll.

* HanpRMep, MaTepRaJI II pa3Mepb1 CTepJKHeii, BXO,li.Silll.RX B Te 11.rm I!Hhle KOHCT• PYKU.RR, BhlonpaJOT TaKRMR, 'ITOnp11 .n.eiiCTBYIO!ll,HX Harpy3Kax creplKHR Yil.,1HHHIOTCH

(HJIII yKopa'IHBaJOTC5!)MeHee '!eMHa Oi!.HY Th!CWIHYIO )l.OJIIO RX nepBOHa'!a.1bHOil,1(,1HHbl. TaKoB JKe rropsr.n.oK .n.orrycKaeMhlX Ae¢opMaU.H~ op11 namoe, Kpy'leHI!!!11 T. rr.

9

CocTOHHIIe paBHoBeCHH HJIH .n;srnKeHHH .n;aHHoro TeJia sasiiCHT OT xa- pmnepa erO MeXaHII'leCKHX B3aHMO,neikTBllll C ,npyrHMII TeJiaMII, T. e. OT TeX ,!l.aBJieHHll, IIpllT5VKeHIIlliiJ!II OTTaJIKHBaHHH, KOTOpble TeJIO IICIIbi-

TblBaeT B pesyJihTaTe .9THX ssaHMOAeiicTBHH. BeJIH'lHHa, HBJIHIOII.J.ancH: OCHOBHOH Mepoii MeXaHII'leCKOfO B3aiiMO)J;elicTBHH MayepnaJibWhlX ten, Ha3biBaeTCH B MeXaHHKe CU./lOU.

PaccMaTpiisaeMble B MexaHHKe BeJIH'lHHbiMOJKHO pa3,'1.MHTbua CKtl-

.tlJlpHble, T. e. TaKHe, KOTOpbre IIOJIHOCTblO xapaKTepnsyiOTCH HX 'liiCJIO-

BbL'>i3Ha'leHIIeM,ll BeKmoprtbte, T. e. TaKHe, KOTOpble f!OMHMO tiiiCJ!OBOro SHaiieHH5I XapaKTepn3yiOTCH ~e II HarrpaBJieHHeM B rrpoCTpaHCTBe.

e,n,HHHUY. Ocuonuoii e,n,nnnueii IISMepeHHH cHJibi B MeJK,'I.yHapO,'I.HOHciic-

CHJibi CJiy.>KaT nssecTHbre II3 cpH3HKH rrpn6opbr, HaSbiBaeMble aurtruw-ttem- paMu.

CHJiy, KaK II see ,'l.pyrrreBeKTOpHbie BeJIH'liiHbl, 6y,n,evr o6osHatraTb

6yKBOii C treptoft Ha,'l.HeiO (HaiipHMep, f), a MO,ll,YJ!b CHJlblCHMBOJIOM

jfl llJIH TOll JKe 6yKBOH, HO 6es 'lepTblHa,'l.HeiO (f). fpacpnqeCKll CHJia,

KaK n .l(pyrne seKTOpbi, nso6pa.lKaeTcH uarrpaBJieHHbiM otpe3KOM (pHC. 1). L{JIJI:Ha Enoro 0Tpe3Ka Bb!paJKaeT B Bbi6paHHOM MaCWTa6e MO·

.ll.YJ!b CllJ!bl, narrpaBJieHHe oTpe3Ka COOTBeTCTByeT HanpaBJieHHIO CHJibl, TO'!KaA Ha puc. 1 HBJIHeTCH TO'!Koft rrpn.'loJKeHHHCH.1bi (CHJIY MO)I{HO

3TH JillHIIII ,neftCTBllH He JieJKaT B O,nHOll fiJ!OCKOCTH,- npocmpartCmBeltHOU. KpoMe Toro, CHJibi, JIHHHH ,'l.eiicTBHHKOTOpbiX rrepeceKaiOTCH B Ok non TOtrKe, Ha3biBaiOTCH cxoaHU{UMUCfl, a CHJibi, JIHHHII .n;eiicTBHH KoTopbiX rrapaJIJieJihHbi .n;pyr .l(pyry,- napa.!l.!le.ilbHbtMU.

2. Te.11o, KOTopoMy H3 ,'l.aHHoronoJioJKeHHH MOJKHo coo6IUHTb JII06oe nepeMeiUeHne B npocTpaHcTBe, nasbmaeTcH ceo6oaHbt.M.

3.ECJIH O,'I.HY CHCTeMy CHJI, ,'l.eHCTByiOUJ,HX Ha CB060,'I.HOe TBep,'l.oe TeJio, MO.>KHO 3aMeHHTb ,'l.pyrOHCHCT€MOH, He H3MeH5I5I IIpH 3TOM COCTOHHll5! IIOKOH IIJIII ,'J.BHJKeHHH, B KOTOpOM HaXO,'I.llTCH TeJio, TO TaKHe )J;Be CHCTeMbl CIIJ! Ha3biBaiOTCH 8K6U6a.!lertmHbtMU.

4.CncTeMa cnJI, rro,'l. .n;eucTBIIeM Kotopo.H cso6o.u,noe TBep.n.oe teJio

10

MO:J{Cem HaXO,l(HThCH B fiOKoe, H83biBaeTCH ypa81W8etueftHOU HJIH fJKfJU~

tK!Ileftmftou HYA/0.

5. ECJIH }l.aHHaH CHCTeMa CHJI 3KBHBaJieHTHa O)l;HOH CHJie, TO ;na CHJia Ha3biBaeTcH paeHooet1cmey10wpl .n;armoil cncTeMhi cnJI.

CnJia, paBHaH paBHo.n.eH:cTBYIOill.eii no Mo.n;yJIIO, npHMO npoTnBonoJJO)KHaH eM: no HarrpasJieHHIO H .n;eficTByiOll.l,aH B)l;OJib ToM: *e npHMofi, Ha-

SblBaeTcH ypG13HOBewuea10wpl CIIJIOH.

6. CnJibi, .n;eii:CTBYIOill.IIe Ha .n;aHHoe TeJio (HJIII cncTeMy TeJI), MO*HO pa3)l;eJIHTb Ha BHeUIHHe II BHyTpeHHHe. 8HeUJHUMU Ha3biBa!OTCH CHJibi, KOTOpbie ,n;effCTBYIOT Ha 3TO TeJIO (HJIH Ha TeJia CHCTeMbi) CO CTOpOHbl .n.pyfHX TeJI, a BHympeHHUMU - CHJihl, C KOTOpbiMH qaCTH ,naHHOfO TeJia (HJIH TeJia }l.aHHOH CHCTeMbi) .n;eHCTBYIOT .n;pyr Ha .n.pyra.

7. CHJia, rrpnJiom:eHHaH K TeJiy B KaKoii-Hn6y.n.h o.n.Hoil era TO'lKe, ua3hlsaeTCH cocpeoomotteHFtoii. CIIJihi, .n.eiicTBYIOill.He Ha nee TO'IKII.n;aHHoro 06beMa llJIH i'l,aHHOHqaCTll IIOBepXHOCTH TeJia, Ha3biBaiOTCH pacnpe-

OeAeflflblMU.

Oomnne 0 COcpe.n;OTOtieHiiOH CIIJie HBJIHtTCH yCJIOBHbiM, TaK KaK

npaKTwrecKII IIpnJiom:HTh cnJiy K TeJiy B o.n.Hoil TO'!KeHeJih3H. CIIJihi, KoTophle B MeXaHHKe pacCMaTpHBaiOT KaK COCpei'l,OTO'!eHHbie,IIpei'l,CTaBJIHlOT co6oif ITO Cylll.eCTBY paBHO}I.efiCTBYIOill.He HeKOTOpbiX CHCTeM pacrrpe- )J.eJieHHbiX CliJI.

B qacTHOCTH, paCCMaTpHBaeMaH B MeXaHHKe CHJia TSOKecTII, ,!l.eHCTnyrom;aH Ha .n;aHHoe TBep.n;oe TeJio, npe.n;cTaBJIHtT co6ofi paBHO,l(eficTBYIDill.YIO cii.rr TsuKeCTH, .n;eiicTBYIOill.HX Ha ero traCTHUhl. JliiHHH .n;eficTBHH

STOll paBHO,!I.eHCTBYIOill.efl npoXO)l;HT qepe3 TO'!Ky,Ha3hiBaeMyiO lfeFtmpo.M

*. 2 основные задачи, которые решает статика

3ai'l,a'laMHCTaTHKH HBJISIIOTCH: 1) npeo6pa3oBaHIIe CHCTeM CHJI, }IeHCTBytall.l,llX Ha TBepi'l,OeTeJIO, B CllCTeMbl liM 3KBHBaJieHTHble, B '!aCTHQCTH npHBei'l,eHHell.aHHOH CHCTeMbi CHJI K npoCTeHIIIeMy BII,ny; 2) onpe.n;e- JieHIIe YCJIO~HH paBHOBeCIISI CHCTeM CHJI, i'l,eHCTBYIOIIJ,HX Ha TBep.n.oe TeJIO.

PeiiiaTb 3ai'l,a'!HCTaTHKH MOJKHO HJIH rryTeM COOTBeTCTBYIOIIJ,HX reo-

MeTpHtieCKHX nocTpoeHIIH (reoMeTpHtieCKHH II rpaqm'leCKHH MeTO)l;hi), HJIH c noMOII~biO '!HCJieHHhiXpac'leTOB(aHaJIHTH'IeCKIIHMeTOi'I,).B Kypce

6y,!l.eT r JiaBHbiM: o6pa30M npHMeHSITbCSI aHaJIHTll'leCKHH MeTOi'l,, Oi'l,HaKO CJiei'l,yeTllMeTb B Blli'l.Y,'ITOHarJISii'l,HbiereoMeTpiiiieCKHe fiOCTpOeHIISI llf-

paiOT npn pewemm 3a.n;aq MexaHIIKII qpe3Bhl'laHHosaJKHYIO poJib.

Чтобы это осуществить - используют ряд положений

§ 2. HCXO,!J;HbiE OOJIO)I(EHH51 CTATHI(H ОНИ ОПРЕДЕЛЯЮТСЯ

НЕКОТОРЫМИ ОСНОВНЫМИ АКСИОМАМИ

npu ll3JIOa<eHHH CTaTHKH MOJKHO lli'l,Tlli'l.BYMH nyTHMH: 1) HCXOi'l,HTb 03 ypaBHeliHH, KOTOpbie noJiytiaiOTCSI B i'l,IIHaMHKeKaK CJie}I.CTBHH OCHOBHhiX 3aKOHOB MeXaHHKH (CM. § 120); 2) H3JiaraTh CTaTHKY He3aBHCHMO OT )J.HHaMHKll HCXO,l(H ll3 HeKOTOpbiX 06ll.I,HX 3aKOHOB MexaHHKll II llOJIOJKe-

* Bonpoc 00 onpell,eJieHHI! UeHTpOB TSilKeCTII TeJI oy.n.er p.accMotpeH B rJI. VIII. Ope.n.BaputeJihHO aaMeTHM, 'ITOec.1u o.n.nopoJIHOe TeJio HMeeT ueHTP cHMMeTprrH (rrpl!- uoyroJihffhlfi 6pyc, U!!JII!F!;:(p, map II T, IT.), TO ueHTp TSIJKeCTII TaKOrO TeJia llaXO,ll.IITCS! B ero IJ.eHTpe CHMMeTpiiR.

11

HHH, H33hiBaeMhiX 3KCHOM3MH HJIH npiiHU.Hna;..m CT3THKH, XOT51 nocyIUeCTBY OHH 5IBJI5IIOTC51 He He3aBHCllMhiMll aKCllOM3\1ll, a cJie,n:CTBH51Mif TeX 2Ke OCHOBHhiX 33KOHOB MeXaHHKll (C~I. § 120). B yqe6Hh!X KypcaX, Kar{ n B .n.aHHOM, o6hiiiHO n,n:yT BTopuM rryTeM, TaK KaK no p5I,n:y rrpiitrHH OK33h!BaeTC51 He06XO.IJ:IIMhiM H3llllH3Th H3ytieHHe MeXaHHKll CO CT3TllKH, T. e • .no Taro, KaK 6y,n:eT H3JI02KeHa JI.HHaMHKa. DoJio2KeHH51 (IIJIJI aKciioMhi), H3 KoTophix rrpn 3ToM ncxo,n:HT, M02KHO ccpopMyJinposaTb cJie.n.yiOIUHM o6pa30M.

1. Ec.llu Ha ceo6oaHoe a6CO.!l10mHO meepaoe me.llo aeiicm6ytom aee CU.Ilbl, mo meAo Mo:JICem HaxoaumbCflB paeHo6ecuu moziJa u moAbKO mo2aa, K02da smu cu.llbt paeHbl no .M.oayAto (F1=F 2) u HanpaeAeHbt 600.Ilb oaHou npJlMOtl e npomueonoAO'JICHbte cmopOHbt (pnc. 2).

2. /l.eucmeue aaHHou cucmeMbt cuA Ha a6coAtomHo m6epaoe mMO He U3olteHJwmcR, ecAu K Heu npu6aeumb uAu om Hee omHHmb ypaeHoeemeHnyto cucmeAty cuA.

Hnbi\!H CJIOKI~H 3To 03HatraeT, tiTO .nne cni::Te:Mbi CHJI, OTJIHIImomue-

CR Ha ypaBHOBeUieHHYlO CHCTe"'IY, 3 KBHB3.1eHTHbl .n.pyr ,n:pyry.

C JI e .IJ: C T B H e: aeiicmeue CU./lbl Ha a6C0/110mHO meepiJoe me.llo He U3MeHUmCJl, eC/IU nepeHecmu moliKY npU.IlOJICeHUJl CU.Ilbt 600./lb ee .llUHUU iJetl-

т.е. для абсолютно твер-го тела действующая сила является

Cm6IUl 8 ;uo6yto opyclj10 moliKlj meAa.вектором скользящим

B caM oM ,n:eJie, rrycTh Ha TBep,n:oe TeJio ,n:efiCTnyeT rrpnJio2KeHHa51 B TO'l:- Ke A CHJia F (pnc. 3). Bo3hMeM Ha JIHHHH ,n:elicTBHH 3TOM CHJibi npous-

BOJihHYIO TOtiKY B u npHJI02KHM B Heii i'l,BeypaBHoBerneHHhre CHJihi F1 n

F2, TaKne, tiTO F1 F nF2 =-F. OT 3Toro,n:eiicTBne CHJihi F Ha TeJio He U3MeHHTCfl. Ho CHJihi F n F 2 TaK2Ke o6pa3yiOT ypanHoserneHHyiO cncTe-

ToqKe B.

TaKHM o6pa3oM, BeKTop, u3o6pa2KaiOmnii cnJiy F, Mo2KHO ctinTaTb rrpHJI02KeHHh!M B JII06oif TOtiKe H3 JIHHHH ,n:eHCTBHfl CHJibi (T3KOH BeK- 'l'OpH33hiBaeTC51 CKOJtb3flU{UM).

IloJiy'leHHbiHpe3y JibT3T crrpaBe)J.JIHB TOJibKO .IJ:JIH CHJI, ,n:eHCTBYIO!UllX

Ha 30COJIIOTHO TBep,n:oe TeJIO. Ilpu llH2KeHepHhiX paC'leTaX HM M02KHO llOJib30B3TbCH JIHUJb TOr,ll,a, Kor,n:a onpei'l,eJIHIOTCfl yCJIOBllH paBHOBeCIIH

Toil HJIH nnoii KOHCTPYKU.HH II He paccMaTpHBaiOTCH B03HHKUJIIe B ee 'laCTHX BHyTpeHHHe ycHJIHH.

* 0T6po111eHHble H.IIH nepeHeCeHHble Cll•1hl oy)leM Ha pHCYIIKaX nepeqepKHBa'fb.

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HarrpnMep, nso6pameHHhiH Ha pnc. 4, a cTep}f(eHh AB 6yp;er HaxoAHTbCH B paBHOsecuu, ecJin F1 =F 2· ITpn rrepeHoce TO'leKrrpnJiomeHHH OOellX CHJ! B KaKyio-HH6y,ll,b TO'IKYC CTepmHH (puc. 4, 6) HJ!ll npn nepe-

cmpoeHHOcO Ha amux cu.llax, KaK rta cmopoHax.

BeKTop R, paBHhiH .n.rraroHaJin napaJIJieJiorpaMMa, rrocTpoeHHoro na BeKTOpax F1 n f 2 (pnc. 5), Ha3hiBaeTCH zeoMemputieCKOu cyMAwu eeKmopoe F1 n F 2: Величина этой силыглавный вектор

R=F1 +F2 • или аналитически