(формат)Тарг
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BaThCH reopeMoli BapHHhOHa (eM. sa)I:aqy KpoMe roro, MOJKHo BbiqucJIHTh MOMeHThi aHaJIHTHtieCKH ITO tPOpMyJiaM (47), KaK, HanpuMep, B sa)I:atJ:e 37.
§ 32. CMJIOBOE TIOJIE.
IJ.EHTP nDKECTM TBEP)J.OfO TEJIA
06JiaCTb, B K3lliLJ:OH TO':IKeKOTOpOH Ha llOMeiiJ.eBHYIO Ty,a:a M3TepH3JlbHYIO qacTm~y ,a:eficTBYeT CHJia, saBHCHIIJ.aH oT no.no.t~<eHHH (Koop,!!.HHaT)
3Tofi TO':IKH, H33bJBaeTCH CU.IlOBbt.M no.lle.M *. DpnMepoM CHJiosoro no.151 HBJIHeTCH llOJie THrOTeHHH (noJie CHJI DpHTHJKeHHH K 3eMJie IIJIH K JIIOOOTY ,!!.pyroMy HeoecHoMy TeJiy).
Ha Kalli.Il.YIO qacTHUY TeJia, Haxo,a:muerocH n6JIH3H seMHofi nosepxuocTH, ,a:eucTnyeT nanpanJieHHaH nepTHI<aJihHO BHH3 CHJia, KoTopyiO H33hiBaiOT cu,wil mRJtcecmu (sonpoc o TOM, 'ITOco6ofi npe,a:cTaBJIHeT cuJia
~ TaKoe cHJiosoe noJie Ha3hiBaiOT cma~uonapHbt.M. EcJIH JKe 3HaqeHHH cHJI MoryT ern.e H3Menl!ThCH c reqeHHeM speMeHH, noJie Ha3hiBaeTcll Hecma~uonapHbt.M. IloHHTHe 0 TIO/Ie MOIKeT 6b1Tb BBe~eHO H lJ,JIH lJ,pyrHX BeKTOpOB, KOTOpbie T3KOe BeKTOpHoe TIOJJe
o6pa3y!OT, H.aTipHMep lJ,JIH CKOpOCTefi HJIH ycKopeHHH TOqeK TeJia (eM.
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T5DKeCTII, oy,a:eT paCCMOTpeH B § 92). 3TH CHJ!bl o6pa3yiOT noJie CHJI THA\e~
CHI.
]LJIH TeJI, p33Mepbl KOTOpbiX oqeHb M3Jibl no cpasHeHHIO C 3eMHbiM paLJ:HyCOM, CH•1bl TKiKeCTH, ,a:eHCTSYIOmHe Ha qaCTHII,bl TeJia, MO.IKHO cqHT3Tb napaJIJieJibHhiMH ,a:pyr ,a:pyry 11 coxpamriOmHMH ,a:JIH KaiK,a:oil qacTHUhi noCTOHHHOe 3HaqeHHe npii JIIOOblX nosopoTaX TeJia. TioJie T51• )KeCTH, S KOTOpOM BbinOJIH5l!OTCH· 3TH ,a:sa z
ycJIOBHH, H33blB3IOT oiJuopOOHbtM nO.IlCM mR- 0/CCCmU.
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l1p11 JIIOOOM ITOBOpore TeJia CHJibl iJ"n OCTaiOT- |
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CH npHJIOJKeHHblMH B OLJ:HHX 11 TeX iKe TOqKaX |
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reJia II rrapaJIJieJibHbiMH ,a:pyr ,a:pyry, H3Mermer- |
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CH TOJibKO HX HanpasJieHHe no |
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,ll,OKasaHHOMY s § 31, pasHo,a:eficrsyromaH P CHJI Pk 6y,a:er npii JIIOObiX noJioiKeHHHX reJia npoxo,a:HTb qepes o,a:ny 11 ry Me neHsMeHHo CBH3auHYIO c reJioM roqKy C, HBJimomyrocH IJ,eHrpoM napaJIJieJihHhiX CHJI TH-
JKecrH Pk· 3-ra roqKa 11 II33hiBaercH ttemnpoM mHO/Cecmu me.lla. TaKHM o6pasoM, ttertmpoM Jwz:Heecmu meepoo2o me.lla rtasbteaemcR rteu3Mertrto C6fl3artrtaJl CBtnUM mC.IlOM mo•UW, iiCpC3 IWmopylO npoxoiJum .llUHUJl iJeficm6UJl paertoiJeficm6ljiOU{CU CU/l mRO/CCCmu, iJeucm6ljiOU{UX rta iiaGnZUlfbl OaHH020 me.lla, npu .llt060M noAoO/CeHuu me.lla e npocmpartcmee. 4ro TaKaH roqKa cymecrsyer, cJie,a:yer us ,ll,OKasaHHoro s § 31.
Koop,ll,HHaTbi IJ,eHrpa THJKeCTH, KaK uempa napaJIJie.1hHbiX CHJI, onpe,ll,eJIHIOTCH cpopMyJI3MH (57), CJie)I:OBaTeJibHO,
(59)
r,11,e x", Y~t. zk- KoopJI:HHaTbi roqeK npHJIOJKeHHH CHJI nvKecrH P~<• ,a:eHCTBYIDmlfX Ha qaCTHU:bl TeJia.
OrMeTHM s saKJiroqemre, qro corJiacHo orrpe,IJ,eJieHHIO ueHrp TH- )KeCTII3TO TOqKa reoMeTpnqecKaH; OHa MOJKeT Jie}K3TL 11 SHe npe,l],C.'IOB
)I:anHoro |
TeJia (nanpHMep, )I:JIH |
KOJihiJ,a). |
§ 33. |
I(OOP,ll,HHATbl U,EHTPOB |
T5J)I(ECTH |
O,ll,HOPO)J,HbiX TEJI |
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llJIH OJI:Hopo:r.Horo TeJia sec Pk JIIOoofi ero qacru nporropii,HOHaJieH |
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OObeMy vk 3TOH qacTII: Ph =yvk, |
a sec p scero TeJia nponopii,l!OHaJieH |
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oo'heMyV 3Toro reJia, T. e. P=yV, r,a:e y - sec e)I:HHHIJ,bl oobeMa. |
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l1oJI:CTaBHB 3TH 3HaqeHHH P 11 |
Pk s cpopMyJILI (59), 3aMeTHM, qro so |
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scex cyMMax y KaK o6m11fr MHOJKHTeJih SbiHOCHTCH 3a cKooKH 11 coKpama-
89
(60)
KaK BH)l.HO, noJio)KeHne neHTpa TmKeCTH o.n.Hopo.n.Horo TeJia saBHCHT TOJibKO 01'ero reoMeTpHqecKOH <l>opMbl, a OT BeJIIIqHHbl y He 3aBHCHT. no .:noii npnqHHe TOqKy C, KOOp,ll.HHaTbl KOTOpoii onpe.n.eJI.HIOTCH <PopMy-
JJaMH (60), Ha3biBaJOT l{eHmpo.M mJtJJCecmu o6'oe.Ma V.
nyTeM aHaJiornqHbiX paccy)K.n.ennii JierKo HaiiTH, ~o ecJIH TeJio npe.n.- CTaBJIHeT COOOH O,!l;HOpO,!l;HyiO llJIOCKYIO H TOHKYJO nJiaCTHHy, TO )l.JIH Hee
(61)
f,!l;e S - llJIOUJ.a)l.b BCeii nJiaCTHHbi; Sk - llJIOIIJ,a)l.H ee qacTeH.
ToqKy, KOOp,ll.HHaTbl KOTOpOH onpe,ll.eJIHIOTC.H <l>OpMyJiaMH (61), Ha-
3biB310T uptmpo.M fl'lJl:JICecmu nAOU{aou S.
ToqHo TaK )Ke noJiyqaiOTCH cpopMyJihi ,!l;JIH Koop,!l;nHaT ttertmpa nlfl-
aJCecmu |
/lUrtuu: |
1::E lkYk• Zc = i ::E lkzk, |
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Xc = i L.tkxk, Yc = |
(62) |
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,!l;JIHHa Bceii JIHHHH; lk - |
,!J;JIHHbi ee tiacTeii. |
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no <PopMyJiaM (62) MO)KHO HaXO)l.HTb u;eHTpbl TH)KeCTH II3)l.eJIIIH II3
TOHKOH npOBOJIOKII ITOCTOHHHOfO ceqeHIIH.
TaKHM o6pasoM, neHTP TH)KeCTH O,!l;Hopo,!l;Horo TeJia onpe)leJI.HeTcH, KaK IJ,eHTp TH)KeCTII COOTBeTCTBYJOlli.ero OObeMa, ITJIOIIJ,a,!l;II HJIH JII1Hllii.
§ 34. CllOCOEbl OllPE,[J,EJIEHH51 I(OOP,[J,HHAT
IJ.EHTPOB T51)1(ECTH TEJI
Hcxo)lH ns noJiyqeHHbiX Bhiiiie oomnx <PopMyJI, MO)KHO yKasaTh KonKpeTHhie cnocoobi onpe.n.eJieHIIH Koop)liiHaT neHTpoB nm:ecTH TeJI.
1. C H M M e T p H 5!. ECJIH O)l.HOpO,ll.HOe TeJIO IIMeeT llJIOCKOCTb, OCb
HJIII IJ,eHTp CIIMMeTpiifl, TO ero u;eHTp TH)KeCTII Jie)KIIT COOTBeTCTBeHHO HJIH B ITJIOCKOCTII CHMMeTpHH, IIJIU Ha OCII CIIMMeTpHH, HJIH B IJ,eHTpe CHMMeTpHH. _
l(onyCTIIM, HanpiiMep, qTo O)l.HOpO,!l;HOe TeJIO HMeeT ITJIOCKOCTb CIIM· MeTpiiii. Tor.n.a ::noii nJioCKOCTbJO OHO pas6rmaeTcH Ha .n.Be TaKIIe qacTn, Beca KOTOpbiX p 1 II p 2 paBHbi .n.pyr )lpyry, a nempbr TH)KeCTH Haxo)lHTcn Ha O)lHHaKOBbiX paccTOHHHHX OT IIJIOCKOCTII CIIMMeTpiiu. CJie,!l;oBa-
TeJihHO, IJ,eHTp TH)KeCTII TeJia KaK TOqKa, qepes KOTOpyiO ITpOXO)liiT paBHO)l.eHCTBYJOUJ.aH )lByx paBHblX H napaJI.~1eJibHbiX CHJI Pt II P2, oy)leT
)leiicTBHTeJibHO Jie)KaTb B ITJIOCIWCTII CHMMeTpHII. AHaJiorJiqHbiH pesyJihTaT llOJiyqaeTC5I II B CJiyqa.HX, K0f)l3 TeJIO HMeeT OCb HJIII :U:eHTp CIIMMeTpHH.
Hs cBoiicrB cHMMeTprrn cJie)lyeT, qTo neHTP TH)KeCTH o)lHopo)lHoro KpyrJIOfO KOJ!b:U:a, KpyrJIOR HJIII npHMOYfOJibHOH llJiaCTHHbl, npHMOyrOJibHOfO napaJIJieJienune,!l;a, mapa u !lpyrrrx O,!l;Hopo,!l;HbiX TeJI, HMei0-
90
IIJ.HX II,eHTp CHMMeTpHH, Jie2KHT 13 reOMeTpnqecKOM u;eHTpe (II,eHTpe CHMMeTpHH) 3THX TeJI.
2. P a 3 6 11 e H 11 e. EcJIH TeJio M02KHo pasoHTb Ha KoHeqHoe qiicJio T3KI1X qacTefi, )l.JIH K3JK)l.OH II3 KOTOpbiX llOJIOJKeHHe II,eHTpa THJKeCTI1 H3BeCTHO, TO KOOp)l.HH3Tbi II,eHTpa THJKeCTH BCero TeJia MOJKHO Henocpe,ll.- CTBeHHO Bblqi1CJIHTb llO <PopMyJiaM (59) - (62). TipH 3TOM qHCJIO CJiaraeMbiX B K3JK)l.Oll H3 cyMM oy)l.eT paBHO qi1CJiy qacTeii, Ha KOTOpbie pa3oi1TO TeJIO.
3a)l.a'la45. OnpetJ,eJIHTb KOOptl,HHaTbl UeHTpa TSDKeCTH Otl,HOpOt(HOH TIJiaCTHHbl, Hso6paJKeHHoii Ha pHc. 106. Bee pasMepbi tJ,aHhi B caHTHMeTpax.
p e Ill e H H e. IlpOBOtl,HM OCH X,![ H pa36HBaeM TIJiaCTifHY Ha Tpll npmmyrOJJbHHKa (JJHHHH paspesa 110Ka3aHbl Ha pHC. 106). 8bi'lHCJilleMKOOptl,HHaTbl UeHTpOB THJKeCTH Ka/Ktl,OfO H3 npHMOyrO;IbHHKOB H HX TIJIOI.l.l,at(H (CM. Ta6JJHUy).
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PHc. 106 |
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rr.~OI.l.l,at(b Bceii TIJ!aCTHHbl S=sl+sz+sa=36 CM2 • |
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I10.l(CTaB,1llli Bbl'lHCJieHHble BeJIH'lHHbl B ¢opMyJibl (6I), |
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Xc= |
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36 |
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2-gcM, |
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4+ 100-f-I08 |
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36 |
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Hafi.l(eHHoe noJIOJKeHHe ueHrpa THJKecrH C noKasano Ha '!epTelKe;TO'!KaC OKa- |
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cnoco6a pa3611eHBH. OH np11MeHHeTcH K TeJiaM, 11MeiOIIJ.HM Bbipe3hi,
ecmr ueHTpbi TH:tKeCTI1 |
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3aJJ.a'·la46. OrrpeAeJJHTb noJioJKeHue |
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KpyrJiofr rrJiaCTHllbl pat~Hyca R c Bbipeso~l |
patJ,Hyca r (puc. |
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107). PacCTOliHHe clC2=a. |
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p e III e H H e. IJ,eurp |
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MeTpHH. IlpOBO)lHM KOOptl,HHaTHble OCH • .Ll.Jill HaXO/Kt(eHHli |
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HOfO Kpyra (43CTb 1), a 3aTeM Bbl'lHTaeM 113 llOJiyqeHHOH |
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TIJIOI.l.l,aAH TI.10I.l.l,atJ,h BbrpesaHHoro Kpyra ('laCTb |
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9TOM JlJIOI.l.l,a)lb '!acTH2, KaK Bbl'lHTaeMal!,t(OJI/KHa 6paTbCll |
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co snaKOM MHHyc. Tor)la |
s =nR 2, |
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2 |
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91
ar 2
R2 - r 2 ; Yc=D.
HaiitJ,€HHbJH ueHTp TlDKeCTH C, K3K Blltl,HM, Jl€iKHT .~eBee T01JKH cl.
4. I1 H T e r p n p o B a H II e. EcJIH reJio HeJih3H pa3oHTb Ha neCKOJihKO KOHeqHbiX lJ:3CTeH, ITOJIOJKeHH.H UeHTpOB T.HJKeCTH KOTOpLrX
H3BeCTHhl, TO TeJIO pa36HBaiOT CHaqaJia Ha npOH3BOJibHhle MaJi:ble OObei1lbi
f1V 10 ,!l,Jl.H KOTOpb!X <J?opMyJibi (60) npHHHMaiOT |
BH,!l, |
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Xc = V |
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r.n.e xu, Y~tt zk - Koop.n,uHaTbi HeKoropoii roqKn, JieJKarueii snyTpn oobeMa 1:1vu. 3areM B paBeHCTBax (63) nepexo.n,HT K npe.n,eJiy, ycTpeMJIBH Bee Livk K HYJIIO, T. e. CTHfHBaH 3TH OObeMbi B TOqKJI. Tor.n,a CTOHIUIIe B pm~encTBax cyMMhl o6paruaiOTCH s HHTerpaJihi, pacnpocTpaHeHHhre I-W Becb o6beM reJia, H <J?opMyJibi (63) .n,aiOT B npe.n.eJ1e:
Xc = ~ SX dv, |
Yc = ~ Sy dv, |
Zc = ~ Sz dv. |
(64) |
(V) |
(V) |
(v) |
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AHa;rwrnqno ,!l,JIH KOOp.n,riHaT ueHTpOB T.HJKeCTH ITJIOI.Ua.n.eii H JIIIHHH no.1yiJaeM B npe.n,eJie H3 <J?opMyJI (61) II (62):
Xc = ~ |
Sx ds, Yc= ~ Sy ds |
(C5) |
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Yc= 1Sydl, |
Zc=f I zdl. |
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Xc= 1Sxdl, |
(66) |
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3TH X qmpMyJI K |
Onpe.n.eJieHI!IO |
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Pnc. 108 |
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ToiJKa nepecel!eHIIH 3THX |
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onpe,n;eJI.HeT |
uernp |
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T.H.IK('CTH HBJI.HeTCH |
MeTO,IJ, B3BelliHBaHHH. I1,n;eH 3TOfO MeTO)J,a |
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HCHa 113 |
paCCMOTpeHHOfO HHJKe npHMepa. |
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flpHMep. IlOKMKeM, K3K MOiKHO .9KC11epHMeHT3JibHO OnpetJ,eJII!Tb Otl,Hy H3 Koor?H-
H3T Ilenrpa nnKecTH caMoJiera (paccrol!lme a), ecmr paccTOl!IIHe AB~ l (pHc. 108)
92
H3BeCTHO. llOCT3BHB KOJieco 8 na TIJiaT<j:JOpMy BeCOB, Hatit~.eM B33eiiJHB3IIIleM CHJIY )\aBJICHHS! KOJieca Ha TIJI3TctJOp\1Y; TeM C3Mb!M 6yAeT H3H)\eHa '!HCJICHHOpaBH3ll :;nofi CHJie peaKIJ,Hll Nl. TO'!HOT3K JKe B3BelliHB3HHeM 113XOAHM peaKIJ,HfO N2. npupaBHHBaH 33TeM HYJ!fO CY'IIMYMOMeHTOB BCeX CII.TI OTHOCHTe.%IIO uenrpa TS!JKeCTH C C3MOJICT3, TIOJiy-
'l3eM N2a-NI(l-a)~O, OTKytJ,a |
uaxo)\HM a~tN1I(N1+N2). |
OtieBHI\HO, N1+N2=P, rt~e |
P - sec caMoJieTa. Ec.ijH 3!la'leHHe BeJIH'!HIIbl P |
Hanepe)\ H3BecTHO, TO ):\Jill onpetJ,eJieHHll a MO.lKHO 060HTHCb TOJ!bKO Ot(H0Kp3TIIbl~l BJBelllHB3HHeM,
§ 35. nEHTPbl ni)KECTH HEKOTOPbiX O)J.HOPO)J.IibiX TEJI
1. IJ. e H T p T H JK e C T H H 0 CT H. PacCMOTpHM)l.yry AB pa.n.11yca R c ueHrpaJihHhiM yrJioM AOB=2a. B cuJiy CHMMeTpHH IJ,eHrp THJKeCTH aroil .n.yr11 JieJKHT Ha oc11 Ox (p11c. 109). Haii.n.e::lol KOOP.lJ.HHary Xc rro cpopMyJiaM (66). l(JIH aToro Bhi)l.eJIHM Ha .n.yre AB 3JieMeHT MM' )l.JIHHoii dl=R dcp, rroJioJKeHHe Koroporo orrpe,ll,eJIHercH yrJIOM cp. Koop,11,11Hara x 3JieMeHTa MM' 6y.n.er x=R cos cp. Do.li.- cTaBJIHH 3TH 3HaqeHH5I X 11 dl B rrepByiO 113 cpopMyJI (66) 11 HMeH B BH,ll,y, ~o HHTerpaJI )l.OJillieH OhiTh pacnpocrpaHeH Ha sc10 .ll.JIHHY .n.yru, noJiyqnM:
|
B |
|
a; |
|
I |
\ |
R2 |
s |
R2 . |
Xc = y |
J x dl = T |
cos qJ dqJ = |
2 T sm a, |
|
|
A |
|
-a |
|
r.n.e L - )l.JIHHa ,11,yr11 AB, pasnaH R ·2a. OrCIO.li.a oKoHqareJihHO Haxo-
.li.HM, qro l{ertmp tnHJICecmu iJy2u oKpy:JicHocmu AeJJCUtn Ha ee ocu CUA1Atempuu Ha paccmoRHUU om qeHmpa 0, pasHo.M
Xc=(R sin a)/a, |
(67) |
r,11,e yroJI a H3~1epHeTCH B pa,ll,HaHax. |
|
2. IJ. e H T p T 5I JK e C T 11 n JI 0 IIJ, a .li. H |
T p e y r 0 JI b H H K a. |
Paso6heM nJioma.li.h rpeyroJihHHKa ABD (pHc. |
110) npHMhiMH, napaJI- |
~X
B
A~D
~
PHc. 110
JieJihHhiMH cropoHe AD, Ha n ysKHX noJiocoK; IJ,eHTphi THJKecTH 3THX noJiocoK 6y.n.yr JieJKaTh Ha Me,ll,HaHe BE rpeyroJihHIIKa. CJie.n.osareJihHO, n ueHTP THllieCTH scero rpeyroJihHHKa JieJKHT Ha 3Toi1 Me.n.naue. AHaJioriiqHhn'ipe3yJihTaT noJiyqaercH ,!l;JIH ,!l;Byx .n.pyr11x Me,!l;HaH. Orc10.n.a 3a-
93
K.moqaeM, liTO uptmp tnfW!Cecmu n.lloll{aau mpeyzoAbHUKa Aea~cum e mott- Ke nepecetteHUfl eco .MeauaH. Dpn 3TOM, KaK H3BecrHo,
CE=BE/3. |
(68) |
3. 11, e H T p T 5I JK e C T 11 ll Jl 0 IIJ; a ~ H K p y I'0 B 0 r 0 |
C e K- |
T o p a • PaccMorpHM Kpyrosofl ceKrop OAB paJI:Hyca R c ueHrpaJibHbiM yrJioM 2a (puc. 111). Pa3o6heM MhicJieHHo nJIOrn;aJI:h ceKropa OAB pa,n,uycaMH, npose~eHHbiMII H3 ueHrpa 0, Ha n ceKropoB.
13 |
e |
|
|
|
8 |
|
0 |
|
PHC. 111 |
Pnc. 112 |
PHC. 113 |
npH HeorpaHII'leHHOMyseJIHqeHHH "!HCJia n, 3TH CeKTOpbl MOJKHO pacCMaTpiiBaTb KaK TIJIOCKHe rpeyrOJibHHKH, IJ,eHTpbi THJKeCTH KOTOpblX JieJKaT Ha .n.yre DE pa.n.uyca 2R/3. CJie)I:osareJibHO, ueHTp THJKeCTH ceKropa OAB cosna~aer c ueHrpoM THJKeCTH .n.yru DE, noJioJKeHHe KOToporo Hafi~eTcH no <PopMyJie (67). 0KoHqareJihHo noJiyqHM, qro uptmp mfla~eecmu nAoU{aau Kpyzoeozo ceKmopa JJ-ea~eum rta ezo ocu cuJ.utempuu Ha paccmoflrtuu om l{ertmpa 0, paertol~t
2 R sin rx |
(69) |
Xc=3-rx-· |
|
Dprme~eM 6e3 JI:OKasaTeJibCTB eme )I:Ba pesyJJbTara. |
|
4. 11, e H T p T 5I JK e C T 11 0 6 b eM a ll II p a M 11 )1. bl |
(11JIII |
K o H y c a). 3Tor ueHrp C JJeJKIIT Ha npHMOH C1 E (p11c. 112), r)I:e E - Bepiii11Ha, a C1 - IJ,eHTp THJKeCTI1 llJIOIIJ;a)I.II OCHOBaHI15I ll11paMI1~hl; np11 3TOM
(70)
PesyJibTaT cnpaBe,li.JII1B )I.JIH .mo6oii MHoroyroJibHOH n11paMI1)1.bi 11 ~JIH
KOHyca.
5. 11, e H T p :T H JK e c T 11 o 6 b e M a n o JI y III a p a. 3TOT ueHrp C JieJKI1T Ha oc11 Ox (ocii CHMMerp1111, p11c. 113), aero Koop~11Hara
x0 =0C=3RI8, |
(71) |
r.n.e R - pa.n.11yc noJiyiiiapa.
¢opMyJihl, onpe~eJI5IIOIIJ,IIe KOop~IIHaTbl IJ,eHTpOB T5I.JKeCTII ,11.pyr11X O)I.HOp0,11.HhiX TeJI, MOJKHO HaHTI1 B pa3JIII'IHbiXTeXHHqeCKHX cnpaBO'IHHKaX.
Pa3.n.eJI BTopoii
KMHEMATHKA TOIJKH H TBEP,li.OfO TEJIA
fJiasa IX
KHHEMATHKA roqKH
§ 36. BBE)J.EHHE B K.HHEMATHI(V
Kurte.Mamwwu H33biBaercH pa3.n,eJI MexaHHKH, B KoropoM HsyqaIOTCH reOMeTpiiiiecKHe CBOifcTBa ,!l,BH:tKeHHR TeJI 6e3 yiieTa HX HHepTHoCTH (MaCChi) H .n,efiCTBYIOIII.HX Ha HHX CllJI.
KHHeMaTHKa npe.n,cTaBJIHeT co6ou, c o.n,Hoft cropoHhi, sse.n,eHHe s ,!l,HHaMHKy, TaK KaK ycTaHOBJieHHe OCHOBHhiX KHHeMaTHIIeCKHX nOHHTHU H 3aBHCHMOCTefi He06XO,!l,HMO ,!l,JIH II3yiieHHH ,!l,BHLKeHHH TeJI C yqeTOM
.n,eHCTBHH CHJI. C .n,pyrofi CTOpOHhl, M€TO,!l,hl KHHeMaTHKH IIMelOT H Ca- l\40CT05ITeJ1bHOe npaKTHIIecKoe 3Haqenue, HanpHMep, npH H3yiiemm nepe.n,atJ: ,!l,BH:tKeHHH B MeXaHH3MaX.
I1o.n, ,!l,BHLKeHHeM Mhl nOHHMaeM B MeXaHHKe H3MeHeHHe C TeiieHHeM speMeHH nOJIO:tKeHHH ,!l,3HHOfO TeJia B npocrpaHCTBe no OTHOIUeHHIO K
.n,pyrHM TeJiaM.
l(JIH onpe,n;eJieHHH nOJIOLKeHHH ,!l,BHLKYIII.erOCH TeJia (HJIH TOIIKH) B pa3Hhle MOMeHThl BpeMeHII C TeJIOM, no OTHOIUeHHIO K KOTOpOMy H3yiiaetCH ,!l,BH:tKeHHe, LKeCTKO CBH3biBaiOT K3KYIO-HH6y.n,h CHCTeMy K00p,D;HH3T, o6pa3YIOIII.YIO BMecre c 3THM TeJioM cucme.My omcttema. B .n,aJihHefirneM 6y.n,eM fOBOpHTb 0 ,!l,BH:tKeHHII TeJia (HJill TOqKH) no OTHOIUeHHIO K ,!l,3HHOH CHCTeMe OTCIIeTa, no.n,pa3yMeBaH no.n, 3THM ,!l,BH:tKeHHe no OTHOIUeHIIIO K TOMY TeJiy, C KOTOpblM 3Ta CHCTeMa OTcqera CBH3aHa. I13o6pa:tKaTb CHCTeMy OTCIIeTa 6y,n;eM B BH.n,e tpeX KOOp,!I,HH3THhiX OCeH (He noKa3hiB3H TeJIO, C KOTOphiM OHH CBH3aHhl). Bbi60p CHCTeMhl OTCIIeTa B KHHeMaTHKe npOII3BOJieH (onpe.n,eJIHeTCH IIeJih!O HCCJie,!l,OBaHH5I), H B OTJIHIIHe OT ,ll;H- HaMHKH (CM. § 74) BCe KHHeMaTHIIeCKHe 3aBHCHMOCTH, noJiyiieHHhie npH H3yqeHHH .n,BmKeHHH B KaKOH-HH6y,n;h O,!l,HOH CHCTeMe OTCIIeTa, 6y.n,yr cnpase.n,JIHBhl H B JII060H .n,pyrofi CHCTeMe OTCIIeTa.
l(BHLKeHHe TeJI COBeprnaeTCH B npocrpaHCTBe C TeiieHHeM BpeMeHH. I1pocrpaHCTBO B MeXaHHKe Mhl paccMaTpHBaeM KaK TpeXMepHoe eBKJIH- ,!l,OBO npocrpaHCTBO. Bee H3MepeHnH B HeM npoH3Bo.n,HTCH Ha ocHoBaHHH MeTO,!l,OB eBKJIH,!l,OBOH reoMeTpiiii. 3a e,!l,HHIII1Y ,!l,JIIIHhl npH H3Mepemm paccTOHHHH npHHHMaeTcH 1 M. BpeMH B MexaHHKe ciiHTaeTCH yHIIBepcaJihHhiM1 T. e. npOTeK310Il!.HM O.Ll,HHaKOBO BO BCeX paCCMaTpHBaeMhiX
95
cncTeMax OTcqeTa. 3a e.n.HRHUY speMeHH ~:pnHHMaeTCH 1c. PasMepHoCTh }I,JIHllhi o6osHaqaeTcH CHMBo.noM L, a speMemr - CHMBOJIOM T.
EBKJIH)l.OBo npocTpaHCTBO H ymmepcaJibHOe speMH OTpaJKaiOT peaJihHhre CBOilcTBa npocTpaHCTBa H BpeMeHH Jlllillh npHOJIHJKeHHO. 0}1,HaKO, KaK IIOKa3biBaeT OIIhlT, )l.JIH )l.BHJKeHHH, KOTOpb!e H3yqaEOTCH B MeXaHHKe (.ll.BHJKeHHH CO CKOpOCTHMH, )l.aJieKHMH OT CKOpOCTH CBeTa), 3TO IIpHOJIHJKeHHe }l,aeT BIIOJIHe )l.OCT3TOqHyEO }I,JIH rrpaKTHKII TQqHOCTb.
BpeMH HBJIHeTCH cKaJIHpHoii:, nerrpepbiBHO nsMeHHIOIII,eiicH semi'lHHOH. B 3a.n.aqax KHHeMaTHKH speMH t npHHHlviaiOT 3a HesasHCHMoe nepeMermoe (apryMer-n). Bee .n.pyrne nepeMeHHble BeJinqHHhi (paccTOHHHH, CKOpOCTH H T. )]..) paCCMaTpHBaiOTCH K3K H3MeHHIOIIJ.HeCH C TeqemreM BpcMeHH, T. e. K3K t1JyHKUHH BpeMeHH t. 0TcqeT BpeMeHH Be)l.eTCH OT HeKOTOpOrO HatW/lbHOcO MO.Mertma (t=O), 0 BhlOOpe KOTOporo B K3)K)J.OM cJiyqae ycJioBJIHBaiOTCH. BcHKHii .n.aHHhiH MOMertm epeMertu t orrpe}l,eJrHeTcH qncJioM ceKyH}I,, rrporne.n.umx OT HaqaJibHoro MOMeHTa .n.o }l,aHHoro; pa3HOCTb MeJK)l.y K3KHMH-HHoy}l,h )J.BYMH llOCJie}J,OBaTeJibHbiMH MOMeHTaMII BpeMeHII H33hiBaeTC51 npoMeJICymKOM epeMertU.
J1oqepiiHYThie H3 OIIb!Ta H IIO)l.TBepJK)l.eHHh!e npaKTHKOH OCHOBbl,
•Ha KOTOph!X CTpoHTCH KHHeMaTHKa, .n.a10T aKCHOMhl reoMeTpHH. HnKaKHX )l.OllOJIHHTeJihHhiX 3aKOHOB H.'IHaKCHOM )l.JIH KHHeMaTIIqecKOfO H3y'IeHHH )l.BHJKeHHH He TpeoyeTCH.
]lJIH perneHHH 3a)l.aq KHHeMaTHKH Ha)l.O, qTQObl YJ3yqaeMOe )J.BHJKeHue
ObiJIO KaK-TO sa,n;aHo (onncaHo).
Kurte.Mamuqectw 3aoamb aeu:J!Certue UJ?U 3a!<OH aeuJICeHUfl tne/la ( mottKu) - srtaiium saiJamb no.llOJICertue amoco me.lla (motaw) omrtocume.llbHO iJamwu cucmeMbt ommemae .ll106ot1 AtOMertm epeMertu. Y cTaHOBJienne Ma-
TeMaTHqecKHX CIIOCOOOB 3a)l,aHHH )J.BHJKeHHH TOqeK HJIII TeJI HBJIHeTCH O)l.HOli H3 BaJKHhiX 3a)J.aq KHHeMaTHKH. fi03TOMY H3yqeHHe )l.BHJK(:'HH5I JIIOooro oobeKTa 6y,n;eM HaqnHaTh c ycTaHoBJieHHH crroco6os 3a)J.aHHH
::noro )J.BHJKeHHH.
0CHOBHaH 3a)l.aqa KHHeMaTHKH TOqKH H TBep)l.OfO TeJia COCTOHT B
TOM, qTOOhl, 3HaH 3aKOH }J,BIIJKeHHH TOqKH (TeJia), ycTaHOBHTh MeTO)l.bl onpe,n;eJieHHH BceX KHHeMaTHqeCKHX BeJIHqHH, XapaKTepH3yiOIII,HX }J,aHHOe )l.BHMeHHe.
HsyqeHHe KHHeMaTHKII HaqHeM c nsyqeHHH )l.BHJKeHHH npoCTelirnero OObeKTaTOqKII (KHHeMaTHKa TOqKH), a 3aTeM rrepeH)J.eM K II3yqeamo
KHHeMaTHKH TBep)J.OfO TeJia.
HerrpephiBHaH JIHIIHH, KoTopyiO onncbmaeT )l.BHJKYIII.aHCH ToqKa OTHOCHTeJihHO )l.aHHOH CHCTeMhl OTCqeTa, Ha3biBaeTCH mpaeKmopueii TO'IKH. Ecm1 TpaeKTopuel1 HBJIHeTCH rrpHMaH JIHHHH, )J.BHJKeHHe ToqKH Ha3biBa·
a eCJIH KpHBaH - KpU60/lUHeUHblM.
§ 37. CllOC06bl 3A,[J,AHIUI ,[J,BH)I(EHIUI TOlJI(H
JlJIH 3a)l.aHHH )l.BHJKeHHH TOqKH MOJKHO IIpHMeHHTb O)l.HH II3 CJie)l.ylO· lUHX Tpex CllOCOOOB: 1) BeKTOpHbiH, 2) KOOp)l.HHaTHh!H, 3) ecTeCTBeHHhiH.
1. BeKTopHbiH crroco6 sa.n.aHHH )l.BIIJKeHuH
T 0 q K H. I1yCTh TOqRa M )l.BHJK(:'TCH ITO OTHOllleHHIO K HeKOTOpOH CHCTeMe oTcqeTa Oxyz. I1oJioJKeHHe 3Tofr ToqKn B JII06ol1 MoMeHT speMeHH·
96