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B Tex CJIY"IaHX, Koni.a npn BbitiHCJieHHH MOMeHTa Bo3HHKaiOT sa- TPY~HeHHH B onpe~eJieHnn npoeKUHH cHJihi Ha coorBercrByroruyro nJiocKOCTh HJIH ITJieqa 3TOll npoeh:UHH, peKOMeH)I:yeTCH pa3JIQ}KHTh CHJIY Ha ,ll.Be B33HMHO nepneH)I:HKYJIHpHbie COCT3BJIHIOIUHe (H3 KOTOpbiX O)I:Ha napaJIJieJibHa I\3!\0ll-HHOYJI:b KOOp)I:HH3THOll OCH), a 3aTeM BOCITOJih30-
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.
&9
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.
PasHo~eiicrsyromyro CIIJI |
THJKecrH |
P1, |
|
p2, ... , Pn. ,a:eficrsyromHx Ha |
qacniUbi |
,a:an- |
|
Horo TeJia, o6osnaquM P (puc. 105) . .Mo,a:yJih |
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::nofi CHJibi Ha3biBaeTCH eeco.lt me.lla 11 onpe,a:e- |
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JIHeTCH paseHCTBOM |
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P='J:.pk. |
(58) |
9 |
|
l1p11 JIIOOOM ITOBOpore TeJia CHJibl iJ"n OCTaiOT- |
PHC. 105 |
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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 |
OTHOIIIeHHIO K TeJiy. CJie,ll,OBaTeJibHO, 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 |
||
llJIH OJI:Hopo:r.Horo TeJia sec Pk JIIOoofi ero qacru nporropii,HOHaJieH |
||
OObeMy vk 3TOH qacTII: Ph =yvk, |
a sec p scero TeJia nponopii,l!OHaJieH |
|
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 |
|
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, |
|
Xc = i L.tkxk, Yc = |
(62) |
||
r,!l;e L - |
,!l;JIHHa Bceii JIHHHH; lk - |
,!J;JIHHbi ee tiacTeii. |
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).
N• |
I |
I |
I |
2 |
l 3 |
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8 |
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Xk |
-1 |
I |
5 |
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Yk |
1 |
5 |
9 |
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SIJ |
4 |
20 |
12 |
2 |
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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), |
JlOJiyqaeM: |
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x1s1 |
-l-x2s2 -l-xasa -4-i-20-i-60 |
I |
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Xc= |
S |
36 |
2-gcM, |
||||||||||
4+ 100-f-I08 |
8 |
||||||||||||
36 |
5-g-CM. |
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Hafi.l(eHHoe noJIOJKeHHe ueHrpa THJKecrH C noKasano Ha '!epTelKe;TO'!KaC OKa- |
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saJiacb |
BHe |
TIJJaCTHHbl. |
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3. |
,lJ. 0 |
ll 0 JI |
II e H 11 e. |
3-roT CllOCOO HBJIHeTCH qaCTHbiM CJiyqaeM |
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cnoco6a pa3611eHBH. OH np11MeHHeTcH K TeJiaM, 11MeiOIIJ.HM Bbipe3hi,
ecmr ueHTpbi TH:tKeCTI1 |
TeJia 6e3 |
Bbrpe3a H Bbipe- |
y |
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33HHOH qaCTH H3BeCTHbh |
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3aJJ.a'·la46. OrrpeAeJJHTb noJioJKeHue |
ueHrpa |
Tl!JKecTH |
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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 |
Tll/KeCTH |
11o1aCTHHbi |
JIC/KHT Ha |
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JIHIIHH C1 C2 , TaK KaK 3Ta JIHHHll |
llBJ!l!eTCll |
OCbiO |
CHM- |
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MeTpHH. IlpOBO)lHM KOOptl,HHaTHble OCH • .Ll.Jill HaXO/Kt(eHHli |
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KOOptl,liHaTbl Xc t(OilOJIHHeM |
llJIOI.l.l,at(b |
JlJiaCTHHbl |
)lO |
llOJJ- |
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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 |
2). |
IlpH |
PHc. |
107 |
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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, |
s |
2 |
=-nr2 |
, x |
1 |
=0, x =a, S=s +s |
=n(R2- r2), |
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1 |
2 |
1 |
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, |
||
I |
~ xk l1vk n T • |
(63) |
|
Xc = V |
.n,., |
||
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) |
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) |
||
(S) |
(S) |
|||
H |
Yc= 1Sydl, |
Zc=f I zdl. |
||
Xc= 1Sxdl, |
(66) |
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(L) |
(L) |
(L) |
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I1pm.vrep np HMeHe!III.H |
3TH X qmpMyJI K |
Onpe.n.eJieHI!IO |
KOOp,!l,l!HaT |
|
uempa T.HJKecTII paccl\10Tpeu B CJie.n,yiOliJ,eM rraparpa¢e. |
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5. 9 K C IT e p II M e H T a Jl h H bl H |
C ri |
0 C 0 0. li,eHTpbl TJDKeCTH |
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HCO,!l,HOpO,!l,HhiX TeJI CJIOJKHOH KoH<J?nrypaUHH (caMOJieT, napoB03 H T. n.)
fl |
1f |
MOJKHO Onpe.n,eJI51Th |
3KCDepHMeHTaJibHO. |
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' |
t |
2 |
O.n,IIH II3 B03MOJKHhiX |
3KCrrepnMeHTa.1b- |
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Hh!X MeTO,ll,OB |
(MeTO,!l, |
ITO,!l,BeiiHIBaH11H) |
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COCTOHT B TOM, qro TeJIO HO,!l,Beiin!BaiOT |
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Ha HIITH |
HJIII |
rpoce |
3a |
pa3JIH'lHhleero |
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TO'lKH. HarrpaBJieHHe HHTH, Ha KOTOpofi |
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no.n,serneHo |
reJio, |
6y.n,er |
KaJK,!l,h!H pa3 |
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,n;asaTh |
HanpaBJiemre |
CIIJihi |
Tmxecnr. |
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Pnc. 108 |
ToiJKa nepecel!eHIIH 3THX |
nanpaBJiemu1 |
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onpe,n;eJI.HeT |
uernp |
THJKecrn |
reJia. |
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]LpyrHM B03MOJKHbiM CUOCOOOM 3KCrrepHMeHTaJihHOrO onpe,ll,eJJei!HH |
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UeHTpa |
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 |
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Xc=(R sin a)/a, |
(67) |
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93