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§ 99. UOH"THE 0 HEBECOMOCTH.
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,l(HHAMHKA CHCTEMbl H TBEP,l(OfO TEJIA
fnasa XXI
BBE)J.EHME B ,IJ.MHAMMKY CMCTEMbl. MOMEHTbl MHEPI.J.MH
§ 100. MEXAHHl-IECKMI CHCTEMA.
CHJlbl BHEIIIHHE H BHYTPEHHHE
CIIcTeMy MaTepiiaJibHhiX TOIIeK IIJIII TeJI, .II:BII:>KeHIIe (IIJIII paBHOBecne) KOTopofi paccMaTpiiBaeTcJJ, 6y.n:eM Ha3biB8Th .uexaHU'iecKOI'i cucme-
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BHyTpeHHIIe cHJibl o6.rra,n,aiOT c.rre.n:yiOlll.HMH csoilcTBaMII:
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B caMOM .n:e.rre, no TpeTbeMy saKoHy .IJ:IIHaMHKH JII06ble .IJ:Be TOqKII CHCTeMhl (piiC. 274) ,n,eikTBYIOT .n:pyr Ha .n:pyra C paBHbiMH ITO MO,ll,yJIIO II rrpOTIIBOITOJIOLKHO HarrpaBJieHHbiMII CIIJiaMH
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263
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§ 101. MACCA CHCTEMbl. I..I,EHTP MACC
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My,HaMH:
Jx=~mk(Yi+z%), Jy=~mk(z~+4), Jz=~mk(4+y%). (3)
4aCTO B XO.l(e paCIJ€TOB il0Jib3YIOTCH ITOHHTHeM pa.n:uyca HHepUHH. uHeplfUU TeJia OTHOCHTeJibHO ocu Oz Ha3biBaeTCH JIHHefmaH
Pz• orrpe.n:eJIHeMaH paBeHCTBOM
(4)
Macca TeJia. I1s orrpe~eJieHHH cJie~yeT, 'ITOpa~uyc HHepuuu
reoMerpuiJeCKH paBeH paccTo.HHHIO or ocu Oz TOM TOIJKH, B Koropoii Ha~o cocpe~OTOIJHTb Macey BCero TeJia, 'IT06blMOMeHT HHepUHH O~HOH 3TOH
TOIJKH 6b!JI paBeH MOMeHTY HHepUHH BCero TeJia. |
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3HaS1 pa~HYC HHepUHH, MO)KHO ITO |
cpopMyJie (4) HaHTH MOMeHT |
IiH€pUHH TeJia H Hao6opOT. |
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<f>opMyJihl (2) H (3) cnpaBe~JIHBbi KaK ~JIH TBep~oro TeJia, TaK H ~JIH |
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JII060H CHCTeMbl MaTepHaJibHb!X TOIJeK. B |
CJiyqae CIIJIOI!lHOrO TeJia, pa3- |
6HBaH ero Ha 3JieMeHTapHbie IJaCTH, Hafi~eM, 'ITOB rrpe.n:eJie CYMMa, CTOH-
IUaH B paBeHcrBe (2), |
o6paTHTCH B HHTerpaJI. B pesyJibTaTe, YIJHThlBaH, |
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'ITO dm=pd V, r~e p - ITJIOTHOCTb, |
a V- 06'beM, ITOJIY'IHM |
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(5) |
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J.1HTerpaJI 3~eCb pacrrpOCTpaHHeTCH Ha BeCb 06'beMV TeJia, a ITJIOTHOCTb p H paccToHHHe h saBHCHT oT Koop~HHaT TOIJeK TeJia. AHaJiorHIJHO cpop-
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Jx= ~ p(y2 +Z2) dV H T. ~. |
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(V) |
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cl>opMyJiaMH (5) H (5')y~o6HO ITOJib30BaTbCH rrpH Bbi'IHCJieHHH MOMeHTOB HHepUHH O~HOpO~Hb!X TeJI rrpaBH.'IbHOHcpopMbi. TipH 3TOM ITJIOTHOCTb p 6y~eT ITOCTOHHHOII H BbiH~eT H3-ITO~ 3HaKa HHTerpaJia.
Hafi~eM |
M 0 M e H T bl H H e p U H H H e K 0 T 0 p bl X 0 ~ H 0- |
p 0 ~ H bi X |
T e JI. |
1. T o H K H if o ~ H o p o ~ H bi ii c T e p * e H b ~JIHHoii l H
Maccoii M. Bhl.IJHCJIHM ero MOMeHT HHepunu OTHOCHTeJibHO ocu Az, rreprreH.l(HKYJIHpHoli crepMHIO II rrpoxo~HIUeii IJepes ero KoHeu A (puc. 275).
HarrpaBHM B~OJib AB Koop~HHaTHYIO ocb Ax. Tor~a ~JIH JII06oro 3JieMeHTapHoro oTpesKa ~JIHHhl dx BeJIHIJHHa h=x, a Macca dm=p1 dx,
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266
3aMeH5UI S,!l.ech p1 ero 3HaLieHueM, HaH)J,eM OKOHLiaTeJibHO
JA=M{2!3. |
(6) |
2. T o H K o e K p y r JI o e o ,11. H o p o ,11. H o e |
K o JI b u o pa- |
.n:uycoM R u Maccoil M. Hail.n:eM ero MOMeHT uHepunn OTHOCUTeJihHO
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PHC. 275 |
PHc. ?.77 |
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ocu Cz, rreprreH,!I.HKYJIHpHoti IIJIOCKOCTU KOJihUa H npoxo,!I.HIIJ,eil qepe3 ero ueHTp C (pHC. 276). TaK KaK BCe TOLIKH KOJibUa HaXO)J,HTCH OT OCII Cz Ha paccTOHHHU hit =R, TO cpopMyJia (2) .n:aeT
J c= ~ mkR2 = (~ mk) R 2 = MR 2 |
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CJie.n:osaTeJihHO, .II:JIH KOJihUa * |
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Jc=MR 2 • |
(7) |
OLiesu.n:Ho, TaKofi *e pesyJihTaT noJiyLIHTCH ,!I.JIH MOMeHTa HHepunu TOHKOH uuJIHHAPHLiecKoli o6oJIOLIKH Maccofi M u pa,11.nycoM R OTHOCH-
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ee OCU. |
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K p y r JI a H o A H o p o A H a H n JI a c T u H a |
uJIH u Ii· |
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p paAuycoM R H Maccoti M. BbiLIHcJiuM MoMeHT uHepuuii Kpyr- |
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IIJiaCTIIHe U |
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sToro KOJihua |
2nr ·dr, a |
Macca dm=p 2 2Jtr· dr, r)l;e p2 =M/nr2 - |
Macca |
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e.n:HHHUh! IIJIOIUaAu rrJiaCTIIHhL Tor)l;a rro cpopMyJie (7) )l;JIH |
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lc=MR 212. |
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(8) |
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* CpaBHHB3ll <flopMyJJbl (4) H (7) MO>KHO eru.e 3aKJJIO'IHTb, 'ITOpa,n:HyC HHepUHH |
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paBeH pa.D.HYCY TOHKOfO |
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C TaKHM >Ke OCeBb!M MOMeHTOM HHepUIIII, |
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267
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M)'JI(1) )!;JIH KOOp,IJ.H.HaT ueHTpa Mace ~mltxk=Mx(:. TaK KaK B HarneM
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TOt.IKa C HBJIHeTC.SI Hat.IaJIOM KOOp.IJ:H.HaT, TO Xc=O H, |
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loz=Jcz•+Md2 |
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(9) |
cl>opMyJia (9) Bbipa)KaeT CJie,IJ.yiOIII.YIO |
T e 0 p e M y f |
IO H r e H- |
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e a *: MOMeHm UHepupu meAa omHOCumeAbHO OaHHOU OCU paeeH MOMeHmy UHepl{UU omHOCUmeAbHO OCU, eu napaAAeAbHOU, npOXOOfltl.{eU ~J£pe3 lfl!Hmp Mace meAa, CAOJFCeHHOAty c npou38eaeliueM AtaCCbl ece~o meAa Ha K8aapam paccmoRHUJl MeJFCay oc.RMu.
l13 <PopMyJihi (9) BH.IJ:HO, 'ITO J oz>J cz•· CJie.n:oBaTeJihHO, us scex oceii .n:aHHOfO HanpaBJI€HH.SI HaHMeHblliHH MOMeHT HHepUHH 6y.n:eT OTHOCHTeJibHO TOH OCH, KOTOpaH npOXO.IJ:HT t.Iepe3 UeHTp Mace.
TeopeMa fiOtireHca II03BOJIHeT HaHTH MOMeHT HHepUHH TeJia OTHOCHTeJibHO .II:aHHOH OCH 0z1 H B TOM CJiyt.Iae, KOr.n:a H3BeCTeH ero MOMeHT HHepuuu OTHOCHTeJihHO Ato6ot1 ocu Az2 , napaJIJieJibHOil Oz1 • Dpu 3TOM Ha.n:o 3HaTb paCCTO.SIHIU! d1 H d2 Ka)K.IJ:OH H3 3Tl!X OCeH OT ueHTpa ,Mace TeJia. Tor.n:a, 3Ha.SI u d2 , Mhi no <PopMyJie (9) onpe.n:eJIHeM
a 3aTeM no TOH )Ke $opMyJie HaXO.IJ:HM HCKOMbiH MOMeHT HHepuHH J oz,·
3a)l.3'13119. Qnpe.IT,eJIHTb MOMeHT HHepuHH TOHKOrO eTep>KHli OTHOeHTeJibHO OeH
Cz, neprreH.IT,HKYJillpHoii cTep>KHIO |
H npoxo.u.mu;eii qepe3 era ueHTp Mace. |
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P e III e H H e. |
ITpose.rr.eM qepe3 Koneu A eTep>KHSI oeb Az (eM. pHe. 275; OCb Cz |
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Ha HeM He noKa3ana). Tor.rr.a no !PopMyJie (9) |
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lc=JA-Md2. |
B .rr.aHHOM eJiyqae d= l/2, r.rr.e l - .IT,JIHHa cTep>KHSI, a BeJIH'IHHaJ A orrpe.rr.eJISieTeli |
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¢opMyJioii (6). CJie.rr.osaTeJibHO, |
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J c= Ml2!3- Ml2j4 = Ml2j12. |
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3a)l.a'la120. Qnpe.IT,eJIHTb MOMeHT HHepUHH UHJIHH)I.pa OTHOeHTeJ!bHO Oell Azl> npo- |
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OCO)I.SIID.eii 'lepe3era |
o6pa3yiOID.YIO |
(eM. pne. 277, 6). |
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p e III e H He. |
no |
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fiOil:renca JAz,=lcz+Md2 • B )l.aHHOM CJiy'lae |
d=R, a no ¢opMyJie (8) |
lcz=MR 212. ITo,n:craBJIHSI 3TH 3Ha'leHHSI,rroJiy'lHM |
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J AZt =MR2!2+MR2 =(3!2) MR 2.
§ 104*. l.I,EHTPOSE)I(HbiE MOMEHTbl HHEPLI,HH. nOH5iTH5i 0 fJIABHbiX OCSIX HHEPLI,HH TEJIA
EcJIH t.Iepes TO'IKY0 nposecTH Koop.n:uHaTHhle ocH Oxyz, TO no OTHollleHHIO K 3'fHMOCHM t{eHmpo6eJICHblMU MOMe/ima.MU UHepl{UU (HJIH npoU38eOeHUJlMU uHepl{uu) HQ3bl8atom eeAUtiUHbl J xu• J yz, J zx. onpeaeA.ReMble paeeHcmeaMu:
J xy = ~ mkxkyk, |
J yz = ~ mkykzk, Jzx = ~ mkzkxk, |
( 10) |
r.n:e mitMaCChi TO'IeK; Xlt, YR.• ZitHX KOOp.IJ:HHaThi; npH 3TOM |
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Bli.IJ:HO, 'ITOJ xy=J yx H T • |
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* XpHCTHaH fiOiireHe (1629-1695)- Bbi,n:aiOID.HiieSI rOJIJiaH)I.eKHii y'leHbiii,MexaHHK, ljJH3HK H acTpOHOM. l1306peJI nepBb!e MaliTHHKOBble 'laebl.B eBSI3H e 3THM H3yqaJI KOJie6aHHH ljJH3H'IeCKOrOMaHTHHKa (eM.§ 129) H BBeJI IIOHliTHe 0 MOMeHTe HHepJ.UIH TeJia ({:aM TepMHH rrpe)VIO>KHJI n03>Ke 8iiJiep).
269
UJIH cnJIOIIIHbiX TeJI $opM)'Jibi(10) rro aHaJiorliH c (5')npHHHMaiOT
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B OTJIWlHe OT OCeBbiX U.eHTpo6eiKHhie MOMeHTbl HHepUHH MoryT 6biTb KaK IIOJIO:>KHTeJihHbiMH, TaK H OTpHUaTeJihHbiMH BeJIH'IHHaMHH, B qacTHOCTH, rrpu orrpe)l;eJieHHhiM o6pasoM Bbi6paHHbiX oc.Six Oxyz MoryT o6pa-
rn.aThC.SI B HYJIH.
f JI a B H bi e 0 C H H H e p U H H. PaCCMOTpHM O)l;HOpO)l;HOe TeJIO, HMeiOrn.ee ocb CHMMeTpHH. DpoBe)l;eM Koop)l;HHaTHhie ocn Oxyz TaK, qTo6bi och Oz 6biJia HarrpasJieHa B)l;OJih ocu cHMMeTpRH (puc. 279). Tor)l;a B CHJIY CHMMeTpHH Ka:>K)l;OH TQqKe TeJia C MaCCOH mk H KOOp)l;HHaTaMII x,, Ylt• ZR. 6y)l;eT COOTBeTCTBOBaTh TQqKa C )l;pyrHM HH,ZJ;eKCOM, HOC TaiWH
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HBJIHeTcH |
ochiO CHMMeTpiiH. |
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PHC. 279 |
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TeJIO, |
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HBJIHeTCH ITJIOCKOCTb abed). |
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DpoBe)l;eM B 3TOH rrJIOCKOCTII KaKne-Hn6y)l;h ocn Ox, Oz H rreprreH)I.HKyJIHpHyiO HM ocb Oy. Tor)l;a B cnJiy CHMMeTpHH Ka:>K.n:.of.I ToqKe c Macco1t mit H KOOp)I.HHaTaMH X~t, y,, Zit 6y)l;eT COOTBeTCTBOBaTh TQqKa C TaKOH :>Ke Maccof.I H KOOp)l;HHaTaMH, paBHhiMH xk, -y,, z". B pesyJihTaTe, KaK H
B rrpe)l;bi)l;YIUeM |
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J xy=0, J lJZ=O, |
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qTO OCb Oy HBJIHeTCH rJiaBHOll OCbiO |
HHepuun )I.JI.SI ToqKH 0. TaKHM o6pasoM, ecAu meAo u.Meem nAocKocmb ou.M.M.empuu, mo Ato6aR ocb, nepneHouKyARpHaR amou nAocKocmu, 6yoem C.Aa8HOU OCbiO UHepl{UU meAa 0//Jl mOf{KU 0, 8 KOmOpOU OCb nepeceKaem nAOCKOCmb.
PaseHcrBa (11) Bbipa:>KaiOT ycJIOBHH Taro, qTo OChOz HBJIHeTC.SI rJiaB· HoM: ocbiO HHepuHn TeJia ).I.JIH TOqKH 0 (HaqaJia Koop.n:.mraT). AHa.'loriiii-
270