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)l,JIH 3H3JIIiTU'!eCKOroIJOJ].C'leTaMOIKHO, UCIJOJ!h3Yll rrepBb!e J].B3 H3 ypaBHeHHii (34), uaiim
Sx=mVo, Sy= -mvl, OTKyAa S=m v-v~+vi.
3a)l.a'la96. fpysy, RMeJOw;eMy Macey m u JietKalll,eMy na ropusoHT3JihHOii IIJiocKoCTH, coo6w;aJOT (TOJI'!KOM)ua'laJibHYIOcKopoeTbv0 I1ocJieAyJOw;ee ABULKeuKe rpysa TopM03HTCH rrocTol!HHoii. cuJioii. F. OrrpeAeJIUTh, qepes CKOJibKO BpeMeHu rpys oeTanoBUTCll,
mv(] |
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Puc. 222 |
Puc. ~23 |
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p e Ill e H li e. ITo J].aHHbiM 3aJ].3'!HBUJ].HO, 'ITO)l,Jill orrpeJ].eJieHUll BpeMellH ABUtKeHUll MO)KHO BOCIJ0.%30BaTbel! JJ.OK3331JHOH TeOpCMOH. J.-IJo6patKaeM rpys B I!pOH3BOJ!bHOM
no,,c;JKeHUii (puc. |
223). Ha Hero AeiicTBYIOT' cuJia THLKeCTii P, peaKUUH IIJIOCKOeTii N |
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u Tcp~I03llW:3ll |
cu.11a F. HarrpaBJillll ocb Ox B eTopouy JJ.BULKeHuH, eocTaBJIHeM nepBoe |
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us ypasueuliii |
(34) |
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(a) |
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B J].3HH01l CJiyqae V1 x=0 (v1 - |
CKopoCTh B MOMeHT OeTallOBKU), a V 0x=V0 • J13 |
e!lJI |
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rrpoeKUUIO na oeb Ox AaeT TOJihKO C!lJia F. TaK KaK oua noeTol!nHa, TO Sx=Fxf1= |
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=-Ff1 , rJ].e f1 |
- |
BpeMl! TOpMotKeHUll. I1o;:I.eTaBJillll Bee STU )J,aHHble B ypaBUeHHe (a), |
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IIOJiyqaeM -mv0=-Ff1 , OTKYJI-3 |
UCKOMOe BpeMH |
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t1=mv0 /F. |
(6) |
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TaKHM o6paso\I, BpeMH TopMotKeaul! paeTeT rrporropuuouaJihHO Ha'!aJibHOii.cKopoCTU.
Pe!IlHM STY IKe 3aA3'!Y,C'IUTall,'ITOTOpM03lll.q3ll CUJia paBH3 Q U He IJOeTOl!HHa, a e MoMenTa na'!aJia TOpMOLKeHul! paeTeT rrporropuuouaJihUO BpeMeHii, T. e. Q=kt, rAe k - HeKOTOpb!H rroeTOliHHb!H K03cjlcpuuueHT, H eT3HOBUTel! p3BHOH F B MOMeHT OCTaHOBKil rpysa. TaK KaK CHJia 33BHCUT OT BpeMeHU, TO OI!l!Tb MOIKHO BOCIIOJ!b30BaTbCH
ypaBneuHe1, (a), orrpe)J.eJIHH Sx no nepBOI1 H3 ¢opMyJI (31). Y'!Tll,'ITO Qx=-Q= = -kt, IIOJIY'IHM
t,
S.-,: = - ~ kt dt= -ktV2.
0
TorAa ypaaueuue (a) JJ.3CT mv0~kt~12. 3uaqeuue k |
uai'IAeMus ycJioBuH, 'ITOrrp!i |
f=t1 Q=F, T. e. kt1=F, on.y;.:_a k=F/11 H OKOH'JaTe,1bl!O6yAeT |
|
t1=-- 2mv0/F. |
(B) |
CJie;I.OBaTeJJbPO, B ::JT01I cJiyqae spewl TopMotKeH!Ill |
YABauBaeTCH, |
§ 85. TEOPEMA 06 H3MEHEHMH MOMEHTA
KOJJH'.JECTBA)J,BH)I{EHH}l T04KH (TEOPEMA MOMEHTOB)
B HeKoropbix 3a,n,aqax B KaqecrBe ,ll.HHaMHqecKofi xapaKTepucTHKH
)I.BIIJKeHHH TOl.JKH BM€CTO caMOrO BeKTOpa KOJIHl.JeCTBa )I.BIDKeHHH tnV paccMaTpHBa!OT ero MOMeHT OTHOCHTeJibHO HeKOTOpOrO UeHTpa HJIH CCII,
204
~H MOMeHThi orrpe,ZJ;eJIHIOTCH TaK me, KaK H MOMeHThl CHJihi (eM. § 8, 14
H 28).
TaKl!M o6pa30M, MOMeHmOM K0J7Uilecmea aeu:J/CeHUfl moliKU OmHOCU· me1tbHO HeKOmOp020 l{eHmpa 0 HCl3btBaemCH BeKmOpHafl BeAUf.WHa tno (mv),
onpeoeAHe.MaH paeeHcmeo.M |
|
mo (mv) = rX mv, |
(35) |
r)~e r - pa)I.IIYC·BeKTOp ABH)I{Yll.\eHCH TOqKH, rrpoBe,ZJ;eHHhiH H3 UeHTpa 0.
llpH 3TOM BeKTOp m0 (mv) HarrpaMeH rreprreH,ZJ;HKYJIHpHO ITJIOCKOCTH,
IIpOXO,ZJ;HIUefi qepe3 mV H ueHTp 0, a imo (mv)\ =mv ·h (pHC. 224; )I.JIH cpaBHeHHH Ha HeM IIOKa3aH H BeKTOp m0 (}')-rXF).
l\1oMeHT KOJIHqecTBa )I.BH)I{eHHH TOqKH OTHOCHTeJihHO KaKOH·HHOY,ZJ;h OCH Oz, rrpOXO)l;HIII,ell: qepe3 uemp 0, oyAeT paBeH rrpoeKUIIH BeKTOpa mo (mv) Ha 3TY OCh:
mz (mv) = [m |
0 |
(mv)]z = 1 0 (mv) 1cosy, |
(36) |
m |
r,ZJ;e y - yroJI Me*)l;Y BeKTopoM m0 (mv) H ochJO Oz.
TeopeMa MOMeHTOB ycTaHasmmaeT, KaK II3MeHHeTCH co BpeMeHeM
BeKTop m0 (mv). lho6hi ,ZJ;OKa3aTh ee, rrpo,ll;HqxpepeHuHpyeM no BpeMeHH Bblpa)l{eHHe (35). lloJiyqHM
:t (rX nW)= (~;Xmv) + (rXm ~n =(VX mV) +(rX ma).
Ho vX mv-0 KaK BeKTOpHoe npoH3Be)l.eHIIe ,ZJ;Byx rrapaJIJieJibHhtX BeKTOpOB, a ma=F, r,ZJ;e rrpH ,ZJ;eHCTBHH HeCKOJihKHX CHJI F='i.Fft.
CJie,ZJ;oBaTeJihHO,
d - |
- |
- - |
d - |
- |
- |
- |
(37) |
d[(r X mv) = |
r X F |
HJIH dt [m0 (mv)] = m0 |
(F). |
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B pesyJihTaTe |
Mhi |
)I.OKa3a.!fn cJie)l.yJOmyJO |
Teo p |
eM y |
MoM e H· |
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T 0 B 0 T H 0 C II T e JI h H 0 |
U e H T p a: npoU3BOOHaH no BpeMeHu om |
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J.WMeHma KOAuiiecmea aeu:J!CeHUH moliKU, 83Hmozo omHocumeAbHO KaKOzoHU6yob Henoaeu:JICH020 l{eHmpa, paeHa Mo.MeHmy OeiicmeyiOU{eU Ha mOiiKY CUAbt OmHOCUmeAbHO m020 :J/Ce l{eHmpa.
CpaBHHBaH ypaBHeHHH (37) H (32), BH)I.HM, qTO MOMeHThl BeKTOpOB mv
llF CBH3aHhl TaKOH me 3aBHCI1MOCThJO, KaKOH CBH3aHbl caMH BeKTOpb! mv
IIF.
ECJIH crrpoeKTHpoBaTh ooe qacTII paBeHCTBa (37) Ha KaKyJO·HIIOY,ZJ;h ochOz, rrpoxo,ZJ;Hll.\YJO qepe3 ueHTp 0, TO, y~H cooTHOilleHHe (36), rroJiy· llJIM
d |
- |
- |
(38) |
dt [mz (mv)] =- m., (F). |
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3'fo paBeHCTBO Bbipa)l{aeT meope.My .MOJteHmOB OmHOCUtne!lbHO OCU.
113 ypaBHeHHH (37) CJie)l.yeT, qTO eCJIH mo(F)=O, TO mo(mv)=const,
205