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T. e. eCAU .MOMeHm aeuCtn6Ytot«eit CU/lbt OmHOCumeAbHO HeKOmopozo tfeHm· pa paseH HYAIO, mo .MOMeHm KOAUt-leCmea aeu:J!CeHUfl moll1W omHocume/lbHO &moeo WHmpa ecrnb ee/lulluHa nocmoJIHHaH. TaKoii pesyJihTaT IIMeeT MecTo B npaKTHqecKII BaJKHOM CJiyqae ABHJKeHIIH IIOA AeiicTBHeM ueHT· paJihHOH CHJihi (CM. § 86),
Puc. 224
§ 87. PAEOTA CI-IJ1bl. MOUJ,HOCTb
,UJIH xapaKTepHCTHK!I ,AeHCTBHJI, OKa3biBaeMoro Cl!JIOH Ha TeJIO rrpll HeKOTOpOM ero rrepeMemeHHll, BBO,A!ITCH IIOHHTHe 0 paooTe CliJlbl, ll!HpOKO liCIIOJib3yeMOe He TOJibKO B MeXaHHKe. CHaqaJia BBe,AeM llOHJITHe 00
3JieMeHTapHoli pa6oTe.
3/leMeHmapHoii pa6omou cHJibi F, |
npHJIOLKeHHoli |
B |
ToqKe |
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(pHC. 228), Ha3b!BaeTCH |
CKaJIHpHaH BeJiuqHHa * |
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dA=F,ds, |
(40) |
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r,Ae |
F't - |
rrpoeKUHH CHJihi F Ha KacareJibHyio M't K rpaeKTOprrH ToqKrr |
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HarrpasJieHHYIO B |
cropoHy rrepeMemeHHR 3Toli ToqKH |
(HJIH |
npo- |
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eKuHH F na HarrpasJieHHe cKopocTH vTOtJKH M); ds- MO.AYJih 3JieMeH- |
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rrepeMemeHHH |
ToqKn M. |
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orrpe.n.eJienrre |
coorsercTsyeT rrpe,rr,crasJieHHIO |
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0 Mepe TOfO JI.eHCTB!IH CHJibi, KOTOpoe |
IIpMBO,AHT K H3MeHeHHIO MO.AYJI51 |
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CKopocnr ToqKrr. EcJIH pa3JIOLKHTb CHJIY F Ha cocTaBJIHIOili,He F-r; rr Fn,
TO H3MeHHTb MO.AYJib CKOpOCTH |
oy.n.eT F't• TaK |
KaK F-r;=ma-r;=m ·dvldt |
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(cocraBJIHIOIUaH |
Fn |
H3MeHHeT |
HJIII HanpasJieHHe BeKTOpa v, HJIH rrpii |
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CHJIY |
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tJTO |
F-r;=F cos a, r.n.e a - yroJI |
MeLKJI.Y F H M't, noJiy- |
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qHM H3 (40) |
,Apyroe |
BbipaLKeHHe |
.AJIH dA: |
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dA =Fds cos a. |
(41) |
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EcJIH yroJI a |
ocTpbili, TO paooTa noJioLKnTeJibHa. B qacTnocrn, npH |
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a=O 3JieMeHTapHaH |
paooTa |
dA =Fds. |
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EcJIH yroJI a Tyrroil:, TO pa6oTa oTpnuaTeJibHa. B qacTHOCTH, rrpH |
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a= 180° 3JieMeHTapHaa paooTa |
dA =-Fds. |
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EcJIH yroJI a =90°, T. e. ec11u |
cu11a Hanpae/leHa nepneHauKyllRpHo ne- |
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pe.!.teU{eHuJO, mo B/leMeHmapHaR pa6oma cu,7bt paeHa HY/110.
3HaK paOOTbl HMeeT CJieJI.yiOJUHH CMb!CJI: pa6oTa IIOJIOLKIITeJibHa, KOf)J.a COCTaBJI5IIOJUaH ~ HarrpaBJieHa B CTOpOHy JI.BHLKeHIIH (CHJia ycKopHeT ,n;BHLKeHue); paooTa oTpuuaTeJibHa, Kor.n.a cocraBJIHIOIUaH
F 't uanpaBJieHa rrpoTHBOIIOJIOLKHO HarrpasJieHHIO JI.BHLKeHHH (cHJia 3aMe)J;JIHeT JI.BHLKeHHe).
EcJIH yqecTh, qTo ds= !dr\, r.n;e dr- se1nop 3JieMeHTapHoro rrepeMe-
memm TOqKH, li BOCI10Jlb30BaTbCH H3BeCTHbiM H3 BeKTOpHOU aJire6pbi IIOHHTHeM 0 CKaJI51pHOM npOH3BeJI.eHHH JI.BYX BeKTOpOB, TO paBeHCTBO
(41) MOLKHO npeJI.CTaBHTb B BHJI.e * ~ |
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dA=F·dr. |
(42) |
* 3AeC:b dA (KaK u"clS B § 83)- CHMBOJI SJie\leHTapHOH BeJIHqHHbl, HO He ]].Hpcpe-
peHUHa,'Ja• .Uu¢¢epeHUHaJioM KaKoli-HH6YAh (PYHKUHH Be.'IHqHHa dA Bo06lll.e MO)KeT He 6b!Tb (CM. § J26).
** CKamiphbi\1 npousBeJJ.eHneM AByx BeKTopoBa nb HashJBaeTCH cKaJIHpHaH BeJiuquHa, onpeJJ.eJIHCM3l! paBeHCTBOM a• b=ab COS a, fJ].e a - yroJI Me)KJJ.Y BeKTOpaMH a
H b Bblp31K€1-1He CKa.'IHpHororrpoil3Be)l,eHHll qepe3 IIpOCKUHH BeKTOpOB aH b Ha KOOp)l,H-
H3T!-lble OCH !!MeeT BRAa•b=a_.bx+ayby+azbz..
208
CJie,IJ;oBaTeJibHO, aAe.MeHmapHWl pa6oma cu!lbl paeHa cJWARpHo.My npouseeaeHUIO CU/lbt Ha eeKmop a!leMeHmapHoco nepe.MeU{eHWl moll1W ee npUAOJJCeHUfl.
EcJirr B ¢opMyJie (42) Bhrpa3HTb cKaJIHpHoe rrpoH3Be,IJ;eHrre tiepes
rrpoeKU.HH BeKTOpOB F li r Ha KOOp,D;HHaTHbie OC!I II ytJeCTb, tiTO rx=X, fy=y, fz=Z, TO IIOJIYtiHM aHa!lUmUiieCKOe BbtpaJJCeHue aAe.MeHmapHOU
pa6ombt |
|
dA =Fxdx+ Fydy+ Fzdz, |
(43) |
B KOTOpOM X, y, Z - KOOp,li;HHaTbl TOtiKH IIpHJIO}!{eHHH |
CHJibl F. |
1
Pnc. 228 |
Pnc. 229 |
PaooTa cHJihi Ha mo6oM KOHetiHOM rrepeMemeHrrH MoM1 (prrc. 228)
BbltJHCJIHeTCH KaK rrpe,D;eJI HHTerpaJibHOH cyMMbl COOTBeTCTBYIOlll,liX S>JieMeHTapHbiX paooT
(M,) |
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A(M,M,) = ~ Fr ds, |
(44) |
(M,) |
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HJIII |
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(1\1 ,) |
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A(M,M,)= ~ (Fxdx+Fydy+Fzdz). |
(44') |
(M,)
CJie,IJ;oBaTeJibHO, pa6oma CU!lbt Ha AI060Jrt nepeMeU{eHuu MoMt paeHa B3RtnOMY eaoAb amoco nepe.Meu~eHUH uHmecpally om &Ae.MeHtnapH.oll
pa6ombt. Dpe,D;eJibl liHTerpaJia cooTBeTcrByiOT sHatieHHHM rrepeMeHHhiX HHTerpHposaHHH B TOtiKax Mo D: Mt (TOtiHee rosop51, HHTerpaJI 6epeTC51
B,IJ;OJih KpHBOH MoM to |
T. e. 51BJIHeTCH KpHBOJIImell:HbiM). |
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EcJirr sem-FmHa Fe rroCTOHHHa (Ft=const), TO 113 (44), |
ooosHatJa51 |
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rrepeMemeHHe MoM 1 |
tiepes s11 rroJiytiHM |
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A(M,M,) = FTh· |
(45) |
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B tJaCTHOCTH, TaKOH cJiytJal! MO}!{eT HMeTb MeCTO, KOr,IJ;a ,IJ;ell:cTBYJOlll,aH CHJia IIOCTOHHHa ITO MO,ll;yJIIO II HarrpaBJieHHIO (F=const), a TOtJKa,
K KOTOpOfi IIp!IJIOLKeHa |
CIIJia, ,li;BH}!{eTCH IIpHMOJIIIHeHHO (pHC. |
229). |
B 3TOM cJiyqae F-r.=Fcos a=const H |
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A< tr, It,)= Fs1 cos a. |
(45') |
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E,IJ;HHHU.efl usMepeHHH paooTbr HBJIHeTcH B CH - 1 ,IJ;21WYJih (1 |
.U2K= |
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=lH ·M=l Kr ·M2/c 2), a |
B CHCTeMe .MKfCC- 1 Kf •M. |
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f p a |
cjJ |
H q e C K H ii: |
C II 0 C 0 6 B bl q li C JI e H li li p a 6 0 T bl, ECJI!i CHJI3 |
33BHCHT |
OT |
paccTOl!Hl'lllS |
H H3BeCTeH rpacjlliK 3aBiiC!iMOCTH f 'fOT S (puc. 230), TOp&- |
6oTy CHJibl MOIKHO BblqliCJIIiTb rpacjluqecKH. |
TiycTb B IIOJIOtKeHHH M 0 TOqKa H3XO,ll;HTCll |
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OT naqaJia oTCqeTa na paccToliHHH So, a B |
rroJiotKenuu M 1 - |
na paccTOliHiili s1. Tpr.D:a |
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no ¢opMyJie (44), yquTb!Ball |
reoMeTp!iqecKKii cMhlCJI |
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liHTerpaJia, |
rroJJyqKM |
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s, |
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Auu.M,>= ~ Fr; ds=a, |
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r,ll;e 0' - |
BeJJHquna 3aii!TpHXOB3HHOK |
Ha p!iC, 230 IIJIO• |
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UJ:3.1IH, YMHOIKeHHOK Ha M3CIIIT36HbiK |
KOScpcpH~HeHT. |
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Puc. 230 |
M o Ill, Hoc T h. |
MomHOCThiO |
Ha3hiBaeTcH |
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BeJmqrma, orrpe.D;eJIHIOUJ,aH paoOTy, cosepwae- |
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MYIO |
CHJIOH B e,!I,HHHUY BpeMeHH. EcJIH pa6oTa |
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coBepwaeTCH paBHOMepHo, TO |
MOili.HOCTh N =Ait1 , r,IJ;e |
t1 - |
BpeMH, B |
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Teqerme KOToporo rrpoH3Be)l.eHa paooTa A. B |
ooUJ,eM cJiyqae |
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N = dA!dt =Fe ds;dt = FT.v. |
(46) |
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CJie.D;oBaTeJihHO, MOU{HOCmb paeHa npou3eeaeHUIO KacamellbHou cocmae/lHIOU{eu cu/lbt Ha cKopocmb.
E.D;HHHuell: II3MepeHHH MOUJ,HOCTH B CI1 HBJIHeTCH eamm (1 BT = =l.U2K/c), a B cHcTeMe MKfCC- 1 Kf ·M/c. B TeXHHKe 3a e,IJ;IIHHUY MOUJ,HOCTH qacTo npHHHMaeTca 1 Jl. c., paBHaH 736 BT (HJIH 75 Kf ·M/c).
PaoOTy, rrpoH3Be,IJ;eHHYIO MarniiHOH, M02KHO H3MepRTh npoH3Be,IJ;eHIIeM ee MOUI,HOCTH Ha BpeMH paoOThi. 0rCIO,IJ;a B03HHKJia yrrOTpe6IITeJibHaH B
TeXHHKe e)l.HHHua II3MepeHHH pa6oThi KHJIOBaTr-qac (1 KBT ·q=3,6x
.il.2K~3671QQ Kf •M).
l13'paBeHcTBa N =FT.v BH.D;HO, qro y )l.BHraTeJIH, HMeiOmero ,IJ;aHHYIO MOUJ,HOCTh N. CHJia THfH F'T: oy.D;eT TeM OOJib-IDe, qeM MeHbUie CKOpOCTh v.
Do::tTOMY, HarrpHMep, Ha no.D;beMe HJIH Ha IIJIOXOM yqacTKe .D;Opom y aBTOMOOHJIH BKJIIQqaiOT IIH3lliHe nepe,D;a qu, ll03BOJIHIOUJ,He rrp H IIOJIHOH MOUJ,HOCTH ,ll,BHraThCH c MeHbUiell: CKopocTh!O H pa3BHBaTh 66.nhUIYIO CHJIY T5Irii.
§. 88. nPHMEPbl BbllfHCJJEHH~ PA60Tbl
PaccMOTpeHHbre mnKe rrpHMephJ.D;aiOT pe3yJihTaTh!, KOTOphlMH MO:lKHO
Herrocpe.n.cTBeHHO |
noJih30BaThCH rrpH peUiemm sa.n.aq, |
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1. P a 6 o T a |
c H JI hi |
T Sf >K e c T H. DycTh TOqKa M, Ha KOTopyiO |
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,IJ;eilcrsyer cHJia |
TH2Kecnr |
P, |
nepeMemae-rc& |
HS. noJio2Kem-IH M 0 (x0 , |
Yo, z0) B IIOJI02KeHHe M 1 (x1o Y1o |
Z1 ). Bbr6epeM |
KOOp,!UIHaTHh!e OCH TaK, |
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qTo6hr och Oz OhiJia HarrpaBJieHa BepTHKaJihHO ssepx (puc. 231). Tor.D;a p x=O, p y=O, p z=-P. Do,IJ;cTaBJIHH 3TH 3HatieHHH B <PopMyJiy (44'),
IIOJiyqHM1 yqHThiBa51, qro rrepeMeHHhiM HHTerpiipOBaHHH HBJIHeTCH z:
z,
A(M 0 M 1 )= ~ (-P)dz=P(z0 - Z1).
2'~
210
EcJIH roqKa M 0 Bhiiiie M 1 , TO z0- Z1 =h, r;I.e h - BepTHKaJihHOe nepeMern.eHne ToqKH; ecJIH 2Ke roqKa M 0 HH2Ke TOqKH M 1 , TO Zo-zl =::;
= - (zl-zo)=-h.
0KOHqaTeJibHO noJiyqaeM
A(M 0 M 1 ) =+Ph, |
(47) |
CJie,IJ;OBaTeJibHO, pa6oma cuAbl tmlJJCecmu paeHa 83flmo.My co 3Ha!WM nAIOC UAU ..MUHYC npou38eaeHUIO ..MoiJyAfl CUAbl Ha BepmUI«<JlbHOe nepe.Me- U{eHUe molltW ee npuAoJJCeHUfl. Pa6ora noJI02KHTeJibHa, ecJIH HaqaJibHaH TOqKa Bblute KOHeqHOH, H OTpHUaTeJibHa, ecJIH HaqaJibHaH TOqKa HUJ/Ce KOHeqHOH.
l-13 llOJiyqeHHOfO pe3yJlbTaTa CJie,IJ;yeT, qTO paooTa CHJibl TH2KeCTH He 3aBHCHT OT BH,IJ;a TOll TpaeKTOpHH, no KOTOpOH nepeMern.aeTCH TOqKa ee npHJI02KeHHH. CIIJihl, ooJia,IJ;arorn.ne TaKHM cBoHCTBOM, H83b!BarorcH nomeH4UaAbHbtMU (eM. § 126).
a} |
t |
Puc. 231 |
Puc. 232 |
2. P a 6 o T a |
c H JI bl _ y n p y roc T H. PaccMoTpliM rpys M, |
Jie2Karn.Hll: Ha ropH30HTaJibHOH nJIOCKOCTH H npHKperrJieHHbiH K cBo6o,IJ;- HOMY KOHUY HeKoropoll: npy2KHH:r.I (pHc. 232, a). Ha nJIOCKOCTH OTMe· THM TOqKOH 0 llOJI02KeHHe, 3aHl!MaeMoe KOHUOM npy2KHHbl, KOr,n,a OHa He HanpH2KeHa (A0=/0 - ,Il,JIHHa HeHanpH2KeHHOH npy2KHHbl), I1 npHMeM 3TY TOqKy 3a HaqaJIO KOOp,D;HHaT. ECJIH Terrepb OTTHHYTb rpy3 OT paBHOBeCHOro llOJI02KeHHH 0, paCTHHyB npy2KHHY ,li;O BeJIHqHHbl [,
TO rrpy2KHHa noJiyqHT Y,IJ;JIHHeHHe 'A=l-lo H Ha rpy3 oy,IJ;eT ,IJ;ell:cTBOBaTb
CHJia ynpyrocrn F, HanpaBJieHHaH K roqKe 0. TaK KaK B HarneM cJiyqae 'A=x, TO no ¢opMyJie (6) 113 § 76
F=c'A=clxl H Fx=-cx.
TiocJie,IJ;Hee paBeHcTBO cnpaBe,IJ;JIIIBO II npll x<O (rpys JieBee TOqKH 0);
ror,IJ;a CHJia F HanpaBJieHa BnpaBo H noJiyqHrca, KaK H ,IJ;OJIJKHO ob!Tb,
Fx>O.
Hall:,n,eM paoory, coBepwaeMyro cnJioll: ynpyroCTH npH nepeMern.e- HHH rpy3a H3 noJio2KeHHH M 0 (xo) B noJio2KeHHe M 1 (x1 ). TaK KaK B ,n,aH-
HOM CJiyqae Fx=-cx, F y=F z=O, TO, llO,IJ;CTaBJIHH 3Tll 3Ha'leHHH B
211
§ 89. TEOPEM.A 06 H3M.EHEHHH KHHETHIJECI(OA
9H EPfHH TOIJ KM
BBe)l.eM IIOHHTHe ern.e 00 O)I.HOH OCHOBHOH )I.HHaMHtieCKOH XapaKTepHCTHKe )I.BH}!{eHHH - o KIIHeTH'-leCIWH 3Hep rmr. KUHemwlecKOu EJHepeueu MamepuaAbHou motfKU HasbteaemcH CKaAHpHaH eeAutiuHa mv2/2, paeHaH noAoeuHe npou38eaeHUR Maccbt motLKu Ha KBaapam ee CKOpocmu.
EAHHHUa H3MepeHHH KHHeTH'-leCKOH 3Heprm-I Ta }!{e, tiTO H paoOTbi (B CI1 - 1 .U}!{). Haii)l.eM 3aBncH:O.IOCTh, KOTopoll. CBH3aHbl 3TH )!.Be BeJIH- ':IHHbi.
213
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