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4. BYJILUKeUKoB llrop& (r. Yc}>a) - JJ;UnJIOM 2-A CTeneRU
5. JIIOKCIOTOB HropL (r. KHeB) - ,lJ,lInJIOM 2-ii CTeneuu
6. KpaBD;OB BJI8,.lI;HMHp (r. rOpLKUii) - ,ZJ;HnJIOM 3-ii CTeneHH
PyKoBoJJ;HTeJIH KON8.HJJ;LI: B.A. HBepoHoBa, r .C. TapsclOK
V M«PO, BOJIrapHH 1971
1. BaPJI8.MOB A.HJU>eA (r. KlIeD) - .IJ;HWlOM I-A CTeneUH
2. A6pHKOCOB A.1IeKceii (r. MOCKBS) - ,AHnJIOM 2-A CTeneHH
3. BY,D;HHK Cepreii (r. )];uenponeTpoBcK) - rpaMOTB
6

4. CaJILAmIOHaC BUTac (r. BUJlbHIOC) - rpSMOTS
5. CHerupeB AneKC8.H,lJ;p (c. OCHBOBKB MorHJIeBcKoii 06J1aCTH) - rps-
NOTa
PYXOBOAHTeJIH KOM8.H,lJ;Ll: M.));. KapaceB, r .C. TapaCIOK
VIMO,PYMMHHH,1972
I. IIpoBOTopoB Cepreit (r. JIeHHHrpa..a;) - ,u;JlWIOM I-it CTeneHH
2. JIarymHu Cepreii (r. ));HenponeTpoBcK) - .n;HnJIOM 3-A CTeneHU
3. MKpTUIH Py6eH (r. EpeBau) - AHnJIOM 3-A CTeneHU
4. BaMau JIeB (r. JIeHHHrpB.,ZJ;) - rpaMOTa
5. IIJleTHeB HropL (r. MocKBa) - rpaMOTa
PyxoBo.n;HTeJIH KOM8.HALl: M. ));. KapaceB, r. C. TapaclOK
VII MO, IIoJILma, 1974
1. KypQ8.HOB AHaTOJIHH (r. :HCHTOMHp) - ,u;HnJIOM 3-it CTeneHH
2. PyAHeB AneKceii (r. MocKBa) - .n;HWIOM 3-A CTeneHH
3. MaclA Cepreii (r. HOBocH6HpcK) - rpaMoTa
4. HaBHit OJIer(r. KHmHueB) - rpaMOT8
5. 8.JILRUH EBreHHit (r. HOBocu6HpcK) - rpaMOTa
PYXOBO,lJ,UTeJIH KOM8.H,lJ;Id: O.. Ka6ap.n;HH, r .C. TapaclOK
VIII MO, repM8.HHH (r));p), 1975
I. KopmyuoB Cepreii (n. MOBHHO MOCKOBCKOH 06JIaCTH) - .n;IIWIOM
I-A CTeneHU
2. A:B.n;eeB JIeOHH.A (r. HOBOCH6HpCK) - ,lI;UUnOM 2-ii CTeneHH
3. maXHOBHtl EBreHuit (r. KaJIHHHBrpa,n;) - .n;HfiJIOM 2-ii CTeneHU
4. BoplO Ba,n;UM (r. Sanop0mLe) - .n;HWION 3-A CTeneUH
5. MaKe.n;oHoB K>pIDi (r.ICaJIHBHB) - ,u;unJlOM 3-A CTeneUH
PyxoBO.n;HTeJIH KOM8.H.n;hI: B.r. Pa3YMoBcKHH, r.c. TapaclOK
IX M«I»O, BearpHB., 1976
1. ByJlaToB BJI8,.lI;HNHp (r. JIeHHHrpa..a;) - ,lJ;UWION I-it CTeneUH
2. KpHBYH BJIa.ztHMHp (r. XapLKoB) - AHnJION I-A CTeneHH
3. XaMHToB HJlL,lI;ap (r. JIeHHHl'pa,n;) - ,lJ;HWION l-ii CTeneHH
4. rOJIy6eHlJ,eB peit (r. CapaToB) - ,lJ;HnJIOM 2-ii CTeneuu
5. CTapmeHKo B8JIepHit (r. SanOpO)KLe) - rpaMOTa
PyxoBoAHTeJIH KOMLI: 0.«1». Ka6ap,lJ;HB, r.c. TapaclOK
X MO, eXOCJIOB8KH.R, 1977
1. illapHnoB PYCJISH (n. KapaKYJlb ByxapcKoit 06JISCTU) - ,u;umrOM
l-ii CTeneHH
2. raaonoJILcKHA AH.zJ;peit (r. MUBCK) - ):tunJIOM 2-ii CTeneHH
3. PemMoB BJI8,.lI;uMHp (r. MocKBa) - ):tHWIOM 2.it CTeneUH
7

4. TpeTh.Jl1leUKO ROHCTaBTHH (r. RueB) - JJ;UlInOM 2-A CTeneHU
5. I.D:YXUH BJl8)I;HNHP (r. JIeuuurpa.n;) - AHnJlOM 2-A CTeneHU
PYKoBoAHTeJlH KOMaHAhI: O.«P. Ka6apJJ;HH, M.B. l»a6UJIeHKOB
XI M«PO, CCCP, 1979
1. nHH MaKCRM Cr. MocKBa) - ,lJ;UUJIOM I-A CTeneHH
2. IIInUJILKHH CepreA (r. MocKBa) - JJ;unJlOM I-A CTeneHR
3. IOYK OJIer (r. KHeB) - ARnJIOM I-A CTeneHU
4. .HCOBOS HropL (r. MocKBa) - ,lJ;HIIJIOM I-A CTeneHH
5. rOPAHeHKo CepreA (r. CMOneSHqu) - .D;HnJIOM 3-A CTeneHH
PYKOBO,lplTeJIU KON8.B,lJ;hI: B.A. OpJIOB, C.If. Ka6ap.D;HH8
XIIM«PO, BOJIrapHH, 1981
1. I)1THH An:eKC8.H)ij> (r. KJIHHItLI BpsmCKOA 06JIaCTu) - .D;HWIOM I-ii
CTeneUH
2. .D;epeBIIHKo BJI8AHCJI8.B (r. Kues) - AHnJIOM I-A CTeneUH
3. COJlOAOBHUKOB HJILR (r. MocKBa) - JJ;HnJIOM I-ii CTeneUH
4. MynmBCKuH AH.zu>eA (r. MUHCK) - AHIL7IOM 2-H CTeneHH
6. IIIy6eHHH HropL (r. Pura) - AHIIJIOM 3-A CTeneHH
PYXOBO,u;HTe.nU KOMaHJJ;hI: O.«P. Ka6apAHH, M.B. rpa6UJIeHKOB
XIII M«PO, repMB.HU$I (<<PPr). 1982
1. M8.KeeB HopHC (r. MOCKBB) - AHIIJIOM l-ii CTeneHU
2. IIauacIOK AneKCBU,ZI;p (r. OJJ;ecca) - AHIIJIOM I-A CTeneHU
3. eTKOB lInen (r. MocKBa) - ,u;HIIJIOM I-it CTeneHH
4. YXOB BJl8,ZI;HMHp (r. JIeHHHrpa.n;) - HnJlOM I-A CTeneHU
5. KaJI,D;a $1M (r. TaJIJIHH) - AUIIJION 2-A CTeIIeHH
PYKOBO,D;HTeJIU KOMaH: O.CP. Ka6apARH, M.B. l'pa6HJIeHKOB
XIV M«PO, P}TMLIHU.II, 1983
1. AneKceeB AuTOR (r. JIeBHHrpa.n;) - AUUJIOM I-it CTeneHH
2. IHHJIOBCKoA AHApeA (r. JIeHHHrp8.,ZJ;) - JJ;UIIJION l-ii CTeneUH
3. MOJI1laHOB BJI8,AHNHp (r. KHeB) - AUnJIOM I-A CTeneHH
4. .D;h.fltlKOB Mux8.RJI (n. qepHOrOJIOBKa MOCKOBCKOH 06JIBCTU) - JJ;BU-
JlOM 2-A CTeueHH
5. BUP3BaJlKC B8.JI)J;uC (r. Pura) - AUUJIOM 3-A CTeneHU
PYXOBo.o;HTeJIH KOMLI: O.«P. Ka6apAuH, B.A. Op.nOB
Ha6JIIOABTeJIh OT CCCP - C.C. KpOTOB
XV McpQ, IIIBeI:\H.II. 1984
1. A.1I:eKceeB AHTOH (r. JIeHHHI'p8.,ZJ;) - AHIIJIOM I-A CTeneHH
2. JJ;eIIIKoBcKmt A.1I:eKCB.B)UJ (r. BapaHoBWlH) - .D;HIIJIOM: I-ii CTeneHH
3. OPJlOB CepreA (r. MocKBa) - .o;unJIOM I-A CTeueHH
8

4. BaKpeBcKHii JIeB (r. MURCK) - .IJ;UnJIOM 2-ii CTeneHH
5. IIoTepsdiKo IfropL (r. KHeB) - .IJ;HnJlOM 2-A CTeneHH
PYKoBo.IJ;HTeJIH KONaIlALI: 0.«1>. Ra6ap.IJ;HH, B.A. OpJIOB
Ha6JIIO.IJ;aTeJIb OT CCCP - B.A. OpJlOB
){VI,}DroCJIaBHS, 1985
1. BapaLIKHH BHRTop (n. ':IepHOrOJIOBR8 MOCKOBCROii 06JIaCTH)-
.IJ;UnJIOM I-A CTeneHH
2. HB8HeHKo Tapac (r. KHeB) - .D;UnJIOM I.ii CTeneHH
3. rpHropbeB reopruH (r. KueB) - ,rJ;HWIOM l-ii CTeneHH
4. mecTKoB IOpHii (r. AnMa-ATa) - AUnJIOM 2-ii CTeneHH
5. qepn OJIer (r. MUHCK) - AHnJIOM 3-ii CTeneHH
PYKoBoJJ;HTeJIH KOMaH.IJ;.hI: 0.«1». Ka6apJJ;uH, B.A. OpJIOB
XVII M4>O, BeJIHK06pHTaHH.R, 1986
1. BOJIKOD OJIer (r. rOphKHii) - .IJ;UnJlON I-A CTeneHH
2. MaThIuH AH.n;peii (r. MocKBa) - ,l];HnJlOM l-ii CTeneBH
3. Mgf'tlHJIOB Cepreii (r. OJJ;ecca.) - .IJ;unJIOM 1-:8 CTeneHU
4. ryHH AHApeH (r. HOBOCH6upCK) - AHnJIOM 2-ii CTeneHH
5. HUKOJISUWBHJIH rpuropHii (r. T6HJIHCH) - .IJ;HnJIOM 3-ii CTeneHH
P}TROBO.IJ;HTeJIU ROMa.H.IJ;LI: O.w. Ks6apAHH, B.A. OpJIOB
XVIII M«I>O, r]J;p, 1987
1. Bu6uKOB AHToH (r. MocRBs) - .IJ;unJIOM 2-ii CTeneHH
2. BY1J;hKO .nMHTpHH (r. BeJlropo.n;) - .IJ;HnJIOM 3-ji CTeneHU
3. rOJILJJ;HH ArreKceii (r. KlieB) - AHnJIOM a-ii CTeneHH
4. BHJIhKJIJIHC rHHTftC (r. BHJILHIOC) - rpaMoTa
5. rJIyeHKoB .nMUTPUH (r. JIeHuHrp8J() - rp8.MOTS
PyKoBoJ];HTeJIH ROM8H)J;LI: 0.«1>. Ka.6ap.IJ;UH, B.A. OPJIOB
XIX M«I>O, ABcTPHs, 1988
1. Kpas'leHKo IOpH:8 (r. MOCKBa.) - ,lJ,UnJIOM 2-:8 CTeneHH
2. MaaypeHKo A.1IeKcaHJij> (r. MUHcK) - .IJ;HnJIOM 2-H CTeneHH
3. neHsHeH KOHCTftHTUH (r. O.n;ecca) - J];unJIOM 2-ii CTeneHii
4. M8JlRUH AHTOH Cr. JIeHHHrpB.ZJ;) - .IJ;UnJIOM 3-ii CTeneHH
5. Mopo3 B8J(HM (r. KueD) - .IJ;HnJJOM 3-A CTeneHIi
PYKoBO}tHTeJIH KOMaH)J;ld: 0.«1>. Ka6ap.IJ;HH, B.A. OpJIOB
XX M«I>O, nOJILWa, 1989
1. KY3LMa HHKOJIaii (r. IIpoTB8 KaJIymcKoH 06JIftCTH) - Jl;HnJJOM 1-ii
CTeneHU
2. KOPWKOB .A.11:eRcaHp (r. M03LIp:b) - .IJ;HnJIOM 2-ii CTeneHH
9

3. ByeB KOHCT8.HTHH (r. BoJIor,lJ;a) - ,lJ;unJIOM 3-it CTeneBH
4. MSKees BJI8JJ;HCJI8B (r. AJIMs-ATa) - rpSMOT8
5. YBapoB IOpmt (r. JIeHDHrp) - rpSMOTS
PYKoBOAHTeJIH KOM8.H)];hI: C.C. KPOTOB, B.A. OPJIOB
XXI MCPO, rOJIJI8H,lJ;HH, 1990
1. KysltMeuKo B8,l1;HM (r. HB8HO-tPPSHKOBCIC) - AunnOM I-H CTene-
HII
2. HB8HOB BaCHJIHi:i (r. Tyns) - ,D;HnJIOM 2-A CTeneHU
3. qOKHH .n;Hap (r. A.n:Ma-ATa) - .lI;HIIJIOM 2-A creneHB
4. mHHICeBII'I Cepreit: (r. Bepe3HHKH llepMcKoH 06JISCTU) - ,lJ,HnJIOM
2-H CTeneUH
5. 9BTHH MnXaHJI (r. TYJI8) - ,lJ;HfiJIOM 3-it CTeneHH
PYKoBoHTeJIH ICOl\ll8H,l];hI: C.C. KpOTOB, A.P. SHJIh6epMBH
XXII MtPO, Ky6a, 1991
1. BamHHcKHit: Cepreit (r. CTepJIHTBl\II8K) - HnJIOM I-it CTeneHH
2. JI.HIIHH An:eKC8B,D;p (r. HaJIh'qIIK) - .n;HIDION I-A CTeneHH
3. myreuKo THMyp (r. MapHyIIoJIh) - .n;HWIOM l-ii CTeneHH
4. .n;06pOBOJIhCKHH Cepreii (r. .n;ueuponeTpoBcK) - ,ltHUJION: 3-A CTe-
neHH
5. TSMomlOusc BUHo;ac (r. BHJIbHIOC) - .n;HnJIOM 3-H CTeneHH
PyKoBoJ(BTeJIH KOM8B,lJ;hI: C.M. K03eJI, A.P. 3HJIh6epMaa
Ha6JIIO.n;sTeJIh OT CCCP - O. tP. Ka6ap.n;HH
XXIII MfI)O, cI>uHJlHH,lJ;IUI, 1992
1. tPpOJIOB A.HJq>eit (r. MOCRBa) - ,ZJ;HWlOM I-A CTeneHn
2. BeapYRoB «Pe.n;op (r. MocRBa) - .n;HnJlOM I-H CTeneUH
3. ryJISIeB HHKOJIaii (r. HumHHit HOBropo.n;) - ]J;unJION I-a CTeneHH
4. IIaHKOB Cepreit (r. TYJIa) - AHWIOM 2-H CTeneHH
5. Ap6sTCKHH )tMHTpnit (r. CaHKT-neTep6ypr) - HnJIOM 3-A CTene-
HU
PYKOBOHTeJIH KOMaH)J;LI: A.P. SUJlh6epM8H, C.C. KpOTOB
XXIV McI>O, CIlIA, 1993
1. BeJIeHOB POMBH (r. HHmHHH HOBropO.n;) - ,lJ;UllJION I-A CTeneHH
2. JIapLKHH BaCHJIuA (r. IIpaB,D;HHcK HU)f(erOpO,ll;cKOH 06JI8CTH)-
,lJ,HnJION I-H CTeneHH
3. AH,lJ;peeHKO CTen8.H (r. MocKBa) - .lI;HnJIOld I-it CTeneHH
4. HrHpeB ));eHHc (r. C8HKT*IIe-rep6ypr) - HIIJIOM 2-H CTeneHH
5. KHCJIOBCKHH )J;eHHc (r. CaHKT-IIeTep6ypr) - HnJIOM 2-ii CTeneHH
PYKoBoHTeJIn I<OM:8H,ll;W: C..M. K03eJI B.A. KOpOBHH
10

XXV MCPO KHTaA 1994
1. C%>H.1IHnnOB BaCHJIHA (r. CB.HKT-neTep6ypr) - HWION 3-:n CTeneUH
2e .n106mHH .n;NHTPHA (r. IlepNL) - rpaMoTB
3. CTpaToHHHKoB MeRcei:i (r. CHCbCTpoii JIeHHHrpa.n;cKoi:i 06nacTH) -
rpa.MOT8
4. IPam;eHKo Ceprei:i (re BapHayn) - rpaMOTa
5. KypamoB )J;MHTpHH (r. MocKBa) - rpaNOTa
PYKoBoJJ;UTeJIH KON8.HALI: C.M. K03eJI BeAe KOPOBHH
Ha6JIIO,o;BTeJIL OT PC%> - BeA. OpJIOB
XXVI MCP07 ABCTp8.JIH.R 7 1995
1. CU6Up.HKOB Cepreit (r. MocKBa) - .D;HnJIOM i-A CTeneHH
2. CaBeJILeB APTeM Cr. C8HKT-IIeTep6ypr) - AHnnON 1-A CTeneHH
3. 3eneaCKHH HJIb.H (r. HH)KHHH HOBropo,lJ,) - AHnnOM 2-A CTeneHH
4e KY3He:Q;oB A.1IeKCaHJJ;p (re TYJIa) - ,u;HnJIOM 2-H CTeneaH
5. KamMeHCKHA EBreHHA Cr. HOBOCH6HpCK) - .D;HnJIOM a-A CTeneHH
PYKOBoAHTeJIH KOM8BAhI: C.M. K03eJI BeA. KOPOBHH
Ha6JIIOABTeJIb OT PCP - B.A. OpJIOB
XXVII MCP07 HOpBerH.R 7 1996
1. BacHJIbeB .n;NHTpHH Cr. RHpOB) - AHUJIOM.l-ii CTeneHH
2. TapBcoB EBreBuii (r. C8.HRT-I1eTep6ypr) - AHWIOM 2-H CTeneHH
3. MHJIH:Q;HH BJI8.n;HMHp Cr. MOCKBR) - .D;HWIOM 2-ii CTeneHH
4e HB8HOB I1aBeJI (r. HH)KHHH HOBropo.u;) - ,lJ,HUJIOM 2-i:i CTeneHH
5e CPOMHH EBreHHfi (r. BepOOHHKH IIepMcKoA 06JIBCTH) - AHMOM 3-it
CTeneHH
PYKoBoJJ;HTeJIH KOMaHALI: C.M. K03eJI B.Ae KOpOBHH
Ha6JIIO,u;aTeJIH OT PC%>: 0.10. OBTlHHHHKOB B.Il. CJI060,u;SIHHH
XXVIII MCP07 R8H8.ZI;B 7 1997
1. ROmeM.RK A.1IeKceii (re CaHKT-neTep{iypr) - AHUJIOM I-ii CTeneHH
2. qYBHKOB A.1IeKcei:i (r. HO.R6pbCK) - AHnJIOM I-A CTeneHH
3.. lleCTYH BacnJlHA (r. CaSKT-neTep6ypr) - JJ;HnJlOM I-it CTeneaH
4. BOpoHOB Ap'reM (r. QeJIH6uHcR) - AHWION I-'A CTeneHH
5. MaKapoB A.1IeKce:n (re CeprHeB-nOca,.II; MOCKOBCKOH 06JIBCTH)-
AHnJIOM 2-A CTeneHH
PyxoBOAHTeJIH KOMbI: C.M. K03eJI 7 B.A. KOpOBHH
Ha6JIIO.n;aTeJIh OT pcp - B.A. OPJIOB
XXIX MCP07 HCJIBHAHJI 7 1998
1. A68HHH .l{MHTpHit (r. POCTOB) - ,EUiIIJIOM }-ii CTeneHH
2. HMaM6eKoB A.n;HJIeT (r. MocRBa) - ,D;HIDIOM l-ii CTeneHH
11

3. Py6D;OB rpnropH:A (n. qepHOrOJIOBKa MOCKOBCKOA 06JIaCTII) - ,lJ;un-
JIOM 1-ii CTeneHH
4. BaphIrlIH HJIL.1I (r. CB..HKTneTep6ypr) - ,D;HllnOM 2-ii: CTeneHH
5. TypHItHH KOHCT8UTHH (r. HOBocu6npcK) - ,lJ;:B:nJIOM 2-H CTelleHH
PYKoBO,D;HTeJIH KOMaH,lI,hI: C.M. K03eJI, B.A. KOpOBHH
Ha6JIIO,o;aTeJIL OT pcp - B.A. OpJIOB
XXX M«I>O HT8JIIUI, 1999
1. Kp8.BOB KOHcTaHTHH (r. MocKBa) - ,[tHIIJIOM 1-H CTeneHH
2. CLIpHlthIH CepreH (r. CapaToB) - ,lJ;HllJIOM I-A CTeneHH
3. 'lY,D;HOBCKHii: A.1IeKCaH,ZI;p (r. QeJI.H6HHcK) - AHllJIOM l-ii CTeneHH
4. IIaHoB EBreHHH (r. qeJIH6HHcK) - ,lJ,HllJIOM I-H CTeneHH
5. IIoJIJIHcKHAlOpHA(r. P8JI;ymHbIH BJI8,lI;HMHPCKOH o6JI.) - ,lJ;HllJIOM
2-A CTeneHH
PYKoBO,D;HTeJIH KOMaH;I];hI: C.M. K03eJI, B.A. KOpOBHH
Y CllemHbIe BhICTYnJIeHH.R COBeTCKIIX II poccHAcKHX mROJIbHIIKOB
a8 Mem.o;ynapo,n;HhIX 4JH3HQeCRHX OJIHMnH8JI;8X 06eCneQHBftIOTC.H BbI-
CORIIM ypOBHeM eCTecTBeaHo-HayqHoro o6pa30B8HHH B cpe.n;Heii mKO-
JIe CHCTeMoA pa60ThI C 0AapeHHLlNH mKOJILHIIK8MH aa 4J8.KYJILTaTHB-
HbIX 38HSlTHBX II B KJI8CCaX C yrJIy6JIeHHbIM Jl3yqeaHeM <pH3HKH, a T8R-
me aB p83JIH'tJHLIX ST8nax BcecolO3Hoii: (}>H3H"ieCKOA OJIHMnHa,D;L1,
KOTOp8H npoBoAHTCH C 1967 r.
BcepoccncK8H (}>H3II'tJecK8..H OJIHMllHa.z:r;8 npOBo,nIITC.H B ll.flTL ST8-
nOB.
UepBLIA STan - WKOJILHhIA. B HeM npHHHM8.IOT yqacTHe Bce me-
JI8JOHe yqeHHKH VII - XI RJI8CCOB, HX o6ee QHCJIO COCTaB.nHeT 60-
.nee MHJI..JI:HOH8 WKOJILHHKOB.
BTopoA 3Tan - paAoHHhIH (ropo,o;CKOii:). B paHoHHLIx OJIHMUH8JI;8X
yq:aCTBYJOT n06e,n;HTeJIH WKOJIbHbIX OJIHMllH8,lJ; no OAHOMY OT KJI8CC8,
-qTO COCTaBJIHeT npHMepHO 200 ThIC. lliKOJIbHHKOB.
TpeTHH STan - 06JIaCTHhIe OJIHMllE8,AhI. B HHX aCTBYIOT KOMBB-
,lJ,hI mKOJILHHKOB, ccl»oPNHpoB8.HHhIe H3 '1HCnD n06e,D;HTeJIeii p8iioHHhlX
H rOpO,lJ;CKHX OJIHNnHa.z:r;. 06x:qee 'tIHCJIO yqacTHHKoB SToro STana OJIHM:
llH8)J;LI OKOJIO 10 ThlC.
qeTBeprLIH STan - 30HaJILHbIe OJIHMllH8,lJ;hI, B KOTOpLIX }"{aCTBY-
lOT n06e,lJ,HTeJIH 06JI8CTHLIX OJIHMfiH8,A H n06e,o;HTeJIH KOHKypC8 myp-
aa.n:a .KBaHT". B STOM STane npHHHMalOT y-qaCTHe npHNepHO 500
WKOJILHHKOB.
II.HTLIH STan - 3aKJIIO'IHTeJIbHhIA. B HeN BhICTynaIOT KOM8.H,U;LI OT
RIDK,D;OH 30HhI POCCHH. 06ag "lHCneHROCTL acTHHKoB SToro STann
OJIHMnU8.,ZI;hI OROJIO 150 'tJeJIOBeK.
12

IIpoBe.n;elmeM OJIHMIIH8,lJ; us BCeX ee 3TanaX pyKOBOP;.RT OpraHbI HS-
pOAHoro 06pa30B8HHH.
,l];nnJIOMaHTLI I-A II 2ii: CTeneHH BcepoccHiicKOH OJIHM:nH8,lJ;LJ cpe-
An IIIKOJILHHKOB X KJIaCCa npHrJIamaIOTCH H8 ABYXHe.n;eJILHLIe JIeTUHe
TpeHHpOBO'IHLle C60PbI, a 38TeM Ha .z:r;ecHTH)J;HeBHLIe oT6opO'llHLIe c60-
PbI, npOBO,lJ,HMhIe B P;HH 3HMHnx KaHHKYJI. K8H,D;n.n;8TaM B KOMRH,ll;Y
POCCHH paCCLIJIaJOTCH P;J1H C8.MOCTOHTeJThHOii pa60ThI 38,lJ;8HIUI, ytIHTbI-
BaIOIIJ;He cneHc}>HKY Mem.n;YHapOJJ;HbIX OJIHMTIH8,lJ;. IIo HToraM JIeTHHX
H 3HMHHX C60POB H BLlcTynJIeHHH HX yqacTHHKOB HR 38KJIIO'IHTeJI&HOM
STane BcepOCCUHCKOH c}>H3H'tIeCKOH OJIHMnn8,lJ;hI B XI RJIRCCe KOMnJIeK-
TyeTcH COCT8B yqacTHHKoB ,lJ,JIH yqe6Ho-TpeHHpOBO'tlHLIX C60POB, npo-
BO,lJ,HMhIX B Te'lleHHe O));HOrO Mecsm;a nepe)]; H8'tlRJIOM Mem,lJ;yuspOJJ;HOll
c}>H3H'tlecKoA OJIHMllHBJJ;L1.
Ha 3THX c60pax npoBOJJ;HTCH TeOpeTl{'QecKaH H 3KcnepHMeHT8JI&H8H
nOArOTOBK8 no nporpaMMe Mem,lJ;YHapO,lJ,HOR cPH3H'IIeCKOii OJIHMnH8AhI
H <POPMHPYeTCH OKOIN8TeJIbHLlH COCTaB C60PHOH KOMaH.n;hI POCCHH.
Ha npOT.aaceHHH p.a.n;a JIeT no.v;rOTOBKa KOM8n,lJ; COBeTCKHX II POC-
CHHCKHX IIIKOJI&.HHKOB npoBO,lJ,HJIaC& aa cPH3H'tleCKOM <p8.KYJlLTeTe Mry 
Ha 68ae sKcnepHMeHTaJILHOH cpe,lJ;Heii IIIKOJILI M 82 B n. qepHOrOJIOBKa
MOCKOBCKOH 06JI8CTH, B ,l];oMe TBOp'llecTB8 r. MOCKBhI, 8 B nOCJIeAHHe
ro.n;LI - Ha 6a3e M«PTII. TIO,n;roTOBK8 yqSCTHHKOB KOM8HA 06ecneQHB8-
nac& qJleUSMH MeTo.n;nqeCKOH KOMHCCHH BceCOI03HOH (a 38TeM Bcepoc-
cHitcKoii) <pH3HTIecKoA OJIHMnH8,lJ;hI WKOJIbHHKOB. H8n60JIee 8KTHBHoe
yqacTHe B SToA pa60Te npHHHN8.JIH JI.II. BaKaHHH8B.E. BeJIOHyq-
KHH, A.P. 3HJIb6epMaH, O.. Ka6ap.t.tHH, C.M. KoaeJI, C.C. Kpo-
TOB B.B. MomaeB, 0.10. OBqHHHHKOB B.A. OPJIOB, IO.A. C8.MapCKHii,
B.II. CJIo60ASlHHH, H.llI. CJI060AertKHH, A.JI. CTaceHKo E.JI. CypKOB,
B.II. HBHJIeB, B.A. IDapoHHH H p.

XVIMeYHGpOAHGg
cl»M3M'IeCKGII OnMMnMGAG
1985 r _, IOrocnaBHII
3a.n;aQa 1
IOHbIH pa.,n;HOJII06HTeJIb nO,lJ,,lJ,epmHBaeT pa,u;HOCBH3b C ,lJ,BYMH ,lJ,eBym-
KaMH npOmHB81OII(HMH B pa3HbIX ropo,u;ax. OR HaMepeR CKOHCTPYH-
pOBaTb TaKYIO CHCTeM:Y aHTeHH, KOTOpa.H n03BOJIHJI8 6hI eMY paarOBa-
pHB8Tb C O,u;HOH H3 ,lJ,eBymeK ()KHByeii B ropO,Ae B) C OrrTHMaJIbHblM
Ka'lleCTBOM: CBH3H, HO TaKHM o6pa30M, 'tlTo6bI BTOp8.H ,lJ,eBynIKa (mHBY-
a.H B ropo,n;eA) HX paaroBop CJIbImaTb He MOrJIa H Ha06opOT. CHCTe-
Ma aHTeHH co6npaeTc.H H3 )J;BYX BepTHKaJIbHbIX aHTeRH, H3JIyqalOII(HX
C O,lJ;HHaKOBOH HHTeHCHBHOCTblO no BceM ropH30HTaJIbHbIM HanpaBJIe-
HIIIIM.
Onpe,lJ,eJIHTe M:HHHMaJIbHOe paCCTOHHHe r Mey 8.HTeHRaMH yrOJI
\110 Mem,lJ;Y llJIOCROCTblO, llpOXO,lJ;Heii qepe3 06e 8HTeHHbI, H HanpaBJle-
HHeM Hft ceBep H paaHOCTb «}Ja3 A SJIeKTpHqeCKHX CHrH8JIOB H3J1yqae-
MhlX aHTeHHaMH.
PemHTe 38,Aaqy B cJlyqae eCJIn IORoma HMeeT pa.zr;HOCTftRnlO pa-
60TalOII:\yJO Hft qaCTOTe v;:;;- 27 MrIJ;, a CHCTeMa aHTeRH YCTaHOBJIeRa B
IIopTopome.
HCnOJIb3YH KapTY IORoma BhIHCHHJI, 'lITO yrJlhI Me)K)J;Y HanpaBJIe-
HHeM Ra ceBep H HanpaBJleHH.HMH Ra ropo,lJ,aA (Konep) H B (EYRe) CO-
CTaBJI.HIOT \II. ;:;;- 72 0 II \112 = 157 0 COOTBeTCTBeHHO.
3aJ];aqa 2
B ,lJ,J1HRHOM: 6pycKe, HMelOII(eM «}JOPMY pa3HOCTopOHHero napaJlJle-
J1enHne.n;a (a > b » c) n C,lJ,eJlaHROM H3 nOJIynpOBO,lJ,HHKOBOro MaTepna-
J1a InSb, TeTleT TOR I B HanpaBJIeHHH pe6pa a. EpYC.,pK Haxo,u;HTCH B Mar-
HHTHOM nOJIe HanpaBJIeHHe BeKTopa HH,lJ,YKIJ;HH B ROToporo COBll8..Zl;a-
eT C pe6poM c. IIepeHocTlHKH TOKa B InSb - 9JIeKTpObI KOTopbIe
)1;BHmYTCH B ::meRTpHqeCKOM nOJIe C Hanp.HmeHHOCTLIO E co cpe,lJ,Heii
CKOpOCTLIO v = uE (KosqxpHIJ;neHT U Ha3bIBaeTCH no.n;BHmHOCTLIO SJIeK-
TpOHOB). IIpH HaJIH'tlHH M8rHHTHoro nOJIS! Heo6xo,lJ,HMO yrIHTbIB8Tb .neii-
CTBHe CHJIbI JIopeH:Q:a aa 3JIeRTpOHhI - HanpaBJIeHHe BeKTopa Hanp.R-
14

meHBOCTII B STOM CJIyqae se 6YJJ;eT nap8JIJIeJILHO sneKTplfllecRON:y TOKY.
&rOT c»eBOllleH H3BecTeH K8.K a4>4>eRT XOJIJI8.
1. Onpep;eJIHTe 1II0AYJIL II HRnpaBJIeSlle BeKTOpa HanpSlmeHHOCTH
SJIeKTplflleCKoro nOJls B 6pycKe.
2. BLftlHcJlHTe paaHocTL nOTellItH8JlOB lIIemAY npoTHBOnOJIOmHLI-
MH TO'qKaMII sa nOBepxHocTH 6pycKa B HanpaBJleHHH pe6pa b.
3. IIoJIytlHTe 8H8J1UTHQeCKOe BLIp8meHHe AJISI nocTosHHoit RON-
nOHeSThI pa3HOCTH UOTeHItH8JlOB, Haii,n;esHoA B n. 3 ecJlH cUJla TOKa II
HH,z:I;YKD;HSI NarHHTHoro nOJIa MeS$IIOTC.H no 3aROHaM: I :::;:: 10 sin rot,
B:::;:: Bo sin (mt + cp).
4. CrrpoeKTupyATe (II 06'bSlCHBTe) SJleRTpINecKyJO nenL, B KOTOpOii:
B03NomHO HCrrOJlL30B&HHe peaYJILTaT8 3 1J,JISI H3NepeHHSI MOJIJ;HOCTH,
nOTpOOJISIeNOH SJIeKTplflleCKHM annapaTON, BKJIIOQeHHhIM B CeTL ne-
peNeBBoro TOKa.
HCrrOJIL3yA:Te TRKHe ,lI;aHShIe: nO,ll;BUmHOCTL 3JIeKTpOHOB B InSb
u :::;:: 7,8 1\12/(B . c); KOHD;eHTpan;H.H 3JIeKTpOHOB n = 25 . 10 22 M- 3 ;
I:::;:: 1 A; B:::;:: 0,1 TJI; b:::;:: 1,0 CM; c:::;:: 1 MM; e = 1,6 . 10- 19 En.
3aJJ;aqa 3
B KOCNH'IeCKON HCCJIe,ll;OB8TeJILCKOM npoeKTe 3anYCRa KOCMH'IeC-
Koro Kopa6JISI 38 npe,o;eJIhI COJIHe'lBOA CHCTeMhI 06Cy)K)taIOTCSI JJ;Be 803-
NomHOCTH:
I. Annap8T 3anYCK8eTCB co CROpOCThlO, P;OcT8To'laoA ,o;JUI BLIXO-
.n;a 38 npe.n;eJIhI COJlHe'lHOH CHCTeMLI, Benocpe.n;CTBeHBO C Op6UThI SeM-
JIll.
II. AnnapaT npU6JIHm8eTC.H K OARoA 113 BHemHHX IIJI8HeT, C ee no-
MO:I:qLIO H3MeUBeT BanpBBJIeSHe .n;BHmeHH.H II npBo6peT8eT CKOpOCTL,
Heo6xoHMyJO JJ.l1SI BhIxo.n;a 3a npe.n;eJILI COJIHequ:oA CHCTeNbI.
MomHo C'IIIITaTL, '1TO BO Beex CJIytl8.RX Kopa6JIL .n;BHmeTCJI no,ll; p;eii-
CTBHeN rpaBHTanHOHHOrO nOJI.R TOJILKO COJlHD;8 HJIH nJI8.HeTLI B 3aBH-
CBMOCTH OT Toro, 'lLe nOJle CHJlLHee B .n;8HHoii TOQKe.
1. OnpeAeJlHTe MHHHM8JILHyJO CKOpOCTL Kopa6JI.H VI II ee HanpaB-
neBDe OTBOCHTeJILHO BeRTopa Op6UT8JILHOH CKOpOCTH 3eMJlH 1I.1I11 pea-
JIH3a:qn:H npoeKT8 I.
2. IIpe1J,UoJIo)KHM, 1lTO Kopa6JIL 3anyn:J;eu B HanpaBJIeHHH onpe,ll;e-
neUHOM B n. 1, HO C 1I.pyrOH CKOpoCTLIO v 2 OTHOCHTeJILHO 3eMJIH. Onpe-
,lI;eJIHTe CKOpOCT:& annapaTa, Kor.n;s 3eMJUl nepeceRaeT Op6HTY Mapca
(T.e. uaA1I.HTe napaJIJIeJILHYIO H nepneE.ItHKYJIJlpHYIO COCTaBJI.RIOIItHe
8TO:A CKOpoCTH OTHOCHTeJIhHO Op6HTbI Mapcs), npH YClIOBHH, 1:ITO B STOT
MOMeHT Mapc H8X01J,HTCJI ,o;a.neKO OT annapaT8.
15

3. IIpeAnOJIOmHM qTO annapaT BOmeJI B rpaBHTalJ;:HOHHoe nOJIe
Mapca. Onpep;eJIHTe MHHHM8.JILHYIO CTapTOBYlO CKOpoCTL annapaTa va
C Op6HTbI 3eMJIH He06xop;HMyro .n;JIH BhIxo.n;a H3 COJIHeTJHOH CHCTeMbI
nOCJJe B03,lJ,eiicTBHH Ha Hero rpaBHTaD;HOHHoro nOJIH Mapca. (Bac He
.n;OJImHO HHTepeCOBaTL TOTJHOe nOJIOmeHHe Mapca 80 BpeMS B38HMOAeii-
CTBHH c annapaTOM.) HaMeK: H3 pe3YJILTaTa 1 BbI 3BaeTe NO,lJ,YJIL H
HanpaBJIeUHe OnTHMaJILHOH CKOpOCTH annapaTa, Heo6xoAHMOH ,lJ,JISI BbI-
xo.n;a 38 npep;eJIbI COJIHe'tlHOH CHCTeMLl. HaHAHTe CBSl3L Mem.n;y sToii
CKOpOCTLIO H COCTaBJISIOII(HMH CKOpOCTH P;O BXOeHH.H annspST8 B
rpSBHTan,HOHHoe nOJIe Mapc8, T.e. COCT8BJI.HIOID;HMH, onpeAeJIeHHhIMH
B ll. 2. Byp;eT JIB COXpBHHTLCH SHeprHH annapaTa?
4. OnpeAeJIHTe MaKCHM8JILHO B03MO>KHYIO qaCTb c6epemeHHoH
SHeprHH B npoeKTe II no cpaBHeHHIO C npoeKToM I.
IIpHMeqaHHH. 1) MomHo CqHTaTb "ITO Bce nJIaHeTLI o6paJI(a-
IOTCSI BOKpyr COJIHna no oKpymHOCT.HM B OAHOM H8np8BJIeHHH H B O,lJ,-
HOH fiJIOCKOCTB. ConpOTHBJIeHBeM B03,o;yxa, BpaeHHeM 3eMJIH 80-
Kpyr CBoeH OCB H SHeprHeii, 38Tpa"IeHHOH npn BLIxo.n;e H3 rp8BHTa:Q,H-
OHHOrOnOJIH3eMJIH,npeHe6pe"IL;2)3Ha"leHH.RBeJIH"lHH:op6HTaJIhHaH
CKOpOCTL 3eMJIH BOKpyr COJIH:Q,8 p8BH8 V o = 30 RN/C OTHomeHHe pac-
CTO.HHHH OT 3eMJIH H Mapca,lJ,o COJIH.Ita paBHO 2/3.
PeweHHIl
3aAa"Ia 1
Hs pnc. 1 nOKaaSHLI nOJIOmeHHH aUTeUR 1 H 2 (paccToSlBHe Mem-
lJ.Y KOTOpLIMH r) H HanpaBJIeUHH Uft ropo,lJ,sA H B (yrOJI Mem.n;y KOTOpLI.
MH \V). IIOJIHhIe c}>ft30BLIe CABHrH ROJIe6aHHH, Boa6ymlJ.aeMLIX o6eHMH
8HTeHHaMH B HeKoTopoii: TO"IKe B HSnpaBJIeUH.HX A H B paBHhI
A(A) = 21t rcosa .+ A<p (1)
J...
A(B) = 21t rcos - a) + Acp, (2)
2
1
A
r.n;e A - AnHRB DOJIBb! nepeAST.
qHKa A - paaHOCTh c}>a3 KOJle.
6a.mrii B 8HTeHH8X, a - yroJI Mem-
1J.Y UJIOCKOCTLIO, npoxo1J.SI:r:q;eA
"lepe3 o6e BHTeHHb!, n BanpDSJIe-
BneM HD ropoAA.
PHC. 1
16

BanumeM YCJlOBHH MHHHMYMa HHTeHCHBHOCTH B ropop;e A H MaK-
cHMYMa HHTeHCHBHOCTH B ropop;c B:
.6 (A) = (2n + 1) 1t,
.6 (B) = 21th,
rp;e k H n - eJIhIC llHCJIa. Ha BhIpa.m.eHHH (1) - (4) CJIc,u;YeT, llTO
(3)
(4)
r
.6(B) - .6(A) = 21t- [cos(cp - ex) - cos a] = [2(k - n) -1] 1[,
A
OTKY,lJ,a
A 2(k - n)-1
r=
4sin('IJ /2) sin(a - 'V /2)
PaCCTO.HHHe r Mcmp;y aUTeuuaMH MHHHMaJIhHO, eCJIH k = n H
a - cp /2 = -x/2. B 3TOM cJIyqac
J...
r mln = 
4sin('V/2)
(5)
\V-x
a=-,
2
(6)
T.C. HOpMaJIb K llJIOCKOCTH, npoxo,u;.Rn::tcii 'lcpca o6e aHTeHHbI, .HBJI.HCT-
CH 6HCCCltTpnCOH yrJIft \II. Ha BLIpameHHii (1), (3), (5) H (6) Haii,ZI;CM
N
1t
.6cp =-.
2
H3MCHCHHC 3H8.Ka.6cp (OT
1[/2 P;O -7t/2) npHBO,D;HT It
npOTHBOnOJIOmHOMY s4>4>eK-
TY: MaKCHMaJIhH8.H HHTeH-
CHBHOCThH3JIyqcHH.H6y,u;eTB
HanpaBJIeHHU A, a MHHU.
MaJIhHa.R - B HanpftBJIeHHH
B. Ha pHC. 2 nOKftaftHId no-
JIOmeHHSI ftUTCUU, HftnpaB-
.JIeHHe Hft cescp N, ft TaKmc
HftnpaBJIeHH.H ua ,lJ,aHHlde ro-
po,u;aA Ii B. H3 pHC}'Hltft BIIJI;-
HO,'lTO
\II ; \V2 - \j/l = 85 0 ,
\11 2 - \VI
2
PUC. 2
17

A C
r.. = = Af 4,1 M
mID 4sin(\V / 2) 4v sin(qI /2)
'Vo = 90° + 'VI + 'V2 = 204,5°.
2
06cY)Kp;eHHe C.rrytI8.JI <1>1  <1>2 npep;OCT8BJUleTC.R QRT8TeJIIO.
3ap;8Qa 2
1. lIop; p;e:AcTBHeM CHJIbI JIopeU'Q,8 npoHCXO,D;HT nepepacnpep;eJIeHHe
5JIeKTpOHOB no nOBepXHOCTH tipycKa. 3JIeKTpoHhI p;BH)K}1Tca.u;o Tex nop,
nOKa CHJIa F' ,;n;eJ:icTBYJOm;a.H ua HUX co CTOpoHhI SJIeKTplftleCROrO nOJIa
C Hanp.H.HCeHHOCTLIO E' , KOTopoe C03p;aeTCH B03HHKmHNH no:aepXHOCT.
HMMIt 3apH)J;aMH., He yp8BHOBecHT CHJIY JIOpeHIJ;8 Ft (pRC. 3):
F, = F', evB = eE',
E' = vB.
TIOJIH8X nanpmKeHHOCTL En 5JIeKTpHtIecKOro nOJIJI B 6pYCKe (pRC. 4)
p8BHa
En = E + E'.
TaK KaK CHJI8 TOKR I B 6pYCKe paBH8 I = nevcb, TO
I
V=-,
necb
E' = vB = IB E _ v _ I
necb ' - U - necbu .
b
It
+
+
+
a
+
+
+
a
e

F' tV' F L
0B
PRe. 3
18
b

1-
-
It E' :=
B0-
PRe. 4

TOr.ltB
Eu =  E 2 + E' 2 =..2-  1 2 + B 2 ::= 41 BIN,
neeb u
a = arctg(E' / E) = arctg(Bu) R$ 38°.
2. B HanpaBJIeHHH poops b B03HHKHeT Hanp.RmeHHe
IB
U=E'b= - =2,5 . 10- 2 B.
nee
3. HanpxmeHHe U 6y,o;eT H3MeH.HTbCH no 38KOHY
U = IB = 1()sinrot. Bosin(rot + cp) =
nee nec
= 10&. costp _ Io cos( 2rot + cp ).
2nec 2nec
TSKHM o6pa30M, HanpSimeHHe HMeeT nocToHHHyJO COCTaBJI.HIOIIJ;YIO
loBo
U() = -coatp.
2nec
4. MOIIJ;HOCTL, nOTpe6n.HeNa.H annapaTOM, paBHa P = IU COS CPl' r,lJ,e
1 H U - .u;eHcTByJOIIJ;He 3HaQeHH.H CHJILI TOKa II HsnpXmeHH.H, CPl - pa3-
HOCTL ft3 KOJIe68HH:A CHJIhl TOKa H HanpJlmeHH.H. IIoCTo.HHH8.f.I COCTRB-
JI.HIOIIJ;aB. HSnp.HmeHH.H B.u;OJIh pe6pa b 6pycKa U 0 - IORo COS tp2' r,D;e CP2 -
pa3HOCTh a3 KOJIe68BHH TOKa H HHJ];yKIJ;HH MSI'HHTHOrO nOJI.H. ECJIH
annapST BKJII01lHTh T8.K, K8.K nOKa38.HO HB pRC. 5, TO npH BhIllOJIHeHHH
YCJIOBH.H R »mL CHJIa TOKa B KBTymKe, a 3UaQHT, II HII)J;YKIJ;HH Mar-
HHTHoro nOJISi 6YAyT nponOpHOH8JIhHhl HanpsmeHHIO fia annapSTe,
lC
annapamy
illynm
PRC. 5
19

C:HJI8 TOK8 B 6pycKe 6Y,ll;eT nponOpHOH8.JIhH8 CHJIe TOKS B snnsp8Te H
CPI = CP2 = <p. B STOM CJIyqse nOCTO.RHHSH COCT8.BJI.RIOn::t8H Hanp.RmeHH.R
B,lI;OJIh pe6pa b 6pYCKS 6Y,ll;eT nponOpHOH8JIbHS MOn::tHOCTH nOTpe6JI.R-
eMoH 8nnapaToM:
u 0 - 1080 COS cP - IV cos cP = P.
3aJ];8"IJa 3
1. ECJIH np:HB.RTh nOTeH:H8JIbayro 3HeprHIO annapaT8 B nOJIe THro-
TeHH.R COJIHI1,8 paBHoH HYJIIO np:H 6eCKOHeQHOM y,lJ,aJIeHHH OT COJIHIt8
TO ,lJ,JI.R BLIXO,lJ;a KOCMH't:leCKOrO annap8T8 38 npe,lJ;eJIhI COJIHeqHOii CHC-
TeMhI C Op6HTLI 3eMJIH He06xo,lJ,HMO, 'ITo6hI B MOMeHT CTapTa ero nOJI-
H8H 3HepI'HSI 6bIJIa 60JIbme HJIH p8BH8 HYJIIO:
E = mu 2 _ G mM  0, v2  20M ,
2 7 1 7 1
(1)
r.n;e m - Macea ann8paT8, 7 1 - pa.z:tHye 3eMHOH Op6HThI, M - Maccs
COJIHna,V-CKopOCTbannapaTaOTHOCHTeJIhHOCOJIH8.0p6HT8JIhHa.R
CKOpOCTh 3eMJIH V o onpeJ];eJIHeTCH H3 BToporo 3aKOH8 HhIOTOHS:
G mM =m v , v=G M .
'i 2 7 1 7 1
(2)
H3BhIpameHHA(I)H(2)nonyqaeM
v 2  2v, v  .J2v o .
C .n;pyroA CTOpOHLI, Mem,n;y cTapToBoH CKOpOCTblO VI H CKOpOCTSIM:H
V H Vo cyII:tecTByer COOTHomeHHe (pHC. 6): V = V o + VI. H3 pHC. 6 HCRO,
qTO V '" V o + VI' T.e.
VI  v - V o  (J2 - 1 )v o .
CJIe,ll;OBaTeJlbHO, MHHHM8JIhHOe 3HaQeHHe HCKOMOA CKOpOCTH
(3)
 = v o ( J2 -1) Rj 12,4 KM/C
PHC. 6
.n;OCTHraeTCSI npH yrJIe q> Mem,n;y BeKTopaMH V
H v o , p8BHOM HYJIIO (cp = 0).
2. }J.JISI HSXO>K,ZJ;eHH.R nap8JIJIeJIhHOH V H
nepneH):{HKYJIHpaoA V.l COCT8BJI.RIOn::tHX CKO.
paCTH annapaT8 V B MOMeHT nepeCeqeHHH op-
20

PHC. 7
PHC. 8
6HThI Mapca (pRC. 7) BOCnOJIh3yeMC.fI 3RKOHaMR coxpaHeHHH MOMeHTS
HMnYJIbCa H 3HeprHH:
m(v o + v 2 )r. = mvr2'
m(v o + v 2 )2
2
o mM = m(vf + Vjf) O mM ,
Ii. 2 r 2
r,ll;e r 2 - pa.nHYC Op6HTbI Mapcs. OTclO.n;a HaxO,l{HM
VII = !l.(v o + v 2 ),
r 2
(4)
u, = ("0 + u,,)'( 1- ;.: ) - (1- 'i /r.).
(5)
3. IIepeH.lI;eM B CHCTeMY OTCqeTa CB.R38.HHYIO c MapcoM. 11:3 nOTeH-
I!;H8JIbHOCTH rpaBHT8.ltHOHHOro nOJUI CJIe,l{yeT, '1TO B 3TOH CHCTeMe an-
napaT. BLIHnH H3 rpaBHTal!;lfOHHOrO nOJIH Mapca, HMeeT TOT me MO,l{YJIb
CKOpOCTH, 't:lTO H npH BXO,lJ,e B nOJIe: v; = VB' H3MeHSIeTC.R TOJIhKO aanpSB-
JIeHHe CKOpoCTH (pHC. 8). J];JIH Toro QTo6hI nOCJIe 06JIeTa Mapca Kopa6JIb
nOKHHYJI COJIHHQHYJO cHcTeuy, aeo6XO,ll;HMO, QTo6LI ero CKOpOCTh v;
y.o;OBJIeTBOpHJIa HepaBeHCTBY, aHWIorH'tJHOMY HepSBeHCTBY (3):
v: > (J2 -l)v M ,
(6)
r.u;e v M - Op6HTaJIhHS.K CKOpOCTb Mapea, paBH8.JI
V M = JGM = Vo M .
r 2 V-,:;
(7)
21

M
c
-
OM
.
D: 0 V;
Vu- V M

Uti
PRC. 9
IIpll Bxo.u;e B rpaBHTaIJ;HOHHoe nOJIe MapCB Kopa6JIh HMee-r CICOpoCTb
v, (pIIC. 9), MOAYJIh ROTOpoii MO.)KHO Onpe,lJ;eJIHTh 113 BhIpameHH.H
v, =  vI + (Vjl - VM)2 .
(8)
H3 COOTHomeHHH (6) - (8), YCJIOBH.H VB == V; II 3HaqeHHH V H V 1.. 113
n. 2 nOJI8elll BhIpa)KeHHe AJIB CTap'l'OBo:A CKOpOCTH Kopa6JI.H V K OTHO-
CHTeJIbHO COJIH:Q,a:
v.  v.[ (;.J' \ (;. J +2-2J2 ;. }
MbI 3anHCBJIH JIHmb OAHO pemeHHe KB8.n;paTHoro Hep8BeaCTBa OT-
HOCHTeJIhHO V K " BTopoe pemeHHe co 3HaKOM . -t nepe.u; KopHeM He HMe-
eT cPH3HQeCKOrO CMbICJI8, TaR RBK npH ,l];a:HHOM B YCJlOBHH OTHomeHHH
Tl 2
-= - v < 0
r2 3 K .
MHHHMftJIbH8..H CKOpOCTb 3BnycKB va OTHOCHTeJIbHO 3eMJIH p8BH8
v 3 = v. -vo = v.[(  r + ( r +2-2J2  -1] 
 0,185 V o  5,5 KM/C.
9HeprH.R annap8Ta B CHCTeMe OTCQeTa CBH3aHHoii C COJIHn.eM:, npH
TaROM cnoco6e 38nYCKB He COXpaHHeTCH - OHa BoapaCT8e-r 38 CqeT H3-
MeHeHHJ1 KHHeTHQeCKOH 3ueprHH Mapca.
22

4. C6epe)KeBBYJO "l8CTh aBeprHH MO)KHO onpe.z:r;eJIHTh no 4>oPMYJ1e
El -E 2
a=
El
v 2 _ v 2
= 1 3  0,8 = 80%.
v 2
1
OP;HSKO NI.R npSKTHtIeCKHX n.eJIe:A 60JIee Ba)KHO p;pyroe OTHomeHHe:
A- ml -m 2
t/- ,
m l
r.u;e m 1 H m 2 - MaCChI TOnJIHBa, ne06xo.u;HMoro ,lJ,JISI pa3rOHa annapaTa B
CJIyq8.HX I Ii II. B COOTBeTCTBIIH C 4>OPMYJIOH IJ;HOJIKOBCKOro
m= (e -1)
(m o - nOJIeaH8.SI Macca annapaTa, U - CKOpOCTL HCTe"leHH.JI ra30B OT-
HOCHTeJILHO snnapaTa) no.nyqaeM
 !!.
e" -ell
P = II
ell -1
OrBeT nepeCT8eT t5LIT!> O)J;H03Ha'lHLIM Ii a8BHCHT OT u. llpH U » vI
p s:=:: 0,55; AJIH COBpeMeHHLIX paKeT, y KOTOpbIX Us:=:: 5 KN./C, P ::= 0,81.

XVII Me)I(AYHGpOAHGJI
cl»M3M'IeCKGR OnMMnHaAa
1986 r., BenMK06pMTaHMII
3aAaqa 1
llJIOCK8.H MOHoxpOMaTHQeCKB.H CBeTOB8.R BOJIaa ,D;JIHHOH A II qaCTO-
TOR f n8,ll;aer nepneB,D;HKYJI.HpEO E8 ABe TO)K,D;eCTBeBHhle Y3KHe IIJ;eJIH L
H M, a8xo,n:HIIJ;HecH Ha p8CCTOHHRH d .u;pyr OT ,n:pyra (pRC. to). CBeTo-
BhIe BOJlHhI, HCxo,n:.HRe H3 K8JK,D;oii
IIJ;eJIH, aB pacCToHHHH X B H8np8BJIe-
HHH e B MOMeHT BpeMeER t OIlliChIB8-
IOTC.H YpaBHeHHeM
}5-
v-
X
y:;:; a CDS (21t (It - 3: )],
r,lJ,e a - aMnnHTy.n;8, OJJ;HH8KOB8.H .n;JISI
06eHX BOJIH; x »d.
1. .l(oK8mRTe, TlTO pe3YJIbTHPym-
yro 8MllJIHTy,lJ,y A ,nBYX BOJIH, BhIXO-
,D;.RIIJ;HX nO,D;yrJIoMB K HOpM8.JIH, Mam-
BO nOJI}"IHTb CJIOmeBHeM ABYX BeKTO-
pOB; MO,lJ,YJIb Kam,lJ,orO BeKTopa paBea a, 8 HanpSBJIeHHe onpe,D;eJIHeTCH
c})830ii CBeTOBO:A BOJIHbI.
llpOBepbTe reOMeTplNecl<H, no Bel<TOpBOH AHBrp8MMe 'tITO
PHC. 10
A = 2a cas 13, r.n;e 13 =  d sin 8.
2. .ABoftB8.JI eJIb 3aMeHeH8 .n;H4>paKnoHHoH pemeTKOH, COCTOSl-
IIJ;eH H3 N O,lJ,HHaKOBhIX eJIeii, p8CCTOHHHe Me)K,D;Y KOTOphIMH O,D;HHa-
KOBoe II paDHO d. HcnoJIb3Y.H BeKTopHhIA MeTOA CJIOmeHHH aMnJIHTY,D;,
n01<8mHTe, 'tITO a81JaJIO H KOHe K8m)];oro BeKTopa JIemaT HS oKpymHo-
CTH pa.z:r;HyCOM R, onpeJ:teJISeMLlM ypBBHeHHeM
R:;:;
2sinP ,
r,n:e a - MO,D;YJIb KWKAOro BeKTopa.
24

rrOKamHTe 1:JTO pe3YJILTHPYIO8S 8MnnHTy,n:8
A:;:: asinNp
sinl} 
H nonyq:HTe pa3HocTL 4»a3 Memp;y pe3YJILTHPYJOe:A BOJIHOH H CBeTO-
BoA BOJIBOA: n8Jl;aJOeA: H8 pemeTKY.
3. H306pa.3HTe a8 O,lJ,HOM B TOM me pHC}THKe rpaHKH BhIp8.>KBJO-
He 3aBHCHMOCTH sin NJ3 H  OT 13. HaoT.u;eJILHOM rp8HKe nOKWKH-
smp
Te, KaR HHTeHCHBHOCTL pe3ynLTHpYJQe:A BOJIHhI 38.BHCHT OT 13.
4. Onpe.n;eJIHTe HHTeHCHBHOCTL rn8BHhIX M8KCHMYMOB.
5. IToK8JKHTe, QTO 'IRCJIO rJI8BHhIX M8KCHMYMOB He MomeT npeBLI-
maTh
2d
T+1.
6. IIoK8mHTe, 'ITO .u;Be BonULI P;JIHHo:ft A H A + I1A r.u;e A'A.« A, 06p8-
3YJQT rJIaBBhIe MaKCHNyMhI Ha yr'nOBhIX paCCTO.ftHHSX
nAA
68 :;::- d e (n::: 0, ::1::1, ::1::2, ...).
cos
BLI'IHCJIHTe 3TO yrJIOBOe paCCTOSIHHe .D;JUI JIHud H8TPHH, COOTBeT-
CTByJOHX .u;nHuaN BOJIE A:;:: 589,0 BM H A + I1A:;:: 589,6 UM P;JIH n ::; 2 H
d :;:: 1,2 . 10-6 M.
38Aaqa 2
B Haq8JIe XX B. 6hIJIa npe,lJ;JlOmeU8 MOp;eJIb 8eMJIH B KOTOpOH npe.n;-
nOJlaraJIOCb 'ITO BeltlJl.H npep;CT8BJIHeT co6oft map pB.ZJ;HYCOM R, COCTOSl-
H:ft H3 O,D;HOpo.u;HOH H30TpoIIHOH TBep.n;oH 060JIO'IKH H mH,D;KorO .RJJ;pa
PB.ZJ;HYCOM Rc.
CKOpOCTH CeHCMJIqecKJlX npo,n:OJIbHLIX V p H nOnepeqHJ:dX VB BOJIH,
TaR H83LIBaeMLIX p- H 8-BOJIH, nOC'I'OHHHhI BHyrpH 060JIOT,lKH. B .ft,l{pe
npOAOJIbHbIe DOJIHId HNelOT CKOpOCTL V cp' a nOnepeqHhIe BOJIHLI He pac-
npOCTp8.H$lJOTCB.
npH 3eMJIeTp.HCeHHH B T01'{Ke E aa nOBepXHOCTH 3eMJIH 06paaYIOT-
C.H ce:AC:MH1'{eCKHe BOJIHLI, KOTOp:LIe np06erftlOT CKB03b 3eMJIJO Ii pe-
rHCTpHPYJOTCS Ha nOBepXHOCTH Ba6JIJO.u;aTeJIeM, KO'I'OphIH MomeT YC-
T8HOBHTL CBoft ce:AcMorpa4» B mo6o:A TOQKe X 3eMHoii: nOBepXHOCTH.
-
YrnOBoe paCCToHUHe Mey TO'llK8MH E H X PURO 2B = EOX, rp,e 0 -
I:!;eHTp 3eMJIH (pRC. 11).
1. IIoKIDKHTe, 'ITO ce:AcMH'leCKHe BOJIHhI pacnpoCTpaHHIOHeCH
TOJILKO B 06oJlo'IKe no npsMoA, .n;OCTHrHyr TOTIKH X 3a BpeM.H t (BpeNs
25

npo6era BOJIH nocJle 3eMJleTpBceHHB),
onpe,ll;eJI.HeNOe 110 opNYJle
2R . e R
t = ;n AJIB e <:: arccos R '
r,ll;e V = V p AJUI P-BOJIH H V =  ,lI;JlB
8-BOJIH.
2. ECJlH nOJIO)KeUHe TOqKH X TaROe,
R
qTO e > arccos -.£., TO CeHCNHqeCKHe
R
P-BOJIHhI .n;OCTHraIOT Ha6JIIO,lI;aTeJI.H DOC-
PKe. 11 JIe P;BYX npeJIOMJIeHd Ha rp8.HHn;e Meat-
,lJ,y 060JIOQKOH H Sl,ZU)OM. HaqepTHTe Tpa-
eRTopHIO pacnpocTpaHeHH.H T8ROit CeHCNl£tlecKoH P-BOJIHhI. IIoJIyqHTe
COOTHomeHHe Meat,lG' yrJIOM e H yrJIOM neHH.H i ceiicecKoii P-BOJI-
RbI H8 rp8.HHn;y Me 060JIOQKOH H H,lU)OIll P;JISI P-BOJIH.
3. HCnOJI:b3YB A8.HH:bIe: R = 6370 KM, Rc = 3470 KM,  = 10,85 KM .
. c-. =6,31 KN. c- 1 , p=9,02KM. c- 1 H peaYJIhT8T, nOJIyq.eBHhIH
B n. 2  Uaqep'rHTe KaQecTBeuHhIii rpaHK 38.BHCHMOCTH e OT i. IIpoKoN-
MeHTHpy:ATe HaHqeCKHe nOCJle.n;CTBH.H 3TOit a8BHCHMOCTH ,lJ;JIH Ha-
6JIIO.n;sTeJIe:A, Haxo.u;SlIItHXCII B P83J1HQHhIX TO'lKaX aeNHo:A nosepx-
ROCTH.
HapHcyATe rpaHKH a8BHCHMOCTH BpeNeHH npo6era p- H 8-BOJIH OT
yrJIa e AJUI 0 <; B <:: 90°.
4. IIocJIe 3eNJIeTp.HCeUHH H86JIIO,n;aTeJIh H3MepSleT 3a,ll;ep)fCKY no
BpeMeUH MemAY npH6hITHeN B-BOJlHhI Bcne.n; 3a P-BOJIHO:H, pasHYIO
2 MIIH II c. Onpep;eJIHTe yrJIOBOe p8ccTOHHHe, H3MepeHHoe H3 n;euTp8
8eNJIH, Me)l(.)J;y TOqKOH 3eNJIeTp.HCeHH.H H Ha6JIIO.n;8TeneM, Hcnonb3YSl
,lI;8.HH:bIe n. 3.
5. Ha6JI1OA8TeJIb, KOTOphIit c.u;eJIaJI H3MepeHHe B D. 4, 38.MeqaeT, 'lITO
'lepea HeKOTopoe BpeMB nOCJIe npH6h1THII p- H S-BOJIH nOCJIe,ll;OB8.JIH erqe
.n;Ba OTCqeTa a8 ceitcMorpa<Pe nOCJIe npoMeatyrK8 speNeHH B 6 NHB 37 c.
06'bHCHHTe 3TOT peaYJILTaT II npOBepbTe, AeHCTBHTeJIbHO JIH OB CB.H38H C
yr'nOBhIM paCCTo.mmeM, onpe.u;eJleHHhIM B npeAhI,ZI;yrn;eM nYHKTe.
3qa 3
TpH'IaCTHn;hI, K8JK.n;WI M8CCOH m, HaXop;$lID;Hec$l B COCTOmlHH pas-
HOBeCH.H, COe,ZJ;HHeBbI ynpyrHMR HepaCT.HHyThIMH JIPY'KHH8.MH, lCam.n;8.Jl
C mecTI<OCTblO k. IIx .n;BIUKeHHe orp8.Hlftleuo no oKpymHocTH, K8.K no-
K83aHO Hft pHC. 12.
1. KWK,DyIO 'qftCTY CNe8IOT II3 paBHoBecHoro nonO)KeHH.H C He-
60JIhmHM CMeeHHeM u.' u2 H Us COOTBeTCTBeHHO H a8TeN OTnYCK8JOT.
26

COCT8BLTe yp8BHeHH.H ABHmeUH.H AJI.H
K&mAOii Tl&CTHl!;hI nOCJIe CMem;eHHSla
MaCC&MH npymHB npeae6pe1Jba
2 a IIoKWKHTe, TlTO CHCTeN8 MomeT
cosepmaTh rapMOHH1JeCKHe KOJIe6aHHH
no 3RKOHY
u = an cos rot
C YCKopeHH.HNH _(02 u, r]1,e an (n = 1,2,
3) - nOCTO.HHHLIe aMUJIHTYALI, 'IITO-
JIHTIecK8.H TlSCTOT& ro MOmeT HMeTb ABa k
3HaQeUH.H: CJ)o.J3 H 0 (ro= hIm) H "ITO
nepBoe 3HaTIeHHe I!;HKJIHqeCKOH qaCTO- PlIca 12
Thl AByKpaTHO BLIpO)K.Q;eHOa
3a CHCTeMS H3 npymHH H 1J&CTHI!; YBeJIHQHBaeTC.H ]1,0 'QHCJI& N  KS:m-
,lJ,8Ji 1JSCTHl!;a COe,lt,HHeH8 npymHHBMH C ,D;BYM.H coceAHHMH. CHSq8.JI8 npy-
mHHhI HepaCT.HHyrhI H H8XOAHTC.H B pasHosecHON COCTOSlHHH. COCTaBh
Te yp8BHeHHe ]1,BH:meUH.H ,lJ,JISI n-it qaCTHl!;hI (n = 1, 2, 3, .u, N) BhIpa-
)ReHUOe qepe3 CNem;eHHe caNOR qaCTHI!;LI, a T8KJKe ,l{JISI COCe,D;HHX
qaCTHI!; naCJIe BLIBe,l{eHH.H HX H3 nOJIOmeHH.H pSBHoseCHSI. IIoKsmHTe,
'ITO BhIpaateUHe 1I.11.H cMerqeHHSI n-ii "IaCTHl!;bI
k
. 2nB1t
un(t);;: assIn (N+ cI» cos roBt
SIBJUleTCSI pemeHHeN ypSBHeHHSI ABHmeHH.H, r 1I.e 8 ;;: 1, 2  a 0 ., N 
n:: 1, 2, .... N, cI> - npOH3BOJIhH8.R 83a, npH YCJIOBHH, "ITO I!;HKJIH-
QeCK8Ji qaCTOTa ,n;aeTC.H BLIpaJKeHHeN
ro.= 2(00 sin (  )
a as (8 = 1, uo ,N) - nOCTOSlBHhle aMIIJIHTyAbI, He 38BHCHMLIe OT n.
YKsmHTe ,D;Hana30H B03MO»CHhIX qaCTOT AJI.H l!;enH COAep)Kae:fi
6eCKOHe'tIHOe 'IHCJIO QaCTH.
4.0npe]1,eJIHTeOTHODIeHHe
"It
Un.]
AJI.H 60JILIDHX qUcen N B ABYX CJIytl8JIX:
a) K0l1e6aHHB COBepmalOTcH C HH3Ko:fi qaCTOTO:H (uH3KOqaCTOTHoe
pemeBHe);
6) KOJIe6aHH.H cosepmSIOTca C M8.KCHMal1bHoH "IaCTOTOH ro = mmax
(BhlcOKOQaCTOTHoe pemeHHe)o
IIoSicHHTe rpalPleCKH pscn0110meHHe qaCTHI!; B I!;enu B MOMeHT
BpeMeHH t ,AJIa 060HX CJIy'QSeB o
27

5. OHY H3 qaCTH 3aNeHHJIH Ha qaCTHY C M8CCO:A M « m. Ce-
naiiTe O:Q;eHKY o:mH,JJ;aeNhlX cecTBeHHhIX H3MeHeHHit B pacnpe.u;eJIe-
HIIH :Q;HKJIJIqeCKHX qaCTOT. OnHmHTe K8QeCTBeaHO tt>OPMY'l8CTOTHOro
cneKTpa ,JJ;JI$I enH C 'l8CTHn;aMH '1epe.u;yroIq,eitcB MaCChI m H M aa OCBO-
Be peaYJIbTaTa n. 4.
PeweHMII
3a.u;aQa 1
1. Pa3HOCTb xoa ,lJ,BYX CBeTOBbIX BOJIR, npoxo.u;HHX Qepe3 eJIH
L II Mt p8BH8 d sin a, CJIeAOBaTeJILHO pa3HOCTL 4>83
11q> == 2).1[ dsinS.
CKJInAhlBftHo6eBOJIHH,nOJIyqaeM
y = a cos( 2ft!t - 2X ) + a cos( 21[{t - 2X + L\fP ) ==
= 2acos "': cos( 211ft _ 2 + "': ).
ECJIH fP = (3, TO pe3YJILTHpylOm;aH BOJIHa HMeeT 8MUJIHTY,o;y
A = 2a cos 13 H 4>a3Y (3.
lIa BeKTopRoit .D;HarpaMMhI (pRC. 13) B8XOARM:
13 == 11q>  A = 2a cos 13.
2
CJIe,o;OB8.TeJILBO t CYMMY STHX ABYX BOJIH MomBO nOJIyqlITb CJIo:me-
BHeM .u;BYX BeKTopOB e MO.AYJIeMaII yrJIOB:hIMH Banp8BJIeHHJlMH, COOT-
BeTCTBYJOm;HNH 4>aa8M BOJIB.
2. CJIO)RHM. BeKTopLl, cooTBeTcTBYIOIqHe 8MnJIHTy.u;aM BOJIB OT
N eJIeit, no npaBHJIY MHoroyrOJILHHKa. Bce CTOpOHhI SToro MHoro-
yrOJILHHKa p8BHLl a, yrJIhI Ne3t.u;y CTO-
pOH8MH p8BHhI 1t - 213, CJIe.u;OB8TeJIbHO
BOKpyr NHoroyroJIhBHK8 MO)KHO OUIl-
eaTL oKPymHocTL (pRC. 14).
HeTPYAHo UOK838TL reOMeTpHtlec-
KH, 'ITO a = 2R sin (3, OTKYAa
a
PRe. 13
28
a A _ sinN(3
R=- a
2 sin (3 , - sin!}.

Pa3HOCTb (}>a3 Memy HCXO,lJ,HOH H
pe3YJIbTHpyJOe:H BOJIHaMH onpe,n;eJISI-
eTCH yrJIOM a:
ex = (  - J3 ) - (- NJ3 ) = (N -1) 13.
3. llOCTPORM rpa4»HKH sin NJ} H
1
sin J3 K8K YHKIUllO P B OAHOM Macm-
Ta6e npH N = 8 (pRC. 15).
¥qHThIB8H, 'lITO HHTeHCHBHOCTb
BOJIHbI nponOpIJ;nOHftJI})HB. KBS,lJ,paTY
aMnJIHTYAbI nOCTpoRM rpaHK
I = 1 sin 2 Np
o sin:.! 13
]!,JIH N = 8 (pRC. 16).
1
sin 13
sin Np
Np
PHC. 14
PIIC. 15
I
o
21t
3n P
1t
PHc.16
29

4. B rJI8..BHbIX MBKCHM)'M8X BeKTOpH8.SI AHarpaNM8 npe.u;CT8BJIHeT
co60ii npHNYIO JIHHHIO (pRC. 17). llpH 3TON 213 = 27tn, r.u;e n = 0, :i:l
:f::2, ... . OTclOAa 13 = nn.
- - -
a a a a a
.1 ..I ..I .' .
PUC. 17
AMnnHTy.u;a pe3YJILTHpyx>e:A BOJIHbI paBHa N a, a HHTeHCUBHOCTL
1= N"l o ' r,n;e 10 - HHTeHCHBHOCTL H3JIyqeHH.R' OT o,o;BoA eJIH.
5. Y CJIOBHe rnaBHLIX MaKCHNYMOB:
p = 1tn HJIH  d sin e = nn, sin e:;::; A , r.u;e n = 0, :1, :i:2, ... .
TaR K8K I sin e 1< 1, TO
I '::" I " 1. Inl " i. n = O. "'I, "'2, _n .
CJIe,Q;OBaTeJIbHO, 'IHCJIO rJIaBHhlX N8KCH:NyMOB He npeBhlmaeT
2d
no =T+l.
6. YCJIOBHe rJI8BHhIX N8KCHMYMOB:
d sin e:;::; nA, n:;::; 0, :i:l, *2, ... .
H<1>4>epeH:Q,HPYH no a, nOJI}"IaeM
"A' (9) = !!cose.
n
CJIe.u;OB8TeJIhHO, npH AA « A
A"!. d a . 8 nA
ill\. = -cos . 88, SID :;::; - 
n d
MA
=> A9 = d I-( r
llo,o;CTaBJI.H.H A = 589,0 HM, 6."J.. = 0,6 HM, n = 2 H d = 1,2 . 10-6 M,
nOJI}"lseM e = 5,2 . 10- 3 p = 0,3°.
30

3aJJ;aqa 2
EX
1. BpeM.R pacnpocTpaHeHH.R BOJIB, HMelO:m;HX CKOpOCTh V, t = V .
T8.K K8.K EX = 2R sin 8 (H3 b EOX, pnc. 18) TO
t = 2Rsin8
V
(.u;JlH P-BOllH V = V p ; 1I.lIH 8-BOJIH V = V.).
IIp.HMoJIHBeAHoe pacnpocTpaHeBHe BOllH B03MO)KHO, eClIH OTpeaOK
EX Bech lIemHT B 06oJIo"lKe. aro BLIIIOJIH.HeTCB npH e < 8 0 , r.u;e 8 0 -
yrOJI, npH KOTOpON OTpeaOK EX KacaeTCSI nOBepxHocTR $I,ZJ;pa. 113
!:J. OHX (pRC. 19) HaxO,JJ;BM:
Re = Rcos8 0 , 8 0 = ftrCCOs  , e <: arccos  .
2. B03MO)KHLIe DYTH P-BOJIH B CJIyqae e > arccos .lt nOKaaftHLI Ha
R
pHC. 20 Ii 21. B 060HX BftpHaHTftX AB nap8JIJIeJIbHO EX.
ECJIH i - yrOJI n8,ZJ;eHU.R BOJlHLI Ba rp8.HH:Q;Y . .RJJ;PO - 060JIO'1lKa., a
r - yrOJI npeJIOMJIeHIUI, TO no 3ftKOHY npe.nOMJIeHHH BOJIH,
smt _ V p
sinr - V ep .
BBapHftHTe pHC. 20 BMeeN:
(1)
---- - --
e = EOC = EOA + AOC; AOC = 90° - OAC = 90 0 - r; EOA = i - AEO.
Pac. 18
PUC. 19
31

PRC. 20
OrclO,lJ,a
PRC. 21
-
e = 90° - r - AEO + i .
TIo TeopeNe CHHYCOB ,D;J1S1h.AEO
Rc
(2)
=
R
- -
sinAEO sinEAO
OTKy.o;8
. A - EO Rc..
SIn = R sm.
H3 BhIpa>KeHHH (1) - (3) nOJIyqHM
(3)
6 = 90" + i - [ arcsin( : Sini) - arCSin(  sini) l (4)
.l{JUI B8PH8HT8 pRC. 21 8H8JIOrHQHO nOJIyqaeM
6 = 90" - i +[ arcsin( : Sini) +arcsi{  sin i)] (5)
(i MomeT MeHHTLCfi OT 0 .0;0 90°).
3. .l{JIH KWK.,lI;Oro 3HRT{eHHH i MLI HMeeM ABa B03MOJKHhIX 3UalleHH.H 8.
Vcp Rc
IIo,D;cTaBJI.R.H B cpOPMYJlLI (4) H (5) - = 0,8313; - = 0,5447, CTpO-
Vp R
HM rp8<})HKH (pHC. 22). BRAHM, T{TO Cym;ecTB}'IOT MHHHMBJIbHOe H M8K-
CHMaJILHOe 3Ha'1JeBH.R yrJI8 8: amin = 75°35' H 8max = 104°25' npH
i = 55°. llOSTOMY npeJIOMJIeBHMe BOJIHM cNoryr 38perHcTpHpOB8Tb
32

9
104°25
100°
90°46'
90°
89°14'
80°
75°35'
f
o 10
50
II
II
..
II
II
II
" ..
90 i, °
30
70
PRe. 22
I
TOJlbKO Ha6JIIO.n;aTeJIH, Haxo.n;.RHeC.R B aOHe M 2 M3 (pHC. 23). .IIpH-
MLle. BOJIHhI .u;OXO.o;.RT.o;O Ha6JIIO.u;RTeJIeii, HaXOp;.RHXC.H Ha yqacTKax
M]E II EM4. Tax Kax
8 0 ::; arccos  = 57° < 9 min :;;; 75°35',
TO Cym;ecTBYIOT aOHN ( M 3 M 4 HM 1 M 2)' r.o;e 3eMJIeTp.HCeHHe om;ym;aeT-
Cft O'lleHh cJIa60, - Ty.u;8 BOJIHbI MOryT .o;OiiTH TOJILKO nOCJ1e OTpameHHSI
OT OAROH Ha rpSHHD; pa HJIH 060JIOTlKH.
):1;0 Ha6JIIO.u;aTeJleH HaxO.o;.HIIJ;HXC.R B aOHe M 2 M 3' BOJIHLI MOryr .u;OH-
TH HeCKOJIhKHMH nyrSlMH: HanpHMep, yrJIye = 9] COOTBeTCTBY1<>T p;Ba
B03MO)KHhIX 3HSQeHH.R yrJIa fi8,D;eHH.R i a yrJIY e = 8 2 - ,It.WKe TpH. no-
E
PRC. 23
2-379
33

STONY Cym;ecTBYIOT ABe (H COOTBeTCTSeHHO TpU) pa3HLIe TpaeKTopHH, a
ceiiCMorpa«}) 38perHCTpHPYeor He O,D;HH, a uecKOJILKO TOJIqJ(OB (,lJ;B8. TOJI1I-
K8 AJI.K 75°35' < 9 1 < 89°14'; TPH TOJI1IKa - 11.JI.K 89°14' <; 8 2 <; 90°). aTO
BH,D;BO H3 pRC. 22.
ITocKOJILKY rpaEK e (i) cHNNeTpHlIeU OTHOCHTeJILHO np.HNoA
e = 90°, ,AOCTaT01lHO paCCNOTpeTL CJIyqaJt 0° <; 9 <; 90°.
rpacPHK aaBaCHNOCTB BpeN:euH npo6era p- a S-BOJIB OT yrJla a AaB
R
aa pac. 24. ]1.JI.K 0° < e <; a o = arccos R BpeMB onpeJJ;eJI.HeTC.H BLIplUKe-
aHeN:
t= 2Rine (V=VpHJmV=V.).
B aOHY 9 0 < e < Groin BOJIHLl npaKTHtleCKH He nona.n;aIOT; B 30UY
Groin <; 9 <; 90° nOn8,Zl;aJOT TOJILKO npeJIONJIeHHLIe BOJIHh1.
IT p H N e'q a R He. .l(JIH TO'tlROro nocrpoeHH.R rpaHKa MO>KHO BLI-
BeCTH «})OPNYJIY
2Re 2p l )
t = y;:-cosr:f: VI' cos\6:t: r ,
Yep smr
rp,e BH)fCBHe 3BaKH COOTBeTCTBytOT nepBON}T SapH8.BTY (CM. pac. 20), a
BepxBHe - BTOPONY(CM. pHC. 21).
4. YqHThIB8JI, QTO BpeNB pacnpoCTp8.HeHHB BOJIUM no npsNoA
t = 2Rine (V= VpHJIU V= V.),
t, c
1693
985
I
I I I
I I I
________ --____  I___I:
------- ----------
I I
I I
::::- _:::::::::-I- p : I
s I I I I
I I I
I I I
I I I I
I I I
I I I
I I I I
I I 89°35'
I I
1413
1404
1304
1247
o
57° 75°35' 90° e
he. 24
34

nO.nyqHM:, "lTO BpeM.H 3anaaB8.H:HH 8-BOnHLI paDHO
M=t.-t p =2R( t. - ;J sm8 (V.<V.
OrClOA8 yrnoBoe paCCTOSIHHe
. V,,t
EOX = 29 = 2arCSln 2R( _ )
 17,84°.
9TO COOTBeTCTBYeT e = 8,92° <
< eo = 57°, T.e. BOJIH:&! p;eiicTBu.
TeJI&HO .n;OXOAIIT /1.0 Ha6JIIO/1.8TeJISI
6es npeJIoNJIemul.
5. Ba nOCJle/1.BHX OTC"IeTa
BhI3B8.H1iI OTpameHHSlMH BonH aft
rp8.HHD;e Mem.u;y of)oJIo"lKoii H SIP:
pOM:. llpoMemyroK BpeN:eBH Mem-
AY STUMM OTC"IeT8MH
At,. = (ED + DX) (  - ..!.. )
 V"
PHC. 25
(pRC. 25), npHQeM ED = DX (H3 CHMMeTPHH) H
ED2 = OE2 + OD2 - 20E. OD COB EOD = R2 + R- 2RR c cOB e
(no TeopeMe I<oCHHYCOB .D;JISI  ODE).
OrcIOA8 nonyqaeM
t84 = 2 v  .J R2 + R - 2RRc cose.
l' .
llO'p;CTaBHB e = 8,92°" U&XO.n;RN tM SI:j: 396" 7 C #1;1 6 NHH 37 c, T.e. Hame
npeAIIOJIOmeUDe nO,lffBepm,nae-rcH peSYJIbT8TaMH SI<CnepHMeHTa.
SaAa..8
1. SanDmeN yp8BHeHHe Bp8IIJ;8TenLHoro 'p;BHmeHH.H .D;J1Si n-A qaCTU.
IUd:
U.
kR [(Un-I - Un) + (U nH - Un)] = I R '
d 2
rAe u:; = dt 2 Un' R - p8,ll;HYC OI<pYmHOCTH" I = mR2 - MOMeHT HHep-
H qaCTHD;LI.
35

TaKHM 06paaoM,
u = ro (U n - 1 + u u1 - 2u n ) ,
(1)
k
r.n;e ro = -.
m
B CJIyq:ae TpeX qaCTH HMeeM CHCTeMY ypaBHeHHH:
l ui = ro(U3 +  - 211J),
u; = roS(U I + Us - 2u 2 ),
u3 = rog(U2 + U 1 - 2U3).
(2)
2. CncTeMa (2) HMeeT pemeHHeM cynepn03HItHIO rapMOHHtIeC:KHX
KOJIe6aHHH (HopMaJIbHhIX 1'40;0;). ):(JIH HRXOm)l;eHH.H HOpMaJIbHbIX 1'40.0;
CHCTeMhI no.o;CTaBHM ypaBHemlSI rapMOHI'NeCKHX :KOJIe6aHHH
u t = at COB rot, U2 = a 2 cos rot, "3 = a 3 cos rot
B BLIpameHHH CHCTeMhI (2):
j -ro2 = (i)(a3 + a 2 - ).
-ro 2 a 2 = CJ)( + as - 2a 2 ),
-ro 2 a 3 = roHlI:! + O:l - 2a a ).
(3) .
CnomHB Bce ypaBHeHHSI (3), nOJIytlHM
ro 2 (a 2 + at + a 3 ) = o.
OTclO.n;a CJIeAyeT:
{ c.o = 0,
D:1 + tl:i + a 3 = o.
STO 03naQaeT, 'lITO CHCTeMa (2) MmKeT HMeTh AB8 B03MO.mHbIX 3Haqe
HHSI qaCTOTbI: ,J3ro o H 0, npHtleM nepBoe 3H8qeHHe )J;BYKpaTHO BhIpom-
AeHHoe.
3. npSIMOH no"ncTanoBKoH HeTpY.AHO y6e.o;HTLcJI, 1lT0 4>YHKH.R
U ( t) = a Sin ( 21tnS + cp ) cos m t
n B N B
.neHCTBHTeJILHO y.n;OBJIeTBOpHeT yp8BHeHHIO (1) IIPH YCJIOBHH, 'tlTO
(J) = 2 (0 Sin ( 1tS )
· 0 N.
36

113 nOCJIe.z:r;aero paBeHCTBa CJIenyeT, 'tITO
0< (0.  2 (00 (npH N ---+ (0),
'tITO cooTBeTcTByeT ,AHana30HY
1 <.s<'Nj2.
4. a) .l(JlH HH3KRX 'QaCTOT
=
U...,
sin (2ns + 4J )
sin [2(n + I)S ; + ]
sin (2ns+ q,)
( 2ns ; + 4J ) + 2n  cos (2ns ; + 4J )
1
1 + 21t ; ctg (2ns > <p) .
sin
llPH 3TOM -qa.rqe Bcero (KpoMe n = ...!......i i e N)
2N
al1.
U n + 1
6) ):{JISI MaKCHMaJlbHhIX qaCTOT ()}::;: co rnax ::;: 2(.00' COOTBeTCTByrom;HX
N
8=-,
2
= sin (nn+q,) =-1.
u n +! sin [n(n + 1) + C1>]
PacnpeAeJIeHHe aMnJIHTY.z:r; B enH nOKa3ftaO Hft pRC. 26, a .n;JIa nep-
Boro CJIyqas. H HB pRe. 26,6 - AJIa BToporo.
5. ECJIH M« m, TO MO)KHO e"IHTBTh, '1TO KOJIe68..HH.H '18eTH Mac-
COR M BhI3B8HhI KOJIe6aSH.HMH 3THX "l8CTHIt; Mem,D;y coce.n;HHMH tlaCTR-
D;aNH M8ccoii m »M, KOTophIe MO>KHO CtlHT8Th Heno.n;BHmHhIMH no
OTHomeHHIO K qaCTRD;aM MaccoH M. C06cTBeHH8H 'l:8CTOTa qaCTHD;bI
MaccoR M B 3TOM npR6JIHmeHHH onpe.z:r;eJI.HeTC.H H3 ypaBseHH.H
.. 2 {2k (2m
Mx = - kx, (1)1 = V ii aI (1)0 V M '
.l{JI.H M « m QaCTOT8 (01 6y.n;eT BaNHoro 60JIhme (Omax = 2(00
(pRC. 27). T01lHhI:fi paCqeT .z:r;JIH CHCTeMhI H3 .z:r;eC.RTH qaeTH, H3 KOTO-
phIX o.z:r;Ba QaCTHD;8 HMeeT Maccy M::;: 0,1 m, P;8eT pe3YJIbTaThI
rob
-=0,677; 1,277; 1727; 1,969; 4,588,
(1)0
KOTopbIe no.n;TBep)K,lJ;8JOT KaqeCTBeHHYJO KapTHHY, npe.z:r;cTaBJIeuuyIO aft
pHC. 27 11.JISI 60JIbmOrO t,lHCJI8 qaCTH.
37

an
n
a
an
t5
n
an
n
hc. 26
o
2(00
(01 (() 0
2(00
(01
(()
PRC. 27
PUC. 28
C YBeJIHTleHHeM: "IHCJIa JIerKHX 'laCTHD; Maccoii M « m YBeJIHTlHBa-
eTC.R qHCJIO 'laCTOT B OOJIaCTH C0 1 ' 't.lTO rrpHBO,U;HT K cneKTpy qaCTOT THrra
.rrOJIoca-nponYCK-rr0J10ca. (pHC. 28).
TO't.lUhIH paC't.leT AJ1S1 CHCTeMbI, COCTOJUIJ;eii: H3 n.aTH q8CTH MaCCOH
m H nSITH qaCTHD; M8Ccoii M, .n;aeT c.neAYJOUJ;He peaYJILT8ThI:
COin
- = 0,804; 1,338; 4,496; 4,621; 4,690.
roo
OuH nOATBepJK KaqeCTBeBHyIO KapTHHY, npeACT8BJIeHHYIO Ha
pliC. 28.
38

XVIII Me-.qYHGpOAHali
.H3M'IeCKali OnMMnMGAa
1987 r., rAP
3a,lJ;a'la 1
rOpUhtH xpe6eT (pHC. 29) 06TeKaeTC.JI a..n;Ha6aTHQecKH BJI8)KHLIM
B03YXOM. .l{aBJIeHHe Boayxa, H3MepeHHoe MeTeOpOJIOrHQeCKHMH
CT81I:r(HSlMH Mo II M3' paBHo 100 KITa, aCT8.H:r(He:A M 2 - 70 KIIa. TeM-
nepaTYps B03,lJ,yxa B TOllKe Mo COCTRBJI.JIeT +200 c.
Kor.n;a Boa.n;yx nO,lJ,HHMaeTCSI, npH ,D;aBJIeHHH 84,5 RITa Ha'lHHaeTC.H
06pa30B8HHe 06naROB. llpH ,lJ,aJIbHeHllieM no.n;HSlTHII B03YX  B KOTO-
pOM npOHCXO,D;HT KOH,lJ;eHCaD;H.H BO,D;.JIHOrO napa (M8CC8 SToro B03JJ;yxa
H8.J]; Kam)];hIM KBBAPSTHhIM MeTpOM nOBepXHOCTH COCTBBJI.HeT 2000 Kr),
.n;OCTHraeT BeplliHHLI xpe6Ta (cTaHItIlH M 2) Qepe3 1500 c. llpH STOM
KaJKJJ;LIH KHnOrpBMM B03.n;yxa OT,lJ,aeT B BH)J;e OCa.u;KOB (,n;o»C)J;JI) 2,45 r
BO,lJ,LI.
1. Onpe,lJ,eJIIITe TeMnepaTYPY T I Ha ypOBHe HH)KHeH rp8.HHD;LI 06JIa-
KOB.
2. Ha KaRoH BLICOTe hI HM CT81I:r(HeH Mo pacnOJIOmeHa HHmH$I.H
rp8.HH 06JI8.KOB, ecJIH ,n;onYCTHTL, 'ITO nJIOTHOCTL Boa,lJ,yxa y6LIBaeT
JIHHeHHO C BhICOTOii?
3. K8.Kaa TeMnepaTypa T 2 6y,n;eT H3Mepea8 aa BeplliHHe xpe6Ta?
.. . .. . .... ...... .... .
,., . II. .. . .....
............. 111 ..:......
-..:. \::. : :: : :.. ":..:.... :" .=.; :.  :...... - ',,:, ":'.:...: :... :.:...
Mo
M g
PHC. 29
39

46 Kaxo:A BhlCOThI OCTHraeT CTOJI6 BOhI (BHCOTB OCAAKOB Ha ep;H-
HHD;Y IIJIOm;8.Q;H), BJaIfi8.Q;alOm;eii H3 nOTOKa B03P;YXB B TeqeHHe 3 q? Y c-
JIOBHS: KOH.D;eHCBD;HH napa CqUTBTb OHHaKOBldMH OT BldCOTLI M 1 AO M 2.
5_ Kax8S1 TeNnepaoryp8 T 3 H3Meps:eTCJI Ha 38.Q;HeA: CTopOHe xpe6Ta Y
CT8HD;HH M 3?
06CYP;HTe COCTO.HHHe B03P;YXB Y CT8HHII M3 no cpaBHeHHIO C ero
COCTommeM Y CT8BD;HH M 06
YKa3aHH.JI H aHHhIe
B03YX CJIe.AyeT paccNBTpHBaTb KBK He8.JIbHldii ra3. BJIHHHHeM
BO,AaHoro napa Ha YAeJIbJIYIO TeWIOeNKOCTJ. H Ha UJIOTHOCTb B03,lJ,yxa a
T81UKe 38.BHCHNOCThlO yp;eJIhHOH TenJlOT:LI HcnapeHHH OT TeMDepaTyp:LI
CJIeP;YeT npeHe6peQb6 TeNnepBTypld CJIep;yeT OnpeAeJIHTh C TOqHOCTblO
,lJ,O 1 K, BLICOTY HHHeA rpaHH:LI 06JIaKOB - C TOqHOCThlO o
10 M, B:LICOTY0C8JU(OB - C TOqHOCThIOO 0,1 MN.
Yp;eJlbH8JI TeDJIOeMKOCTb B03AYJ[.a B HHTepecyrorqeN HHTepB8Jle TeM-
nepaTyp c p = 1005 J];m-Kr- 1 .K-l; nJIOTHOCTh B03p;yxa y CTaHHH Mo
IIpH aTNoc<}JepHoN aBJIeHHH. Po H TeMIIepaType To = to + 273 == = 293 K
pUHa Po:;::: 1,189 Kr-M- 3 ; yeJIbHBSI TenJIOTB HcnapeHHH BOhI B 06JIac-
Cl'
TH 06JIaKa qv= 2500 KJ];m. Kr- I ; - = y = 1,4.
C v
3ap;aWJa 2
1f3 TOqeQHOrO HCT01:lHHKa P BbIJIaeT nyqoK 3JIeKTpoHOB II BJIeTa-
eT B M8J'HHTHOe I10JIe C HHyI<ItHefi. B KonbD;eBoA: (TOPOH,lJ;B..1IhHofi.) Ka-
TymKH BOJIh JIHHHA nong. IIycTL yrOJI paCKpldTHII sneKTpOHBoro II}"I-
Ka 2no OqeHh N8JI (no « 1 P8.JJ;). HHmeKD;HH 3JIeKTpOHOB npOHCXO,lJ,T
H8 cpeHeM p8)];Hyce R TopOHa C nOMorqLIO YCKOpIlJOm;ero HanpJDKe-
HHH Uo. llpep;nonaraeTcH, 'rITO MOYJIL B nOCTOJlHeH.
SJIeKTpOCT8TlttleCKHM B3aRMop;eAcTBHeM SJIeKTpoBOB B nyqKe npe-
He6peQL6
1. )t1I.JI BH.)KeHHH IIyqKa no JIHHHH M8l'HHTBOrO nOJIg TOpOH,A8 He06-
XO):l;HMO I1pHJIOmHTL O,D;HopO)tHOe NRrHHTHOe none C HH,lI;YKD;Hefi. B I' OT-
KJIOHmoee nyqOK6 BLNHCJIHTe HHJWKJO Bl WIll SJIeKTpoHa, KOTOpLIii
ABR)KeTcg no KpyroBOA TJ)8eKTopHH paJUlyCON R BHYTpH TOpoH,[t8.
2. BLttlHCJIHTe HH.AYKIO TOpOIl)];8JIhHOrO M8.J'HHTHorO nOJIa B, 4>0-
KYCHpyIOm;ero 8JIeKTpoHHHfi. nyqOK B qeTldpex T01:lKax, CP;BHHyrhIX Ha
1t
'2 APyr OTHoCHTenhHO APyra (pRC. 30).
Y K a 3 a H He. llpH Hccne,n;OB8.HHH TpaeKTopHH sneKTpoHOB HCKPHB-
neHReN BURBH MarHHTHoro nOJIH NomBO npeae6peQL.
40

p
PRC. 30
3. Be3 OTKJIOn$.UOm;ero nonSl C HHP;YKe:H .8 1 nyqoK 3JIeKTpOHOB
He OCTaeTC.R B TopoH)J;e, a nOKHp;aeT ero Apefi.cl>OBhIM ABH)KeHHeM nep-
neHKYJI$.[PHO nJIOCKOCTH TOpOHAa.
a) llOKamnTe, 'qTO paAHaJIhHOe OTKJIOHeHHe SJIeKTpOHOB OT P8AHY-
ca R HH)KeKHH KOHellHO.
6) Onpe.D;eJIHTe HanpaBJIeHHe ,ApeHcl>OBO:H CROpOCTH.
Y K a 3 a H He. YrJIOM paCKphITHII sneKTpoHHoro n}"lKa CJIe,AYeT npe-
He6pfNb. HCnOJIh3yi\Te 3aKOHhI COXpaHeHHSI SHepI'HH H MOMeHTa HM-
nYJILCS.
.D;aaBhIe
.!!.... = 1,76.10 11 KJI. Kr- 1 , U o = 3 KB, R = 50 MM.
m
3aAa-qa 3
IIpH pacnpOCTpaHeUHH CHHYCOHAMbHLIX KOJIe6aHHH CHJIbI TOKa H
HanpSl.meHHH no 6eCKOUellHOH L-C.:r(enOQKe (pRC. 31) cl>a3bI Kone6a-
HHfi. HSnp.RmeHHH Ha KOHAeHcaTOpax P;BYX nOCJIeAOBaTenLHhIX SNeeK
OTJIH'tUUOTCSI HS nOCTOSl.HHhIH yron cp.
1. Onpep;eJIHTe 3aBHCHMOCTL cp OT ro, L II C, f'Ae (t) - n,HKJlwteCK8JI
qaCTOTa SJIeKTpOMarHHTHhIX KOJIe6SHHA.
2. OnpeAenHTe CKOpOCTL pacnpocTpSBeaH.H BOJIa, eCJIH Kamp;8SI
.fItIeiiKS IIMeeT AJIHHY I.
41

"  C
I C
J C
. . .
.
I
I
PRe. 31
3. TIpu KaKOM yc.nOBBH CKOpOCT:b pacnpoCTp8HeHIUI BOJID cJIa60 3a-
BHCHT OT (0 H KUODO 3HaqeHHe CKOpoCTH B STOM c.rryqae?
4. DpeJJ;JIO)KHTe npocTYJO MeX8HH1IeCKyJO NOeJIL, aBaJIOrH'IH}'1O
OIIHC8HHOA BLIme 3JIeKTplAecKoA Itenu, H BLIBe,n;HTe ypaBHeHIUl, 1<0-
TopLIe 060cHoBLIB81OT np8BHJIbHOCTL sToA MOJJ;eJIH.
PeweHHII
Sap;aqa 1
1. li3 COOTBOmeHH.R Ne)K)Q1 8BJIeUHeM H 06"heMOM npH 8,l:tHa6aT-
BOM B3MeueHHH COCTOBHHSI
Po = ( Vi ) '
PI V o
H yp8BUeBUSl COCTO.JIHHH
PoV o _ PlVi
To - 7i
nonyqaeN
71 _ ( PJ. ) 1-!
_ _ J
To - Po .
HCnOJIh3YS JJ;8HHHe Po = 100 KIIa, PI = 84,5 HITS, Y = 1,4 H to =
= 20 °C (To = 293 K), M())KHO H8:ATB
Tl ::: 279,2 K.
2. TaR K8.K A8BJleHHe B03.u;yxa onpeJJ;eJlHeTC.II BeCON B03JJ;ymaoro
cToJI6a H TO'1lRoA H3MepeBB.R, BIIrICOTa hi H8.XOJJ;HTC.H U3 pa3BOCTH ,n;aB-
neRd Boap;yxa
Po- PI =6 phi'
rJJ;e p - cpeAHSft IIJIOTBOCTh B03a.
B CHJIY npeIIOJIOJKeHHB 0 JIHHeihloM YMeHLmeHHH WIOTHOCTH 803-
JJ;yxa
- Po + PI
p::: 2 .
42

3aa'rleUHe UJIOTHOCTH PI MOmBO BhI'rlHCJIHTL H3 ypBBHeBH.R COCTO,H-
HH.R
..l!9- = J!L
PoTo PI71.
HCnOJIb3ya p;aHHhIe 3a.n,aQH, nOJlyqaeM
PI =: 1,054 Kr. N- 3 .
.n;JI.H cpep;Heii WIOTHOCTH no.nyqaeTca
p= 1,122 Kr.l\4-3.
TorAB C nOMOLIO nepBoro ypaBBeHH.R HaXOAHM
 = Po __PI ;
gp
hI = 1409 N.
3. H3MeHeuHe TeMnepaTyphI npH nOp;H.RTHH B03,lJ,yxa OT rpB.HH:QhI
06JIaKOB P;O BepmHHLI ropHoro M8CCHS8 06YCJIOBJIeHO P;BYM.H nponecca-
NH:
- OXJI8.JKp;eHHeM /1..0 TeMlIepaTyphI T % llpH 8,AHa6aTHoM nOp;H.HTHH;
- nOBhImeHHeM TeMnepaTyphI aft IJ.T 6JIaro/1..apH Koup;eHcaICHH.
CJIeP;OBaTeJILHO, T 2 = T" + IJ.T .
BhIllHCJIea He T".
H3 COOTBomeBHH i. = ( = J - 
TI = 279,2 K, Pi = 84,5 RlIa, P2 = 70 KlIa Ii g = 1,4 nOJIyqaeM
Tx = 264,6 K.
BLI'rI H CJI eUBe IJ.T.
BhIp;eJIHJOeecH npH KOH,lJ,eHC8.IUlH KOJIH'tleCTBO TeIIJIOThI BAeT aa
uarpeB8BHe B03/I..yxa, T .e. qvm = c lI IJ.T m 1 , rp;e m - Macca BOP;hI, o6pa3Y-
lOm;eiica H3 1 Kr B03p;yxa (m l = 1 Kr). TB.KHM 06pa30M,
C yqeTOM 3HaQeHHH BeJIHYlHH
I1T = qv m .
C p11lt
HCnOJIb3Y.H 8.HHhIe 3UaQeHUa BeJIH'IHH, MO)l(BO BaATH tJ.T = 6,1 K.
CJIe/1..oBaTeJILHO, T 2 = 270,7 K.
4. 113 nOABHMaJOeroca cToJI6a Boa,n;yxa /1..0 p;OCTHmeHH.H BepmHHLI
2000 . 245 .10- 3
Ha nOBepXHOCTL llJI01IJ;8,lJ;LIO 1 1'4 2 3a 1 C BhIeT 1500 Rr oC8J];-
KD ():I;oa). 3a 3 q aTO COCT8BHT 35,3 Rr.
8aaYleUHe 1 Kr. M- 2 03HaQaeT p;Jla BOP;hI BhICOTY OC8.)I;Ka 1 MM. CJle-
]l;OB8TM:bHO, BLIcOTa cToJI6a BLIna.n.mo1IJ;eA BOP;hI COCTaBJI.ReT 35,3 MM.
43

5. TIocKonLKY p;BHmeRHe B03,n;yxa HB 38,lJ;He:ft CTopOHe XPOOTB npo-
HCXO,lJ,HTa.AHB6aTHO,H3cooTHomeuHH
T. =T.( ::t
C eTOM 38.Q;8.HHLIX BeJIHTIHH nonyqaeN T 3 = 300 K.
06cymp;euHe
npH p;aHHhIX YCJIOBH.HX C'l'aH.R M 3 H3Mep.HeT 60JILmYJO TeMnepa-
TYPY, 'lieN: CT8.H:r(H.H Mo. IIpH o6TeKBHHH ropaoro MBCCHBa B03P;YX CTa-
BOBHTCB 60JIee TellJlhIM H CYXH:M. IToBLImeHHe TeMITepaTypLI 06YCJIOB-
neRO KOa,D;eHCB:r(HeA Bop;BHoro napa. ECJIH me, npH npO'llHX paBHLIX
YCJIOBH.R.X, :MaCCRB 06TeKaJICH 6L1 COBepmeHHO CYXHM B03.n;YXOM, TO TeM-
nepaTypLl, H3MepHeMLle CT8.HI.tHHMH Ma HMo, 6hIJIH 6hI O,lJ,HHB.KOBbl-
MIl.
3ap;aQa 2
1. J];BH.?KeHHe SJIeKTpOHB no KpyroBoH TpSel<TOpHH o6eCne'tlHBae'l'-
CSI cmIOA JIopeH:r(B
F L = evo .
Cnep;OB8TeJILHO,
mv. 2
R O ;;; evo&,
oTKyp;a
 ;;; mu o .
eR
(1)
CKOPOCTb SJIeKTpOaa, npH06peTeHHSH B SJIeKTpH'tlecKOM none, 1'40-
meT 6L1TL Hd,n;eHB H3 COOTHomeHHH
mu 2
-!!.. = eU o :
2
V o = J2o .
(2)
113 BLlpamemdt (1) H (2) uaxO,n;HM
13, =  J2eUo = .! 2Uom ; B, = 3,7.10- 3 Tn.
eR m R e
2. 8J1eKTpOH, BLL1IeT8.1OHfi. nop; yrJION cp < a o K JIHHHH HH,n;yI<:r(HH
HMeeT COCTaBJUIIOYJO CKOpOCTH, nepneH)J;HKYJIJlpHytO HB.Q;yK:r(HH '1'0-
POHJJ;aJILHOro MarHHTHOrO nOJUI (puc. 32): ii ..L B, U = V o cpo
44

+
+
+B
+
+
+B
+
+
+
V;rX
+ +
PHC. 32
PRC. 33
B CHCTeMe OTC'leTa ,lJ,BHmym;eHC.R co CKOpOCTblO V B,lJ,OJIb JIHHn:H
HHP;}TKD;I1H nOJI.H, aJIeKTpOH J];BHmeTC.H no oKpymHocTH, CJIe,lJ,OB8TeJIb-
HO, eBu = maqC rJJ;e a QC = uro.
CJIep;OBBTeJIbHO, F = maV.c HJIH eBu = mum, OTKy.n8
ro = eB . a T = 21t = 21tm .
m ro eB
Qepe3 npOMemYTOK BpeMeHH, P8BHhIH T, 3JIeKTpOHhI cf}JoKycnpy-
IOTCH, 3a 3TO BpeMS! OHH CMecTaTC.H HB paCCTO.HHHe
L = voTcosq>  voT, TaK KaK COS<P = 1.
7tR
113 pRC. 30 BRJJ;HO ':ITO L = - noaToN)'
2
7tR V o 21tm
- = voT =
2 eB
oTKyp;a
B = 4mq, = 4B. =  J2U,,", ; B = 1,48 _10- 2 Tn.
eR R e
3. 0603H8't1HM M8JIOe pa)J;HBJIhHOe CMeeHHe 3JIeKTpOH8 npH ero
,lJ;BHmeHHR co CKOpoCTLIO v(v r , VII) B MBrHHTHOM nOJIe seKTOp HH,lJ;YK-
D;HH KOTOpOro B nepneHJJ;HKYJISIpeH nJIOCKOCTH (pHC. 33), Qepe3 tJ. x.
Torp;a F II = Bev %'
tJ.t Bev;rtJ.t Be
tJ.v = - = = -tJ.x TaK Ka.K!!.X = V x Ill.
"m m m
CYMMHpya (HHTerpHPY.R), nOJIyqHM
v u :::: vO v + : (R-Ro)= : (R-Ro) TaKK8K VOy=O.
45

z
CPaBHHM KHHeTHqeCKUe 3Hep-
rUH 3JIeKTpOHa ,lJ,O H nOCJIe CMem;e-
HRB.:
[ E - m ( 2 2 ) ....... m 2
· - 2" . + V. .. 2" V. ·
E = mvo
H 2.
H3 38.KOHft coxpaaeHHS 3HeprHH
CJIeP;YeT. qTO vi' <: vO. T.e.
Be vom
-(R-Ro)  v o ;;::) R " Be +-.
m Be
CJIeP;OB8TenLHO,pH8JILHOeOTKJIOHeHHeKOHeqHo.
CKOpocTh Ape:A.pa onpep;eJIHM H3 BTOporo 38.KOH8 HLIOTOH8:
he. 34
mv 2
eBvAf) ;::: R.
PaHbme 6HJIO P;OK838HO, qTO
Be
vnp = -(R - Ro). (2)
m
(I)
IIpHMeHHM 38KOH COXpaHeHHH MOMeHTa RMIIYJILCa (pUC. 34):
v" R;:; voRo. (3)
H3 BHpa)KeHH:A (1) - (3) H8XOP;HM
v R m 2
(R - Ro) ,R3 ;:; 2 2 .
B e
T8KKaK
_ Ro
v. - VOR
 = -eBv r ,
e
a z = --Bv r ,
m
e
/).V Z = --Bl1r,
m
TO
e
V z = --B(R - Ro}.
m
TaK K8.K R> Ro, TO /).V. < 0  T .e. ro>efi.cl> npOHCXOP;HT npoTHE OCR OZ.
46

3aaWJ8 3
1. Ha BeKTopHoA ,AHarpaMMhI HanpameBHA 11.1111 O,AHOrO KOHTypa
(puc. 35) cJle,ZJ;yer
U L = 2U e sin i. (1)
Ha BeKTopHo:li ,ZJ;HarpaNMhI AJUI CHJI TOKOD B yaJle (pRC. 36) CJle,AyeT
Ie = 2I L sin i.
nepeNHO)IUiB BhIpWKeBH.R (1) B (2), nOJI}"lHM
U L = 4 U C sin.a !...
I L Ie 2
(2)
OrCIO,A8
. .2 <p ZL 00 2
8m -=-=-1£
2 4Z c 4
H
r,Ae
<p = 2 arcsin  .J LC ,
2
o  m  JiC '
2. 3anHmeN ypaBHeHHe BOJIBhl
rr . ( 27tX )
 =vOSffi oot+'A .
C,ABRr cl>s3 Mey cocep;HHMH .ft1le:liKaMH paseH
G';_ 27t% .
A
(8)
(4)
-
U CJ
1;1
-
U L
-
U cz
1;
-
I LZ
he. 85
PRe. 36
47

m
k
k
m
k
m
he. 37
Haiip;eM cl>a30Byro CKOpOCTL paCrrpOCTpaHeBH.H KOJIe6aHHii:
21tx
cot +  = const,
coAt + 21tAx = 0
A '
_ Ax _ COA
v - Ai - - 21t .
H3 BLIpameHHH (4) H (5) npH x = I nOJI}"lHM
col
V=-.
«p
IIO,lJ,CT8BHB 3H8qeHHe <p H3 BLIp8.meHH.H (3), H8eM
col
v=
2 arcsin ; oJ LC
wi
3. IIo;ll;cTaBHM B cl>OPMYJIY v = - 3H8TIeHHe
<p
2sin .!
2
co = -
..fiE
nOJryqeHHOe H3 BLIp8meHH.H (3). Torp;a ;l];JI.H v 6y.v;eM HMeTb
sin .!
1 2
V=--
..fiE ! .
2
I
CKOpoCTL pacnpocTpaHeHH.H BOJIHLI nOCTO.HHHa H pasHa v = r;:;:;
-vLC
sin!
2
npH BLIIIOJIHeHHH YCJIOBHH -= const, T.e. IIPH MaJILIX 3HaQeHH.HX
!.
2
C)J;BHra (}>a3LI (f' (<p « 1 pa,n;).
48
(5)
(6)
(7)
(8)
(9)

I
113 BhIpameHHH (8) CJIeP;YeT, "lTO V = (";";; IIPH BhInOJIHeHRH YCJIO-
"i LC
BRR ro .J LC «1. (.n;eAcTBHTeJIhHO ro .J LC = 2 sin   tp « 1 p8,[t.)
4. IIpoCT8.H MeXftHH'IeCKftSl MOp;eJIh, ftHftJIorH'lH8S1 38JJ;ftHHOH SJIeR-
TpH'lecKOH !:tenn, npep;CTftBJIeHft Hft pRc.3 7.
113 BeRTopHoii p;HarpftMMhI 1J.1ISI CMem;eHuii rpY30B (pRC. 38) nOJIY-
'lHM
/::,,1 = 2x sin.! . (10)
2
113 BeKTopHOH ,n;nsrpSMMhI .D;JUI CHJI ynpyrOCTU (pHC. 39) 3anumeM:
mx" = 2k Al sin !. .
2
(11)
113 yp8.BHeHHSI rapMOHHtleCRRX KOJIe6811H:A x" = _(02X, HCnOJIb3Y.H
BhIpSmeHH.R (10) II (11), no.nyqfteM
2kMsin! 2sin!
2 _ x.':ux _ 2 2
m ---
Xmu: m l!J
4ksin 2 !
2
m
. <P corm
sm 2 = 2VJ;.
CpaBHHBSa. nOJIyqeHHhIe BhIpWKeHH.H C BhIpameHII.HMH ,n;JIJI 9neKT-
pH1:leCKOii !:tenH, BIf.AHM nOJIHYJO SHaJIOrH'tiHOCTh BeJIH1:lHB  H LC, a8-
KOHOB ,n;JIJI 9JIeRTpH1:leCRHX 3ap.H.D;OB H cMem;eHRii npymllH, 9JIeKTpn-
"leCKHX HSnpJlmeHUH II CRn ynpyrocTeii.
oTRy,n;a
(12)
Xl
kAI
-
Al
-
F
X 2
kAl
PRC. 38
PRC. 39
49

XIX MeJKAYHGpOAHGII
cl»M3M'IeCKGR OIlMMnMGAG
1988 r., ABCTpMJI
381';8'18 1
IIpoqecchI H3J1yqeallJl II nOrJIOm;eHHH cftoTOHOB 06paTHMhl; BCJIeA
38 npoqeccoM nOrJlOeHHJI 4»OTOH8 8TOMON NOJKeT Ua6nlO,D;8TbCSI ero
CnOHT8HHOe H3JIyqeHHe (4)>nyopecn,eHn,H.H). 9T0 SiBJleHHe HcnoJlbsyeT-
CSi B COBpeMeHHhIX MeTO)tax 06a8PY'KeHWI H H;J;eHTHc}JHK qaCTU:r(,
a T8Kme ,D;JlH cneKTpoCKonHH C:KopoCTe:A aTOMHhIX '18CTII.tt.
B H,ZJ;e8JIbBo:ft SKcnepHMeHT8JIhBoA: YCT8HOBKe (puc. (0) )tBllm}'TCJI
qaCTH:r(hI (O,D;aOKpaTHo HOHH80B8HHhIe HOBhI) co CKOpocTbJO V H8BCTpe
qy nyqy Jla3epa c H3MeHJleMo:A ,D;JIHBoit BOJlELl. ITOKO.mJJ;SecS qaCTHIthI
(v = 0) MOryr 6hITL B036ymp;eBht H3JIyqeHHeM C P;JIHHOft BOJlHLI Ao ==
= 600 aM. ,n;BHJKHec.H qaCTH:r(hI B COOTBeTCTBHH C npHanOM .D;on-
nepa Bos6ym.u;WOTCB H3J1yqeHueM C p;pyroA: 'IRCTOTOA, nOSTOMY' ,D;JlH IIX
B036ym.u;eHHII n83ep p;oJlmeu n8CTpaHB8TLCJI Ba ,ApyryIO ,D;JIHHY BOJlBM
A{v). CneKTp CKopocTeA: HOBOB B ,n;Hana30He OT VI = 0 AO V 2 = 6000 M/c
npHBep;ea aR PIlC. 41.
1. HcnoJID3Ys KJIaCCH'lecKH:A a4>QJeKT .D;onnepa, onpe,ll;eJIHTe, B Ka-
KON ,AHanSaOae P;JIHB BOJIH ,D;OJImao HQXOJP{TLC.R H3JIyqeHue .na3epa .n;JIH
B036ymeHB.R Bcex 'laCTH:r(. IIpep;cTaBLTe KaqecTBeBBo pacnpeeJIeHHe
'tJHcna norJlOm;aeMhtX ToaOB B 38.BHCHNOCTH OT AJlHHhI BOJIHLI Jlaaepa.
II oiJcmpauBaeMbtU
J1a3ep
J...
-
v
(]JJlYOpeCqeHqUJl
l1emelCmop
he. 40
50

1;:3
(II) 

Q:I 0 C'I
0 g,.;:)
 o
0 
;::s
0 ;:)
 E D.V
  0
 
t:r- 

::!
('t:)
0
I
I
I
15 000 v. M/C
5000
10 000
PHC. 41
Bonee cTporn:A paC1leT Tpe6YeT npnMeHeHHH peJIHTHBncTcKoft f}>op-
MYnLI
, l+v/c
v ;;;;;; v -.
1- vIe
On;eHnTe norpemHocTh npH HcnOnh30B8.HHH KJI8CCn1leCKOrO npH-
6J1HmeHHHa
2. IIycTh HOHLI nepe):l; HX Boa6ym):l;eHHeN YCKopalOTCSl3JIeKTpn'lec-
KHM nOJIeN npn HanpJDKeHnn U a K8.KOBa KOJIJrqecTBeHHaH a8.BHCHMOCTh
mHpllHLI cneKTpa CKopocTeft /)..v OT YCKopHlOero HanpameHIIH U?
YseJIHTlHBaeTCJI HJIH }TMeHhmaeTCSI mHpHH8 cneKTpa CKopOCTeii qac-
TUIt B pe3YJIhT8Te IIX YCKOpeHII.R?
e
3a HOH C OTHomeHHeM 3apH,ZJ;8 K Macce - = 4 . 10 6 KJI/Kr HNee'!' ,n;Be
m
,n;JIHHhI BOJIHhI Boa6ym.zteHH.H: Al = 600 HM II A2 = Al + 10- 3 HM. IIOKRmH-
Te t qTO ,n;Ba nHTepBaJla ,n;JIHH BonH cneKTpa JIaaepa npn B035Y)I(,lJ;eHHH
Beex HOHOB nepeKpLlB8lOTCSI. Bo3MO)KHO JIB paa,n;eJIeHHe CneKTpOB B03-
6Y)K)I;eHHSI )tBYX ypOBHe:A B peaYJIhTaTe YCKOpeHH.R HOHOB B 3JIeKTpn-
1leCKOM DOJIe npR HanpH>KeHHH U? ECJIR )t8, TO Onpe,AeJIHTe MHHHMaJIh-
HOe 3H8'1eHHe HanpSl>KeHH.R.
3a,n;a"lla 2
O.lJ;HOPO)tHhI:A ItHJIHHp;pnqeCKHH HCK (Nacc8 M;;;;;; 0,40 Kr pa.n;HYC
R = 0,060 N, TOJIHH8 d = OOI0 14) DO.lJ;BemeH aB ,n;BYX npllKpeOJIeH-
HhIX K era ocn (p8JJ;HYCON r) HHT.HX OAHB8KOBO:A: AJIHHLI. MaCCOH II TOJI-
ID;HHOH HHTeft t 8 T8K>Ke NaccoA: OCB MomBo npeue6pe1Jh. IIo,n;H.RB IteHTp
T.RmecTH )tHCK8 HaMaThIB8.HHeM HHTeii aa BLICOTY H = 1,0 M, ,n;B1OT ,AHC-
KY B03MomHOCTh OOYCK8ThCS: (pRC. (2). ,l];ocTHrH}'B caMoro HB)I(Hero
nOJlOmeHH.H, ,ABCK OO.HTL H81lHH8eT nO,D;HHM:8ThCH. IIpe,n;noJIar8.S1 AJI.H
51

i 
L1ff
r
I
I
t
I
I
I

./{---.j.,
t,
fl 1.>
H
S I
,--1...
/ I'
, .1 ,
, ..... I
, /
...
, f
..:
PHC. 42
PHC. 43
PHC. 44
ynpom;eHHH, 'lITO ToutaA BCer.D;B HftXO.D;HTCSI fiB O.D;HO:H BepTHKBJIH C TOq-
RoA P (pHC. (3) pemHTe cJIe,nyrom;lIe 38.J];SQH.
1. K8KYlO yrJIOByJO CROpOCTL pS3BHBaeT ,lJ;HCR nOCJIe Toro, KaK ero
eHTp TSlmeCTH COBepmHT nepeMem;eHHe 8?
2. KaKYIO RHHeTl{'qecK}TIO 3HepI'H1O nOCTYII8TeJIbHoro ,lJ;BH>KeHHHHMeeT
,lJ;HCK nOCJIe npOXO)KJ:1;eHHH ero :u;eHTpoM TSlmecTH nyrH 8 = 0,50 M? B K8-
ROM COOTHomeHHH HB.XO,lJ;HTCH 3T8 3HeprHH C ,u;pyrHMH BH,lJ;BMH &Hep-
rHB, ecJIH P8.,Q;HYC OCR r:;: 0,0030 M?
3. K8.KOB8 CRJIB H8TameHHSI K8)KJ:1;0:H 113 ,lJ;BYX HIITe:A BO BpeM.H onyc-
RSHHSI ,lJ;HCK8?
4. Haii,lJ;HTe yrJIOByro CROpOCTL ,lJ;IICR8 BO BpeMs era nO,lJ;'beMa KaK
f}:)YUKHIO yrJIa nOBopOTa q> (pHC. 44). YCTBHoBHTe, no MeHLmeii Mepe
RBQeCTBeHHO, KOMnoueHTLI nepeMem;eHHSI H CKOpOCTH IteUTpa THmec-
TH ,lJ;KCRB B ,lJ,eKapTOBoH CHCTeMe ROOP,lJ;HHBT K8K cl>YHRIUllO yrJIa Ip.
5. MHHHMBJILUOe 3U8QeHKe CHJILI H8T.R:meHHH, npH ROTOpOM HHTL
paapLIB8eTca, paBUO T min:;: 10 H. KaKos8 MaKCH'M8JILH8.a BLICOT8 onyc-
R8HHH ,lJ;HCKB, npn KOTOpoii HHTL BO Bpe'MH nO,lJ;'be'M8 ,lJ;HCR8 em;e He
p8aopBeTCB?
3a,lJ;B'IB 3
B BLIcoKoTeMnepBTYPUO:H IIJI83Me r8aOBoro paSp.R)J;8 H8XO,lJ;SlTC.H
(Z - 1)-Kp8THo HOHH30BBUHLIe HOUhI BTOMOD, cO,lJ;ep)KaHX Z npOTO-
HOB B .RAPe. 0603HBQHM TBKHe BTOMLI A (Z - 1) +.
1. PBCCMOTPHM cJIyqaA, KorA8 e,lJ;HHCTBeuHhI:A 3JIeKTpOH B 8TOMe
A (Z - 1) + H8XO,lJ;HTCB B OCHOBHOM: COCTOSIHHH. B &TOM COCTOHHHH cpe,lJ;-
HRB: KBa.zq>aT paccToSlHHH r sneKTpoH8 OT s.zq>8 onpeAeJIHM KBK CYM-
MY KB8JU>8TOB Heonpe,lJ;eneHHOCTeB: ero KOOp,lJ;HHBT (AX)2, (Ay)2 H (AZ)2.
Cpe,lJ;HHfi. Ksa.n;paT HMnynLcB p  SJIeKTpOH8 onpeAeJIHM R8R CYMMY
RBa.z:tpBTOB Heonpe,lJ;eJIeHHOCTeH npoeKI\HB: HMrryJILC8 HB KOOp,lJ;HHBTHLIe
52

OCI! (.6.pX)2, (l:1p,)2 II {6.pZ)2. K8KOMY HepaBeHCTBY Y,lI;OJUIeTBOpSIeT npOH3-
Bep;eHHe p  r  ?
2. HOH A (Z - 1) + MomeT 3aXBaTHTL BTOpoii 3JIeKTpOH H npH STOM HC-
nYCTHTh ctJOTOH. 3anHmHTe ypaBHeHHSI., onpep;eJISI.IOID;He 'IlaCTOTY SToro
H3JIeHHH. (BhI'IlHCJleUHH qaCTOTbI npOH3BOP;HTh He Hymao.)
3. BbI't{HCJIHTe BHyrpeaHlO1O 3HeprHIO aTOM8 A (Z - 1)'" HCXO.D;H H3
Toro €}>aKTS, qTO B OCBOBHOM: COCTOHHHH OHa HMeeT MHHHM8.JIhHOe 3H8-
'Ileane. IIpHMHTe npH aTOM CJIe,lJ,YIOID;He npH6JIHmeuHH: a) HCnOJlL3yii-
Te B BLIpameHHH p;JUI nOTeHD;HaJIbHOH SHeprHH SJIeKTpOH8 3HaqeHIle
r = r 0 H3 II. 1; 6) HCIIOJIb3yHTe B BLIpameHHH P;JI.H KHUeTH'tIeCKoA: 3Hep-
rHH 3HaqeHHe Cpe,ll;HerO KBa.n;p8Ta HMnYJIhCa p H3 npH6JIHmeHHOro
- -2
p8BeHCTB8 p  r  h 2 .
4. BHCJIHTe 8.HaJIOrHQHLIM 06paaoM 3HeprHIO CHCTeMLl A (Z - 2) + ,
B03HHKmeH B pe3YJILT8Te peKOM6HH8n;HH, B OCHOBHOM COCTOHHHH. 060-
3U8QhTe cpe,lJ,HHe paCCTOKHHSI SJIeKTpOHOB OT s.zq>8 T. II T 2 (8H8JIOrHQ-
HO TOH3 11.3). MomBo npHHHTh, 'liTO B38HMHoe paCCTO.HHHe Mem.n;y 3JIeK-
TpOH8MH paDHO r 1 + 7 2 , Cpe,lJ,Hee 3H8qeHHe }tam,ll;oro KBpaTa HMnYJIL-
-2 -2 2
ca Y,lI;OBJIeTBOpSI.eT COOTHomeHHIO Heonpe,u;eJIeHHOCTeH: PI r 1 = Ii II
-2 -2 2
P2 7 2 = 11. .
Y It 838 H II e. HcnoJIb3yHTe TOT «}>8.KT, '1TO aHeprHH aToMa B OC-
HOBHOM COCTOSIHHII MIIHHM8..1IhH8 npJI YCJIOBHH r. = r 2 .
5. KaRoe 3U8QeHHe HMeeT Z, ecJIH IIPH peKoM6uH8n;HH H3JIyqae-rCH
«}>OTOH C qaCTOTOH COo = 2,5 . 10 17 c- 1 ? qTO aTO 38 HOH?
II p II M e Q a H He. B aTOM IIYHKTe paCCMOTpHTe TOJIhKO CJIyqaii pe-
KOM6HH8.IJ;IIH HOHa B OCROBHOM COCTO.HHHH C HeIIO,ll;BJUKHLlM 3JIeKTpO-
HOM.
PeweHMII
3a.n;aQa 1
1. H3 «}>OPMYJIhI KJI8CCH'tIecKoro a(}>cpeKT8 J];oIIJIepa ,u;JISJ. 3JIeKTpOMar-
HHTHLIX BOHH
v' :: v o ( 1- )
npH YCJIOBHH, 'liTO 't,IaCTH:r(Ll ,u;OJImHLI BocnpHHlIMaTb BOJIHLl C qaCTOTOH
{o ' H8XOP;HN: P;JIHHY BOJIHLl JIa3epa A(V):
 = kV) ( l-): A(v) = J.,,( 1 +;).
53

  :e
 0 :3
:
o tS ::t
 ::t -=t
Ct)ft)
q)
o
600 600,012
A(V), HM
PRe. 45
OTCIO,JJ;D ]l;Hana30H .D;JIHH BOJIH JIDaepa T8.KOH:
A (v) E [ A (VI); A.(v 2 )] = [600 11M; 600 HM + 0,012 HN].
TaK !CaK 38BHCHNOCTb ,v;JIHHLI BOJIHhl A( v) OT CKOpOCTH qaCTHIUd
JIHHeHHa.H, TO nOrJIOW;aeMble f}>OTOHhl B ,JJ;HanS30He JIHH BOJIH JI83epa
6YmrT pacnpe.n;eJIeHLI paBHOMepHO (pHC. 45).
no peJUITHBHCTCKOA «l>oPMYJIe nOJIyqaeN
l+v/e A(V)
Ap (v):;;;;; AD = . TaR KaK V 2 /C 2 <: 4 . 10-lO TO
1 - v / c .J l _ v 2 / c 2
Ap(V) = '" (v) . (1 + :;2 )-
H3MeHeHHe JIHHbl BOJIHhI JIa3epa npH HCnOJlh30BaHHH KJIaCClftlec-
Koro a«l>f}>eKTa ):{onJIepa COCT8.BJlHeT
fj.').. = ')..(v) - Ag= Ao vie,
a npH HcnOJIb30BaHHH peJI.RTHBIICTc!Coro -
8.A p = Ap{V} - Ao= Ao vie - Ao v 2 /{2c 2 ).
OTHOCHTeJIbHaS omH6Ka B CJIyqae KJIDCCHQeCKOrO npH6JIHmeHHH
paBHa
1l1.A. - .6.A. p I A.oV2/(2)
£ = = v (; 10- 5 .
l1.A A.ov/c 2c
KaK BH,ZI;HM, OHa MaJID!
2. nYCTb v II V - CKOpoCTH HOHa11.0 H nocne YCKOpeHHSI COOTBeT-
CTBeHHO. no 38.KOHY COXpBHeHH.H SHeprHH,
miJ2 mv 2 J e
 =2+ eU , OTKy,v;8 V = v 2 +2 m U.
54

.D;JI.JI mHpUHhI cneKTpa CKopOCTen HOHOB nocne YCKOpeBH.H HMeeM
-2 -2 2 2
A - - - V 2 - VI V 2 - VI
LlV = V2 - VI = _ _ = .
V 2 + V,  V + 2 ;, U +  V: + 2 ;, U
IIpH YBeJIII'tJeHHH U 'lHCJIHTenb sToii .rq>o6H He H3MeH.HeTCB, a 3Ha-
MeHaTenb YBeJIHlJHB8eTC.H, n03TOMy C poCTOM U mHpUHa cneKTpa Av
YMeHbmaeTC.JI.
3. AHa.TIoruo TOMy KaK 3TO .D;eJIa.TIOCh BU. 1, H8XO,D;HM ABa AHa-
usaoua ,D;JIHH BonH JI83epa:
Al (v) E P" I (VI); Al (v 2 )] = [600 HM; 600 HM + 0,012 UM]
Ii
1.. 2 (v) E [A 2 (V I ); 1.. 2 (V 2 )] =[600 UM + 0,001 HM; 600 UM + 0,013 HM].
TaR KaK mHpHH8 HHTepB8JI8,lJ;JIHH BOJIH paDua
AA = A.( V 2)- A(V I ) = AV ,
c
TO npH YBeJIU'IeHHH U 3H81leHHe AA YMeHhmaeTCJl H npH HeKOTophIX
3UaTIeHH.HX U HHTepBWIbI AJIHH BOJIH na3epa Al ( V ) II A2 (v) MOryT He
nepeKpLIB8ThC.H. H8.HMeHhmHN: U min Uanp.IDKeHHe 6YAeT B TOM CJIyqae,
KorA8 HHTepBaJIbI Al (v) H A 2 (v) TOJIbKO HallHH8.JOT paCXOAHTbCB, T.e.
Ror.n;a Al ( V2) = 1.. 2 (VI).
[  V + 2 ;, U mm ] [  v: + 2 ;, U mm ]
OrClO.n;8 Al 1 + = 1.. 2 1 + .
C C
ECJIH yqecTL, TIT O VI = 0, H B3 .RTh A 2 - Al = 10- 3 HN: = A I, TO nOJIyqHM
A,  v + 2 ;, UmJn = c!J.A + A2 2 ;, UmJn .
Bo3Be.n;.R o6e qaCTH ypaBueHH.R B KB8.ApaT, nOJIyqHM KB8.,lJ;pSTUOe YP8B-
BeURe OTBOCHTeJIbBD ,JU m1n :
e ( 2 2 ) - r::; ITr- ( 2 2 2 -2 )
U mm .2-;;; 1.. 2 - Al + 2cA2 2;;; VUmin - V 2 A I - c AJ... = o.
55

TaK 1<8K .JU min > 0 , TO B03MOmHO TOJILJ(O O)1;UO pemeUHe:
C 2 f1):.2 '). + (). _). )( V 2 "-: _ c 2 f1I2 ) - cf1IAa
p;:;; = .
nun J2 ()'-)':)
-2 .,.i
TaK R8K c 2 6. 'A 'A ::: 324 .10 16 (HM)4 . 2'
c
a (- Ai')(r4Ai' _C 2 AA2) = 1,54-10'" (mlt -  .
TO c 2 A):2'A: »( A - A:) (V:A: - c 2 A):2) H n03TOMY
[ ] 2 [ ] 2
( 2 2 ) ( 2 2 2 -2 )
Aa - Al V 2 A 1 -c A), v:'). _c 2 f1I2
U min = = = 159,8 B.
2Cf1IA2 J2: (A - ').) 2 J2  CM A2
3ap;8'18 2
1. 3annmeN YCJIOBHe }1.JISI TO'lKH KacaHH.R A (mecTJ(oe cr.cenJIesHe):
v - coor= 0, v = COo r. (1)
H3 3aKOB8 coxpaHeUHII 3ueprHH CJIeAyeT (CN. pRC. 48):
Mv 2 lco 2
MgH ::::Mg(H - 8) ++T => 2Mgs =Mr CO + lco.
MR2
'Yqor.a, 'lTO NOMeHT HHepn,HH WlCKa I =  (OCLIO npeue6peraeN),
nonyqHN
2Mgs = Mr2ro + lro = Mr2ro + M:2 co ::) 4g8 = R2ffi( 1 + : ).
r 2 2 .ji';
Ho R 2 «1, CJIep;OBRTenhHO, ffio = .
2.
M v 2 M r2()} 2gsr 2 M
E"u ::::  :::: 2 ;:; R2 9,8 . 10- 3 >K
ElL ;:;:: I(j,) = MR2ro = Mgs= R2 E..
.- 2 4 2r 2 -'
R2
E" = Mgs == 2r 2 E"u.
56

3. Ha 38Koa8 COXpa.HeHHH 3HeprHH HMeeM (pRC. 46) E" + Ep := Eo =
=con8t,cHeoBaTenhHo,
( MR2 2 )
dE dE d ( Mv2 ) d _ 2 co
 p
-+-=0::) +
*  2* 2
d(Mgs) dv R 2 du
= Mv-+M-v-=Mgv::)
dt dt 2r 2 dt
dv 1 ( r2 ) 2r2
a=-=g - « 1, a=g-.
dt R 2 R 2 R 2
1+-
2r 2
113 BTOpOrO 38KOHft HhlOTOaa B npOeKIJ;,RH aft OCL OY (c yqeTOM 3H8-
qeHHSI npOeKIJ;Hii) CJIeyeT (pRC. 47):
r 2 Mg ( 2r2 )
Mg- 2T=Ma = 2Mg R2 :::) T = 2"""" 1- R2 = 1,95H.
4. T8.l( K8K r « H, a 2T  M g, TO, KaK CJIeyeT H3 pRC. 48, Max :=
. r r r R
=2TSlna 2T H  Mg H => aJ: -g H -all H «all'
n03TOMY p;BHmeHReM no OCR OX MomBO npeHe6peTlh: V x = O.
Ha CRMMeTpHH (MLI TaR BLI6npaeM x = 0): x:= O.
I..{eHTp AHCR8 BcerJl;8 )J;BHJKeTCH no npSlMoA, a B HHm-
BeA TO'IKe HRTh nepexop;HT C O,AHo:A CTOpOHhI HeK8 aa
JU>yrY1<>, H a8'IHH8eTCH nO'beM. B CHCTeMe OTC'IeT8 IJ;ea-
Tpa Mace HCKa Jl;JUI TO'IKH K8C8HHSI A 3anumeM:
vI/ := ror sin (<<p + 90°) = ror cos q>.
-
v
PRe. 46
a
2T
Ja
Mg
Mg
Y
PRe. 47
PRe. 48
57

wr
wi
PRe. 49
Kale ULl BH.o;eJIH Ell" «EA.,,; E.... == Ell' TaR
KaK 38 BpeMH nepexo,n;a i1H  r« H  TO MO)KHO
a8nUC8TL:
Ep = const
E. = E = const =:> OJ = const = J4gH = 2,fiii ,
..... R 2 R
V... = 2 J gH .!:..coscp.
R
Ha pHC. 49 BHO qTO BO BpeMH nepexo,na
i1y = r sin q> :::) y = H + r sin cpo
nO'heM llH8JlOrlNeH cnycKY (BCJIeJJ;cTBHe 06paTHMocTH BH)KeHH.H):
CnycK ITepexo,n; IIoA"DeM
x=o x=o x=o
vx=O vx==O vx=O
y = H + r y = H + r sin <p y = H - r (cp - 1t)
v y == rro == 2r .J g(H + np) V v = 2 R r .JgH cos<p Vv == 2r .J g[H - r(q> -1t))
R R
ro =  ..Jg (H + r<p)
R
ro =  .JgH
R
ro ==  .J g[H - r(q> - it)]
R
5. 3anHmeM BTOpOH 38KOH fuIOTOH8. B npoeK:Q;HH Uft OCL OY (so
BpeM.H nepexo,P;a):
2T-MK=-Ma.=M( - d;; )=> 2T = M[ g+( _ d;; )}
dUll dv v d<p dv v r;-::;; r. .J 4gH 4gH r .
-- = ---:= --CJ) = v4gH -Slnq> = -Slnq>.
dt dcp dt d£p R R R2
CHJIa T MaKCHMIlJILHa, npH 1IUIKCHl\mJILIIOM 3HaqeHRH (- ; J
58
(- d;;L
_ 4gHr
- fi2'

n
0603HS'IIHB'IIepe3 T 2 H TI CHJILI H8T.RmeHH.R HHTH npH yrJI8X cp = 2' H
7t
<P = -'2' nOJIytIHM
T 2 = M ( g + 4gHr ) .
2 R 2
Mg
MLI BH)J;HM, qTO T 2> 2 > T I.
IIpH Hmu T 2 = T roax =::)
2Tmu _ 4gH m u. r H
=>--g+ 2 => mQ
M R
J-gr _ (2T_-Mg)R'
4gr 4Mgr
1,23 M.
3a.n;a 3
1. BCJIe,lJ;CTBHe D;,eHTpaJIhHOH CHMMeTpHH
;: = (.6.x)2 + (Lly)2 + (.1z)2 = 3(.6.x)2;
p  = (LlPx)2 + (Llp lI )2 + (6.pz)2 = 3 (f¥J,,)2 .
_ 9h 2
EcJIH Llp" .6.x :;t h /2 , TO r  p  = 9(6.%)2 (6.Px)2 :;t 4.
2. IIycTh M - Maccs HOHSA z +, v - CKOpOCTL SJIeKTpOHa nepe pe-
KON6HHaI{,HeA, U - CKOpOCTL HOHa A (ZI)+, W - CKOpOCTh HOHB
A (Z-2)+ nOCJIe peKoM6HHHH. Tor,n;a
mv 2 (M+m) u 2 (M+2m)w 2
2 + 2 + E..(z_l) = 2 + E-tZ_2) + hmo
H
mv + (M + m)u =(M + 2m) w + hmo/c.
8HaeHH.R E aB (Z-l) H E BH (z-2) onpeeJImoTC.R B nOCJIe,lJ,YJOHX nYHK-
Tax.
3. nOJIH8H 3HeprH.R 8JIeKTpOHa
E = d _ kZe 2 .
2m r
59

IIOCKOJIbKY por = 11., TO
E = P _ kZe 2 po
2m h.
IIpH E = E min
dE = O.
dpo
Tor,lJ;8
2po _ kZe 2 = 0 P _ kZe 2 m
2m Ii ,0 - Ii .
TIOSTOMY
k 2 Z 2 e 4 m
E =
_(Z -1) 21i 2
h 2 Z 2 e 4 m
11 2
=
k 2 Z 2 e4. m
21i 2
4. IIoJIHaH SHeprHB ,lJ,BYX SJIeKTpOHOB
d  e 2 e 2 he 2
E=-+--kZ--kZ-+-.
2m 2m 'i 7 2 li2
Ii Ii
TaR KaK P=Pl =P2= - = -,TO
1j 7 2
2 2 h ,17 2 L'7 2 h 2
E = .!i.. + .!!!... _  _  e P2 +....!..l!...;
2m 2m Ii Ii 21i
E == p2 _ he 2 p ( 2Z _!. ) .
m Ii 2 '
dE = 2p _ he 2 ( 2Z _! ) = 0  p = ke 2 m( Z -) ;
dp m Ii 2 Ii
E = k'e'm( Z - U 2k'e'm( Z - r = k'e'm( Z - )'
aB(Z-2) li2 li2 li 2
5.
_ k 2 Z 2 e 4 m = (M + 2m)w 2
2li 2 2
hcoo
O={M +2m)w+-.
c
k2e4m( Z _ )2
- hro o ,
li 2
60

IIonyqHM /1.1l1I onpe/1.eJIeHHH Z ypaBHeHHe
k 2 e 4 m [ 2 ( z - ! ) 2 _ Z2 ] - h(iJo ( 1 + hco o J = o.
212 2 4 (M + 2m)c 2
TaKKaK (M+2m)c:> mc 2 , TO
#ico O
(M + m) C 2 < 1,7 .10- 7 « 1,
Tor/1.a
2,179 . 10- 18 [2( Z -tY -Z.] - 2,642 · 10- 17 = 0,
1
Z2_Z+g -12,1248=0,
Z2 - Z -11,9998 = O.
OrCIO/1.ft Zl = 4; Z2 = -8.
HCKOMhIH MORA (Z-2)+ eCTb Be 2 + (6epHJIllHH).

Po
Po
C
.
B
-
.
- - - - -
- - A = -
- - - - -
Po
PRC. 50
t
t 2
t 1
o
'tl
T
PRC. 51
MeUH "(1 H TeNllepaTypLI t 1 H t 2 , COOTBeTCTBYJOIItHe rOpH30HTaJIhHhIM
OTpe31C8M rpa4>uKa. (3aaqeHHB t 1 H t 2 HymHo BaATH B BH,lI;e Q,eJIoro 'IIHC-
JI8 rP8.AYCOB, a MaCChI )KHAKOCTeH - C TO'llHOCThIO 11.0 11.ecSITLIX rpaM-
Ma.)
n p II Me'll a H He. ITpeAnoJIaraeTc.H, 'lTO nap:hi: paCC:M8TpHBaeMLIx
2KRAKOCTeA c xopomuM npm>JIH)KeHReN: no.a;1:JBBfiIOTC.H 38.K0HY )];8JIh-
TOBa, cornaCBO KOTOpoNY p;aBJIeHHe CMecR ra30B pasHO C}TMMe napu-
aJIltHLIX p;8.BneHuA ra30B, COCT8.BJIJlIOIItRX CMech, H 'liTO B HHTepBaJle
8.BJIeHHA, COOTBeTCTBYIOIItHX p;8.BJIeHH.RM H8C!.Im;eUHLIX napOB, EX
MO)I(HO paCC:M8TpHBaTh IC8I( H,lte8JIbUhIe r8.3hI.
3aAa-qa 2
Tpu MATepR8JIhHhIe TO'lKH P l' P 2' Pa (He nemam;He aa oHoA np$l-
MoA) MaCCaNH COOTBeTCTBeHHO m l , m 2 . ma. B38HMOeAcTBYSl Me)K,Q;Y
co6oA TOJlhKO rpaBHT8.ItHOHHO, C ApymMH 06"heKTaMH He B38HMOAeA-
CTBYIOT. nYCTIt cr 0603H8'laeT OCh. npoX011.S1m;ylO Qepe3 eHTp Macc 8THX
63

TO'tleK nepneH,lJ,HKYJI.RpHO IIJIOCI<.OCTH TpeyrOJILHHKa P 1 P 2 P a . Kal<.HM
YCJIOBH.RM ,lJ;OJI>KHbI y.u;OBJIeTBOpHTL paCCTO.RHHH Mem.u;y COOTBeTCTBYIO-
HMH TOql<.aMH: P 1 P 2 = a 12 , P 2 P a = a23 P I P 3 = au - H yrJIOBaH CKO-
pOCTh ro BpaIIteHH.R CHCTeMbI paCCMaTpHBae:MhIX TOqeK (OTHOCHTeJILHO
OCH cr), 'tJ:T06bI BO BpeMJI aToro BpaIqeHHH cl>opMa TpeyrOJIbHHK8 PIP 2 P 3
He H3MeHHJIaCh, T.e. "tITo6hI CHCTeMa BpaIItB.JIaCh RaK TBep.o;oe TeJIO?
3a.o;8'183
HccJIeAYU:Te B03MO:iKHOCTb H CJIe,lJ,CTBHSI npeo6pa30BB.HHH 3JIeKTpoH-
Boro MHKpOCI<Ona (c MarHHTRbIM ynpaBJleHHeM 3JIel<.TpoHHhIM nyqKoM
R YCI<OpHIOIItHM HanpSl:iKeHHeM U = 511 KB) B npOToHHbIH, B KOTOpOM
nyqOI< npoToHoB YCI<OpHeTC.H H8np.R>KeHHeM (U). ,Z:J;JI.H aToro BLInOJI-
HHTe ,n:B8 CJIe,n:YIOII..:J;HX 38.)J;8HHH.
1. ::bleKTpoH nOCJIe BblXO,lJ;R H3 YCTpoiiCTB8, YCI<OpHBmero ero C
nOMOLIO HanpH.meHHSi U, non8.n;aeT B 06JIaCTb HeO,lJ;HOpo,n:Horo
MarHHTHoro nOJla C HH,n:YKD;neH 13 . C03,lJ;aBaeMoro CHCTeMoii HenO,lJ;-
BH>KHhlX K8TyIIIeI< Ll  L2 L8, .... Ln. CHJIa TORa B RftTymKax paBHa
COOTBeTCTBeHHO ip i2 ia. ... in- B STOH 06JIaCTH 3J1eRTpOH ,lJ;BHmeTC.H
no HeKoTopo:A TpaeKTopHH T. qeMY OJI)KHa 6LITL paSHa CHJIa TOKa
4, i, i. ... i B K8TymK8x L1, L2, L3. ..., Ln. 'tlTo6LI npoToH, yCKO-
peHHbIH nanp.H»\eHHeM (-U). noa.H B 06JIaCTh M8I'HHTHOro nOJISi Mor
,l];BHraThCSI no TOH me TpaeKTopnH T (n B TY me CTOPOHY), no KaKon
,lJ;BHranCH8JIeKTpoH?
¥ K a 3 a H He. IIonytIHTh OTBeT HR 3TOT Bonpoc MOHCHO, HaH,n:H yc-
JIOBlle, IIPH KOTOpOM ypaBHeHH.R onHcLIBalOHe TpaeKTopHIO, B 060-
HX CJ1yqaax O,lJ;HHaKOBhl. MO>KHO HCIIOJIh30BaTb COOTHOllieHHe
_ d _ I d -2 1 d 2
P dt P = 2 dt P = 2 dt P .
2. BbIHcHHTe. BO CKOJIhKO pa3 H3MeHHJIaCh 6bI pa3pelIIalO8.JI cno-
C06HOCTb paCCM8TpHB8eMoro MHRpOCKona (T .e. MHHHMaJIbHOe paCCTO-
HHRe Me)K.ny ,lJ;BYM.H TOqe'tlHbIMH npe,lJ;MeTaMH. H306pameHHSI KOTOpLIX
B BHJ];e ,lJ;HCKOB pa3JIH'tIHMbI) ecJIH 3BMeHHTb nOK SJIeKTpOHOB n-
KOM npoTOHOB npe,lJ;nOJIaraJI. "ITO pa3pem8lO8SI cnoco6HOCTL 06YCJIOB-
JIeHa TOJIbKO BOJlHOBhIMH CBOHCTBRMH MRTepHH_ IIpHHHMaeM CJIe,lJ,yro-
ee: CKOpOCTH SJIeKTpoHOB H npOTOHOB nepe,lJ; HX YCKopeHHeM B YCT-
po:AcTBe paBHbI HYllIO; B38HMO,lJ;eiicTSHe co6CTBeHHLIX Mal'HHTHhIX
MOMeHTOB 9JIeKTpOHOB H npOTOHOB C MaTHIITIThlM nOJIeM npeHe6pe)KH-
Mo MaJIO; paCCMaTpHBaeMhIe 'tl8CTHIJ;LI ,1J;BHraHCb B MarHHTHOM nOJIe,
He HcnYCKBIOT SJ1eKTpoNaI'HHTHOI'O H3JIyqeHHH.
II p H M e q a H He_ B paCqeTftX CJIe,lJ,yeT HMeTb B BH,lI;y, 'liTO: 1) 3Hep-
I'HH nOKOH SJIeKTpOHB Ee:;: mf!c2 = 511 KsB; 2) SHeprH.R nOKO.R npOTOHa
Ep:: m p c 2 :;: 938 MsB.
64

PeweHMA
3a.n;aQa 1
1. KHlIeHHe IIpOHCXOHT Tora, Kora ,n:8BJIeHHe HaChIeHHLIX na-
poB p8.BHO aTMoctPepHOMY:
P,=Po  In () = o.
ITO,[l;CT8.BJISUI B q,oPMYJIY (1) H3 YCJIOBH.R
(1)
InE!.. = a, + b
T I'
Po
nOJI}"IHM
at
TKHfi' = -b.
I
(2)
qHCJIOBLIe 3HaqeHHH at H b t MOjKHO nOJIyqHTb, no,n:CTaBHB ,lJ;8HHLIe
H3 Ta6JIhI (CM. c. 62). nJI.H Kam,AoH .>KH,n:KOCTH HMeeM p;Ba HeH3BeCT-
HhIX (at H b , ) H ABa yp8BHeHH.H. PemaSI CHCTer.ry ypftBHeHHH, H8XOHM
3H8QeHHSI a,H hi.
2. HcnapeHHe c nOBepxHocTH oTcyrcTByeT. PaccMoTpHM nY3hIpeK,
06p830B8BIDHHCH BHYTpH »CH,lJ;l<OCTH A, »CH,lJ,KOCTH B HJ1H H8 HX rpaHH-
D;.e. OH Mo.meT B03HHKHyTh, HanpHMep, BCJIeCTBHe q,JIYKTyaD;.HH. ECJIH
.o;aBJIeHHe B HeM MeHhme Po, nY3hIpeK HCTIe3HeT. ECJlH OH B03HHK BHYT-
pH )KHOCTH A, TO p;8BJIeHHe B HeM p = P A (T), ecJIH BHYTPH .)KH,D;KOCTH
B, TO P = PB (T), eCJIH HB rpaHHD;e, TO p;; P A (T) + PB (T). ITpn HeKOTopOH
Tel\mepaType T%CYMMapHoe.u;aBJleHHePA(T%) + PB (T x ) =Po' PA(T%) <Po
It PB (T %) < Po' T .e. nY3LIpLKH B .mH,lJ;KOCT.RxA H B HCQe3HYT, a aa rpaHH-
e - HeT. STOT II}T3LIpeK U8''lUeT nOHHMaTLC.R. ITapD;.HaJIbHOe 8B-
JIeHHe nspoB mHKoCTH B B nY3LIpbKe paBRO PB (T ,), napOD mn,D;KO-
CTHA - PA(T%), T8R K8.K mHp;KOCThA He p8CTBOpaeTCSI B mHKOCTH B
H, CJIe.n;OBaTeJIbHO, KOJIUqeCTBO napa B IIY3hIp&Ke He H3MeHHTCJI. IIo-
STOMY 3TOT nY3LIpeK ,lJ;oii,lJ;eT,lJ;O nOBepXHOCTH mHp;KOCTeH. By.n;eT HR-
6JIIO,lJ;8ThCH nOrpaHH'I:IHOe KHneHHe. T % H8XOJ];HM H3 ycJIOBH.R
PA(T%)+PB(T%)=po =>
eJA -t-flA all -t-lls
=> e T . + e T 2' = 1.
PemUTb 3TO ypaBHeHHe TO'l:lHO .n;OBOJILHO CJ10mHO. Ho TaR KaK B YC-
JIOBHH Tpe6ye-rcs TO'l:lHOCTh 1 K, TO aTO ypaBHeHHe MomBO pemHTh Me-
TO.o;OM 1I0JIOBHHHOro .o;eJIeHHSI C nOMom.;LIO KaJILKYJISITOpa. IIoJIyqaeM
T % = 340 K.
3 -379
65

Bo SpeMSI TaKOra KHneHHSI T ::;;: const. IIorpaHHtlHoMY KHneBHIO CO-
OTBeTCTBye-r nepBhI:A ropH30HT8JILHbIH yqaCTOK, T.e. i. ::;;: Tx::; 340 K.
IIY3LIpeK 06'beMOM V co.n;epmHT BeeCTBO A MaccoH m A H BeeCTBO B
Macco:A m B .
PA V ::; m A R; PB V ::; m B R::::::) m A ::;;: PA M A .
T,w: M A T. M B m B PB M B
(3)
P m
 uaXO,lJ;HM, nO,lJ,CTSBHB T % B OPMYJIY (2). nOJIyq:aeM  = 22,4,
PB mB
T .e. Macca >KH)J;KOCTH A YMeHbmseTCSI B 22,4 pa3a 6h1CTpee, 'IleM Macca
mH.n;KOCTH B. KOTAs )KHAKOCTb A nOJIHOCTbIO HcnapHTCSI JKH,lJ,KOCTb B
m B
q:aCTH'IHO BhIKHnHT: t!..m n ::;;: m -  a qaCTHTIHO OCT8HeTCSI: mB =
m A
mB
::; m (1- -)::; 95,5 r. TenepL norpBHKtlHOe KHneHHe KOH'IlHJIOCL, H
m A
CJIe,l{yIOHH HaKJIOHHLIH yq8.CTOK COOTBeTcTByeT HarpeBY »tH.n;KOCTH B
AO TeMnepaTyphI ee KHneHHH. KOJllflIecTBo sem;eCTBa B npH STOM He
MeRHeTCR H B MOMeHT t 1 m A ::;;: 0, m B ::; 95,5 r. .HCRO, TITO t 2 - TeMuepa-
Typa KHneHHH mH,T.J;KOCTH B; t 2 :;: 373 K.
3a.u;B'Ia 2
B HHepItH8JILHOA CHCTeMe OTCqeTB, CBH3aHHOH C OCLIO (J (a TaKas:
CHCTeMS cecTByeT, nOCKOJlbKY CRCTeNS TOTIeK H30JIHpOB8ua), pac-
CMOTpHM ,lJ;BHmeHHe 3THX TOqeK, C'IHTaS HX Kouc}n:frypaUHIO ueH3-
MeUHO:H, a yrJIOBYIO CKOpOCTL CHCTeMbI - 0). 0603UaQHM BeKTopLI
II  ::;;: 42  ::;;: b 13 , O::;;: Ill' r.n;e TO'lKB 0 - eHTp M8CC.
no onpeAeJIeHHIO ueHTps M.aCC,
R 1 :;:  (m a b 13 + m22)'
r.n;e M::; m 1 + m2 + mso Y CKopeHHe TOTlKH MBCCO:H m 1 paBHO
Q) ::; CD 2  :;:  (m33 + b.2).
CUJIhI, AeHCTBYJOHe UB TOqKY MBCCOH m 1 co CTOpOHbI ,lI;BYX ltPy-
I'HX TOqeK, COOTBeTCTBeHHO p8BHhI
- l12t-
1\2 :: G---a-2 
bJ.2
- l12t ma -
 = G-a-43 .
bi3
66

IIo BTOPOMY 38KOHY H&IOTOHa
2 + Fi3 = Tn.t .
IIocJIe apn4>MeTK1IeCKHX npeo6pa30BaUHH nonyqaeM
m 2 - ma - (j)2 - (02 -
bf2 2 + bfa 3 = GM m2 b 12 + GM mhta.
nOCKOJIbKY TO'lKH P l' P 2 H P 3 He JIemaT ua OHOH npBMoA, BeKTOp
;;'2 HeKOJIJIHHeapeH BeRTOpy B' T .e. H3 npeAhIym;eI'O BLIpameHHB
CJIeyeT:
m - (02 -
bP: 2 = GM m22
H
m 3 - co 2 -
bPa 3 = GM m33'
OTKY8
b,'= V '
b,'= V .
no 8U8.JIOI'HH paCCM8TpHBElH TO'lKY P 2 HJIH P 3' nOJIyqaeM
b 23 =  .
HT8K, Eeo6xOHMhIM YCJIOBH eM .HBJI$leTC.H
b.. = bu= b..= V::' (m, + m. + "'" 
TOT me pe3YJI&TaT MamBO nOJIyqHT&, HCnOJIh3ya 38KOB COxp8He-
BHa MOMenTa HMIIYJIbca.
3a.u;a-..:a 3
3aMeTHM, 'iTO KHUeM8TH'ieCKft.R SHepI'H.R SJIeKTpoH8 U= 511 KsB
pasHa ero BHYTPeHHeH 8HeprHU. n03TOMY AJI.R 3JIeKTpOa8 Heo6xoHMO
npHMeH.HTb peJISlTHBHCTCKHe 4>oPMYJILI.
1. 11-11$1 "iaCTHIthl B M:arHHTHOM no.ne
dP _i
dt - L
67

- "'0 V
P :: - HMIIyJIbC,
.J l- V 2 /C 2
FL = q[ v, R(X)] - CHJI8 JIopeHIJ;a,
B (x) - HHJJ;YKU-HH M8rHHTHOrO nOJIII (3aBHCHT TOJIbKO OT KOOpH-
H8T).
CHJI8 nepneH.o;HKYJI.Rpa8 CI(OpOCTH n03TOMY MOAYJIb CKOpOCTH,
HMnYJlbCa II saepr:nSl 'lSCTHIJ;bI He 1I3MeH.RIOTC.R:
_ djJ Id(p)2 Idp2 - _
p. dt = '2---;U- = '2 dt = F L . P = 0, T.e. p = conat.
ITYCTb 8 - nyrb , npoiiAeHHblH Q8CTHIJ;eit. TOr.o;8
ds == V; djJ = V dP == ; V dP = q [ V, B(X) ] .
dt dt ds ds
Pa3eJIHM 06e 'laCTH nOCJIe,ltHerO BblpameHH.H us pV:
d ( p ) q [( v ) - ] d - q [ _ - ]
ds P = P -;; , B(x) WIH dB r = p r, B(x) .
3.o;eCb i - e1];HHH'IlHbIH BeKTOp, COBna;:r;81OIIJ;1I:A no HanpaBJIeHHIO co
CKOpoCTbIO. 5TO ypaBHeHHe 38,AaeT TPseKTopHH BHmeHHH 'l8CTHIJ; 01];H-
H8KOBLIMH, eCJIH
qBo (x) = qpB'(x) ; .8.(x) = _ P p B(x).
p P p P
B(x) HBJIHeTCH reOMeTPJrtleCKOH CYMMoit HHJ];yKIJ;HH nOJIeA , C03A8H-
HblX K8TymKaMH, nOSTOMY HH,lI;YKU-UIO nOJI.H KWK,lJ,O:H KaTymKH HMO
H3MeHHTb B - Pp p8a: Bj - i j , T.e. AJISI sneKTpOH8
p
E2 = (m o c 2 + Ue)2= mc4 + p 2 C 2 ,
U 2 e 2 P
2 T 1 II 35 .. 35 .
Pe = mtVe+; -::: ; "i = - "J'
c p
2. KWK)J;OH qaCTHIJ;e COOTBeTCTsyeT BOJIHS JJ,e BpoibIH: A = !!, yroJI
P
pS3pemeHHH, npOnOPItHOH8JIbHbI:H A -1., yMeHbmHTCJI B p P = 35 pa3.
d p Pe
PaapemaJOaH cnoco6HoCTb YJIyqmHTC.H B 35 pa3.
68

XXIMeYHapOAHaR
.M3M'IeCKGII OnMMnHGAG
1990 r., ronnaHAMIi
3a.a;aQa 1
1. Mid XOTHM H3yqHTb .n;H4»paKItHIO peHTreHOBCKHX JIyqeH H8 npa-
:MoyrOJILHOH KPHCTaJIJIUqeCKOH pemeTKe. C STOH n;eJIblO HCCJIe.n;yeM
,P;Hcl>paKItHJO IDIOCKOA: MOHoxpOMaTH"lecKoit BOJIHId, na.zr;8JOeA: nepneH-
.n;HKYJI.HpHO Ha ,P;BYMepHYJO pemeTKY, npe,lJ;CT8BJI.mo:rqy1O C060H CHCTe-
MY B38.H:MHO nepneHKYJUIpHLIX rqeJIeH, OTCTOHIqHX .n;pyr OT .n;pyra
COOTBeTCTBeHHO H8 paCCTOHHH.HX d. H d 2 a 06m;ee 'IRCJIO m;eneH O'p;Horo
HanpBBJIeHH.H N l' npyroro - N 2. .nHpaKOBH8..H K8:pTHHa na6JIIO,P;a-
eTCH Ha 3Kp8He, KOTOphI:H paCnOJIOmeH nap8.JIJIeJIbBO pemeTKe HS pac-
CTOSiHHH L OT Bee (L » N.d. H N 2d2 ).
Onpe.n;eJIHTe nOJIOmeHHe H mHpHHY rJIBBHhIX MaKCHNYMOB HS 3K-
paBe (mHpHHoH M8KCHMYM8 U83hIB8eTCSI paCCTOSIHHe Mey 6JIHmaii-
mHMH NHHHMYMBMB C 06eBX CTOpOH M8KCHMYM8).
2. P8CCMOTpHM Teneph TOHK}'IO nJlSCTBBY Ky6ecKoro KpHCTaJIJIa
C nOCTO.HHHOH pemeTKB U H paaMepoM Noa x Nou x N.a, npH"leM
N 1 « No. KpnCTaJIJI pa3:MelI\8eTCa nOA M8JlhlM yrJIOM K S8npaBJIeHHIO
n8AeHHR peHTreHOBCKoro H3JIyqeHH.B BP;OJIL OCB Z (pHC. 52). )];Hp8.R-
HOHH8.H KapTHHa Ha6JIIO.n;seorC.R ua 3Kpane, KOTOphIH H8XO,P;HTca H8
60JIhIDOM paCCTOHHBH L OT KpHCTaJIJIa.
y
PRC. 52
69

Onpe.o;eJIHTe nOJlO)KeUBe II mUpHHY NaKCIINyMOB JC8.K 4>YHKItHIO
yr JIB 9. B 1:1aM npo.RBJISleTCS TOT c}J8KT, "'ITO N I « No?
3. )J;H4>paKI(HOBH8S1 KapTHH8 MO)KeT 6h1TL 06"bSlCHeHB HCXO.o;H H3
TeopEB Bparra. ITpeADOJlaraeTC.R, "ITO peHTreHOBCJCoe II3JIyqeHHe OT-
pa)KaeTCSI OT aTOMBhIX nJIOCKocTe:A pemeTJCH. HSTep4>epeanHeH T8.KHX
OTpIDKeHHLIX nyqKOB H C03.o;aeTCSI .o;HctJP8.KnHOHHM KapTHHa.
)];oJ<a)KHTe, qTO .OTpameHHe Bparra. ,{taeT T8KHe me COOTHomeHHH
AJUI MaKCRMYMOB, J(8.J( B n. 2.
4. ITpR T8.J( HaaLIBaeNOM nopomKOBOM MeTOe HCCJIe,u;OB8BRSI KPH-
CT8JIJIOB nyqOK peHTreBoBcKHX .nyqe:A paCCeHB8eTCH sa O"'leSL 6oJlLmoM
qUCJle M8JIeUhKHX KPUCTaJIJIHKOB, o6pa3YJOII:tHX nopomoJC. (Koue'IJHo,
pa3N:eplioI OTAeJIIIHOrO KpHCT8JlJIHKa MHoro 6oJlLme, 1:IeM nOCTO.HHBBa
pemeTKH a.) PaCCeJIHHe peSTreSOBCKoro H3JIyqeBH8 C ,lVIUSOH BOJIHhI
0,15 EM HB nopomKe COJIH RCI (HMmoeH Ky6UQecKYK> pemeTKY, no-
J<a3aBHYJO sa pRC. 53) aeT Ha 4>oTOnJI8CTHHKe (paccToSlHHe .0;0 Hee
L = 0,10 N) CHCTeNY KOHu,eHTpWlecKRX TeMBhIX KOJIeD;,. PMHYC HaH-
MeUIJmero KOJILI(8 COCTaBJIJleT 0,053 M (puc. 54). HOHhI KaJIHH H XJIOpa
RMelOT npaKTH1lecKU OAHB H Te me p83MepH H MOryT paCCM8TpHB8TLca
KaK OIniHaKOBWe pacceBB8IOID;He I(eHTpH.
BLr'ClBCJIHTe paCCTO.RHUe Ne)K,IW BYX.H COCe,u;HHMH HOHaMH K8JlH.H
B KpHCT8JIJIe RCL
..
.
.* 
.,
.
.
_.
PRe. 53
I ((E))) I
PIIc. 54
70

3aAa'la 2
npe.o;nOJIaraeTC.R BhlBO,It KOCMHtlecKoro Kopa6JI.R .ATJI8HTHC. HB
OP6HTY BOKpyr 3e:MJIH. H3secTHO, "ITO 3T8 0p6UT8 KpyrOBSsr H pacno.
JIOmeU8 B nnOCKOCTH 3KB8Topa. B HeKoTophl:A NO:MeHT .ATJIaHTHC. BLI-
nycTHT cnyrHHK S H8 Hanp8Bn.H1Oe:A H30JIHPOB8.HHO:A mTaHre AJIHHOH
L. MO)KHO C'lHTaTL, 'liTO OB8 mecTK8.R H HMeeT npeHe6pe)l(H.MO M8JI}TIO
Maccy. TpeHHe T8K)I(e 1\(0)l(HO CqHTaTL MaJIhlM. nyCTb a - yroJI, KOTO-
pJd:A COCT8BJI.HeT mT8Br8 C JIHBHeH .ATJI8HTHC. - ueBTp 3eNJIH
(pRC. 55), TaKme pacnOJIOJKeu B UJlOCKOCTH 3KBRTopa. MOmBO npHBHTL,
1:ITO MaCCR cnyrHHRa ropa3Ao MeUbme N8CCW .ATJIBBTBca.. a AJIURa
mTBBrH ropa3,D;O MeBLme pa.n;Hyca Op6UThI.
1. PaCC1lHT8HTe. npl'! K8.KHX 3HaQeHH.HX yrJla (I CRCTeN8 COXpa1I.ReT
CBoe nOJIOmeHHe OTHOCHTeJILBO 3eMJIH. RJIH. .u;pyrHMH CJIOBaMH. npH
KaKBX CBOHX 3ua-qeHHSiX yroJI (I He H3:MeHSleTCJI co speMeHeM.
2. I1p08.HaJIH3HpYHTe YCToiiqUBOCTL CHCTeMbIA -8 (.ATJl8.BTHC. -
CllYTHHK) B 3TOM nOJIo*eHHH 1J)Isr. KWK,Q,Oro p8BHOBecHoro COCTOHHH.R.
3. ITYCTb B aeKoTophIii NOMeHT BpeMesu mTaHra uOJlyqaeT ae60JIL-
moe OTKJ10HeHUe OT paBBoBecHoro COCTORHII.R. OCT8B8JlCL B UJIOCKOCTH
sKsaTopa. npH 8TOM B CHCTeNeA -8 B03HHKHYT KOJIe6UHsr.. BhIpa3UTe
nepHO,D; 3THX KOJIe6UHH 'Jepe3 SpeMsr. O,lUloro ooopoTa BORpyr 8eMJIH.
4. HH.n;YKH.R MarBHTHoro nOJIS 8eMJIH nepneUJJ;HKyngpH8 nJIOCKO-
CTH pHCyBK8 H HanpaBJIeaa sa QUT8TeJI.H (CM. pHC. 55). BCJIe,l.tCTBHe
Op6HTaJIbHOro ABHateHRH mTaHrH MeJK,ZW ee KOHaMH nOIIBJI.ReTCSI: pa3-
HOCTL nOTeHH8.JIOB. O:KPY)K81OaJ1 Cpe,o.8 (MarHHTocc})epa) npe,ncTaB-
JISleT co6oii pa3pe*eHBhIH HOHH30BaHHLIH ra3 C O"leHL BLICOKOH 8JIeKT-
pHtIeCKOA npOSO,D;HMOCThlO. C nO:MOIIJ;LIO KOHT8.KTHLIX UJIaCTHH BA H S
mT8Hra 3JIeKTpHtlecKH COeAHB.ReTC.R C HOHH30B8HHbIM ra30M. B pe3YJlL-
TBTe1:lero no BeA: UaqHHaeT HJJ;TH TOR (CHJIa TOKa I).
a) Onpe1J;eJIHTe HBnp8.BJIeHHe SToro TOKa.
Heo6xo,ItHMhIe ,It8UBhle: nepHO.u; 06paIIJ;eHHH BOKpyr 3eMJIH T ::
= 5.4 . 10 3 c; JVIHHa WT8HI'H L = 2.0 . 10 4 M; M8I'HHTH8snm;:J;YKItH.R 3eM-
JI:H H8Bh1COTeCnyTHHK8. B=5.0. 10-5TJI;
Macca .ATJIaBTHca. M = 100 T.
6) 3aTeM K IDTaHre nO,ItKJIIOQ8IOT
HCTOqHHK TOKa. pacnOJIO)l(eHHLIH
BHYTPH Kopa6JI.R, o6eCne'IJHBBIOIIJ;H:A
no.u;.n,ep»taHHe CHJIhI TOKa 0.1 A (UpH.
qeM TOK H8.llpaBJIeH npoTHBOnOJIO)KHO
nepBOHaqaJIbHoNY). Onpe.n;eJIHTe. K8K
,D;OJIro HyH<HO nOMep.)KJIB8Tb 8TOT TOK
"lT06LI ShICOTa Op6UThI KOCMJAeCKOrO
Kopa6JIH H3MeHHJIaCL DR 10 M. ClIHTaii.
o lL 0
B 
 . 
PRe. 55
71

Te a. = O. BJIHHHHeMTOI<.OB, npoTeIC8IOnprx B npOBo.o;eH MarHHTOCe-
pe, MO)l(HO npeHoopeqb. YBeJIINHTCH HJIH }TMeHbmHTC.H BhICO'T8 cnyT-
HHK8?
3ap;aQa 3
.MU.J1.D:HCeKYHJtHLIH nYJIbCap. - HCTOqHHIC H3JIyq:eHH.R: DO BCeJleH-
BoA, HcnYCKaIOm;Hft OTleUh KO'poTKHe KNnyJIhCLI C nepHO,lJ;O'M OT O,u;BOH
11.0 HecKOJII,KHX NHJIJlHceJ<YHll.. IIOCKOJIDKY qQCTOTQ SToro H3JIyqeHH.fI
HaxO,ltllTC.R B pap;HOAHana30ae, xopomUM pa,zJ;HonpHeMHHKOM MO)l(BO
npHHIlMaTL erc OTll.e.nbHLIe HNnyJIbCLI H OTleUb TOTlHO' onpell.eJI.HTL ne-
pHOll. KOJIe6aHHA.
1. Pa.ztHOHMUyJIhCLI B03HHKaIOT OT nOBepxHocTH oco(ioro THna 3Be3-
,lJ;M, TaK na3LIBaeMoA ueHTpoHHoA. T8.KHe 3Be3.o;hI npe,nCTaBJI.KIOT co-
60A OlleRL IIJIOTBhIe 3BeSJJ;UhIe 06pa30Bamt.fl p8.)J;HYCON B UecKOJIhKO 11..&-
CHTKOB KHJlOMeTpos H M8CCO:A, npH6JIH3HTeJlhHO p8BHOfi Nacce COJIU-
Its, 6111CTpo BparqalOm;Hecs BOKpyr cBoeA OCH. li3-38 SToro Bpatq;eHHH
seATpoBH8JI 3Be311.8 BeMBoro paCnJIIOIItHB8.eTCH. ITpe.u;nOJIO)l(BM, qTO
Tp - paCCTO.HBHe OT I(eHTpa 3Be3,lJ;hI 11.0 ee IIoBepxHocTH Y nOJIlOca Bpa-
m;eHHH, a r - paccTO.HBHe OT TOro )lte eBTpa.o;o nOBepXHOCTH Y 8KB8-
TOpa. TOr,lJ;8 Ko&4JHItHeBT CIIJIIOID;HBaHH.R 3Bea.D;LI MomBO BhIpa3HTL
r. -T.
T8.K: E = ..!-..!!.... PaCcMOTpHM HeHTpoHByIO 3BesJ{y, HMelOm;yIO Macey

2,0. I030lCr,cpeAHHi!:l,O .I04M,nepHo.u;BeHHSl2,O. IO-2 c .
BLl'lHCJIHTe Koa4>4>HHeBT CnJIIOHBaHHS TaKo:A BefiTpoHuOH
3Bea,lJ;hI.
(1), paJl./e
314,164
314,163
314162
314,161
314,160
314,159 1(00
I
314,158  :
,.1. . I J I I J . I .a
-10 -0,50,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 t. CYT
he. 56
72

2. }fa-38 nOTeph 3EeprHH BeHHe aBea,lXhI nOCTeneSHO 3aMeJJ;JI.R-
eTCH, n03TONY CWIJOID;HB8BHe 6y.n;er YMeHbmaTbC.R. O,AH8.KO 3Bea.u;a HMe-
(IT TBep,lXYJO KOPY, KOTopa.a .UJI8.B8eT. aft nOBepXHOCTH mIlJJ;K():H cepp;-
IJ;eBHHhI (H,lXpa). BpeM.H OT BpeMeHH B Rope npoHCXOJJ;.aT COTpSCeHHJl,
'liTO npHBOJJ;HT K CKaQK006pa3HOMY H3MeHeHHIO ee 4>OPMhI. 06H8pyme-
HO, 'liTO BO BpeM.R H nOCJIe TRKOrO COTp.HCeHHSI yrJIOB8.J1 CKOpOCTh KOphl
H3MeH.HeTC.H. 38.BHCHMOCTb yrJIOBOH CKOpOCTH OT BpeMeBH H306pame-
E8 Eft pRC. 56, rp,e 0)0 - Hftq8JIhHa.H yrJIOB8a CKOpOCTL 3Se3,lJ,hI, COI -
yr JIOBa.a CKOpoCTh BO BpeNSI COTI)JIceHH.H, 0>2 - nOCJIe COTpHceHHSI.
BLIqHCJIHTe pa,zJ;HYC )KH,lJ,KOro JI.zq>a R. U8 OCBOBe ,AaHHLIX pHC. 56.
CQnT8.iiTe, T.JTO nJIOTHOCTH KOphl H HJJ;pa op,HHaKOBhI, a H3MeHeUHeM
€}>OpMhI s.zq>a MOmBO npeHe6peT.JL.
PeweHHIl
3a.n;aQa 1
1. P8CCMOTpHN CUSQ8JI8 H8npaBJIeHRe no OCR OX. ECJIH BOJIHhI npH-
XO,lJ,.RT H3 coce.n;HHX lII;eJIeH (paCCTOHHHJI Mem.AY KOTOphIMH d t ), p83HOCTh
xop;a JIyqeii paDua
6 1 = nlA,
YCJIOBHe M8KcHMYMa d 1 sin.p1 = nlA.
ITOJIOmeUHe HX Ha sKpaHe onpe,AeJIHeTCB KoopHHaTaMH
X n = n}AL , C'IlHT8.a n}A «1.
I d:a llt
Pa3HocTh xo,na Mem.n;y q;eHTpaJIhHOH eJIhlO H eJIgMH us KP810
pemeTKH paDHa
6 ( N} J = N I nt A .
2 max 2
C ]l;pyroA: CTOpOHhI, paaaOCTh xo.n;s ,D;JIH nepBoro MHHHMYMa, CJIe]l;Y-
lOID;ero 38 rJI8BHLIM MSKCHM}'MOM,
A( l L = 1 IltA+ ; .
nOJlo»ceHHe MHHHMYMOB Hft 3Kp8He onpe.n;eJISIeTCSI BhIpameHHeM
( Nt A + !: ) L
2 2 J..L 'AL
xn" +6x= =-+-.
Nl   Nl
2
73

OrCIOlt8 CJle,n;yeT:
J..L
Ax = N} .
IIIHpHH8 rJIaDHOrO MaKCHM}TM8 paDBa
2J..L
2Ax:: Nl .
A:a8.JIOI'H't{BhIe BLIKJI8,ll;KU MomBO C,{teJI8Tb AJIa Hanp8.BJIeBHH IIO OCII
OY.
nOJIOmeBUe rJI8BHOrO M8KCHNYM8 olIpe)l;eJI.HeTCSI KOOp,nHH8TaMH
( ) (  ').L n 2 ').L )
X IJI ,Y n 2 = T'T '
8 mHpHH8 rJI8BHOro M8KCUMYM8 -
2AL 2'AL
2Ax = Nl ; 2Ay = N2 .
2. B Hanp8BJIeHHH OCH OX MhI 4BHHM. pemeTKY c marOM a TS-
KHN o6paaoM, NhI UMeeM
_  'J...L . _ 2U
x --, Ax--.
Ita a Noa
B Hanp8BJIeHHH OCII OY NhI 4BHp;Hl'tU pemeTKY C a4K}>eKTHBHl:dM ma-
roM aCDSe.
no 8B8.JIOrHH 381IHWeM:
y :: nzA.L ; Ay = 2A.L
n, acose Noacose
B H8lIp8BJIeHHB OCB OZ MhI 4BH,AHM. pemeTKY c 34>cPeKTIIBHLIM
marON a sine:
_ ')..L. A y ' _ 2')..L
Ynl - asine ' - N}asine.
ITOCKOJIhKY sine - OqeHh Ma.n:a.a BeJIHH8, 6YlteT TOJI:bKO BH,ZJ;eH
HYJIesoA: nopSiAOK AH4»paKHH, npHqeM OqeHh WHpOKO, TaR R8R
N. sine« No-
3. Ha TeopHH OTp8)KeBH.H Bparr&. CJIe,nYeT  '1TO npoUCXO"D;UT HHTep.
tl>epeJlItHSI peHTreHOBCKHX JIyqeA, oTpWKeHHhIx 8TOMHhIMH IIJIOCKOCTH-
74

MR, KOTOpLIe C03,lJ,aIOT ,AHf}>P8.K:QHOHHYIO pemeTKY. )J;n.H eoceAHHX nJIOc.
ROCTeH 3anHweM:
II = 2a sin c.p  2ac.p = nA r,lJ;e
c.p - yrOJI ,D;n4»paRD;HH. OTCIO,lJ;a CJIeAyeT:
 :=::: 2m ==:: n'A
L "t' a'
x= nJ...L .
a
MhI nOJI}"IHJIH TaRoe me BLIpameHJle, RaR B n. 2.
4. ,l{JIH paCCTOHHHH .J2a Mem.n;y coceHHMH HOnB.MH K MLI HMeeM:
x O053
tg2cp = - == - = 0,53,
L O10
_ /.. _ O15. 10- 9 0 31
a--- s==, HM.
2sincp 2 . 0.24
PaCCTO.HHHe Me)K)J;Y coce,nHHMH HOH8MH KaJIHH B RpHCT8.JIJIe KCI
pSBHO
-K ;;.J2a = J2 . 0,31 HM = 0,44 HM.
3aJ(aqa 2
1. Tax RBK MaCC8 m l .ATJl8HTHca. Muoro 60JlLme M8CCLI m 2 ero
CnYTHHK8 (ml »m 2 ), CKOpoCTb .ATJl8HTHca. npH ero .l];BHmeUHH no
KpyroBoii op6HTe BORpyr 8eMJIH nOCTOHHH8.. ):J;BH)KeHHe CfiYTHHRa
CKJl8)J;LIB8eTCH 113 RpyroBoro ,lJ;BHmeHHJI B:MeCTe C .ATJlaHTHCO:M. BO-
Kpyr 3eMJIH II (B03Momuo) KpyrOBOro ABHmeHH.R cn}1THHK8 BOKpyr .AT-
JIaHTHca. .
3anHmeM BTOpO:A 38KOH HhIOTOHa:
- ]J,J1H .ATJI8HTHca.
Gm '"-
(J)2 R = l ".:f
R 2
(m s - Macea 3eMJIH, R - pa,zJ;HYC Op6HTLI .ATJIaHTHca. ), OTCIO,na CJIe-
,AyeT,qTO
(J)2 = Gms .
If'
(1)
75

A
I L casa
I
I
I S
L________________ -
, m2
,
,
R I
I
,........ -
, ........ F.
_t___
PHC. 57
- AJIB cnYTHHKft (CM. pllC. 57)
ma" = - (F 6 - Fc) Bin a 
[')Xe F. H Fe - CHJI8 TSImeCTH R eHTpo6emH8R CRJIS, .n;eHeTBYJOHe Hft
cnyrHHK. CJIe.o;OBaTeJIhHO,
"'-2 W " ::; _ [ G "'-2(jJ2 (R - LCOBa > ] Bina.
(R-LcosaY
HCnOJIh3YH npn6JIHmeHHe
1  ....!...- + 2L COB a
(R-Lcosa)2 R2 R3
R Bh[pameHHe (1) nOJIyqHM
Ln" = _ [ Gina + 2GIna Leoaa _ Gma R + Gina Lcosa ] sina
R 2 R 3 R 3 R 3
HJIH
a" + 3 (02 sin a . cas a = O.
(2)
ECJI:H a::;: const, a"= 0, TO sina = 0 => a::;: 0, a = 1t;
it 3
cas a. = 0 =>Q= 2' a = 2'1t.
76

TaKHM 06paaOM, nOJIO.H<eHHH paBBOBeCHH ,lJ;OCTHraIOTCH npH yr-
7t
JIax a, 1<paTHhIX "'2.
2. PftBHOBeCHbIe COCTO.HHHH CHCTeMbI 6y,lJ;YT YCTOHqHBbI, eCJIJI 3Ha-
1<H MOMeHTa CHJlbI, ,u;eHcTBYIOIqerO Uft cnYTHHR, M::;: m 2 La" L ::;: m 2 L 2 a"
II 3HaRli H3MeHeBHH yrJI8 (a - (Xo) 6y,D;yT npOTHBOnOJIO»\HbI.
H3 Ta6JIHIJ;bI BII)J;HO, qTO YCTO:WIHBLIMH COCTOHHHSI CHCTeMLI 6y,lJ,YT
npH a ::;: 0 II a = 7t, a HeycTOfiqHBbIMH - npH a :; 1t/2 Ii a = 37t/2.
Ta6J1,ur.;a
3aal< (a - a o ) -+ -+ -+ -+ -+
0 7t 3 27t
a 2 1t i 1t
3H8.K M +- -+ +- -+ +-
3. ,l{JUI M8.JIhIX aHaqeHnn a BhIpameHHe (2) MomHO 3anHCRTb B BRne
a" + 3m 2 a' = O.
aTO ypaBHeHHe rapMOHUqeCKHX I<OJIe6SHHH C IJ;HI<JIH'1eCKOH qaCTO-
Ton, pSBHOH ro.J3 .
27t To
OTCIO,lJ,S T. = r;; =   058 To,
ID'V3 'V3
r,u;e To - BpeMH O,lJ,HOrO 06opoTa CIICTeM:bI BORpyr 3eMJIn.
4. a) COrJIaCHO npaBHJIY JIeHq,s, TOR HSnpaBJIeH OT cnyTHHKa K .AT-
JISHTHCY. npH a = 0 II OT .ATJISHTHca. K cnyrHHKY npH a: = 7t.
6) 3anHmeM BbIpameHHe JI.JISI nOJIHOH 3HeprHH CHCTeMbI:
E = Ek + E ::;: .!:.mw 2 R 2 _ Gmma ;;;; _!. Gmma
p 2 R 2 R 
r.n;e m - MaCCR CHCTeMhI.
MaJIoe H3MeHeHHe p8,ll;uyca Op6HThI CBSl3aHO C H3MeHeHHeM 3Hep-
rHH CHCTeMbI COOTHomeHHeM
M ::;: .!:. Gm M ::;: .!:.mro 2 RAIl
2R 2 2 '
rp;;eM= 10 M.
H3 cHTyaItHH, 3aHHO:H B YCJIOBHH n. 4, CJIe,lJ;yeT, qTO 3HeprHH CH-
CTeNhI .n;HccnnHpyeT, BCJIe,D;CTBHe SToro }TMeHbmaeTCH pa.n;HYC op6HTbI.
77

ITpH BRl1IO''IeHHH pacnOnOmeHHOrO BHyrpH Kopa6JIH HCTO''lHHKa
TOKa, ROTOpLIH: C03,n;aeT TOR B npoTHBOnOnO)KHON SanpaBJIeHHH, 3Hep
rH.R nOrJIo:rqae-rclI CHCTeNOH H, K8R CJIe,n;CTBHe, PMRYC Op6HThI 6Y,lJ,eT
YBeJIHtlHBaTLC.R.
BpeN.R nOAAepm8HHII CHJILI TOKa I = 0,1 A OT HCTO'lHHKa uaii.ll;eN
H3 BLlpameHHH
{ AE = .F;vt = BILwRt,
=> t = mmA]l _ 1tm/!Jl .
M=.!mro2&R 2BIL BILT
2
nOp;CT8BJIIISl3HaQeHHSI BeJIKtlHH, nOJIytlHN t  5,8 . 10 3 C.
3ap;a-.:a 3
1. HCnOJIL3yeN ssepreTH'lecKIrli no,[J;XO,lJ,. ,ZJ;JI.R TOllKH M8.CCOH m = 1 Kr
Ha nOBepxHocTH HeiiTpoHHoA: 3Bea.o;hI:
GM
nOTeHD;H8JI nOJIH T.RroTeHHII \VT = --,
r
1
nOTeHD;H8JI nOJIII D;euTpot'ie)KHhlX CHJI HHepD;HH \Vd = -jro 2 r 2 cos 2 Q)
r.o;e M - Macca 3Be3)];hI, r - ee HYC Ha wHpoTe cpo
nOBepxHocTL .o;OJImU8 ti:b1Th 8KBHIIOTeuD;H8JIhHoA: \V T + \V ql5 = const.
,ZJ;JI.H 3KBaTOpa cp = 0, r:::: r e ,
1t
.o;JISI nOJIIOCa cp = 2" r = rp.
n03TOMY
GM :::: GM +.! m2r I!2.
rp re 2
OTCIO.o;a CJIeJ];yeT:
2 3
T, =I+ cor l! ;
rp 2GM
ffJ2 r. 3
l+--!--l 2 3 2 3
E = t;. - rp = 2GM A:I (i) r;. = 2n r, :::: 3,7.10-4.
r. (J]2 r 8 2GM GMT!
I! l+--L
2GM
2. K8K CJIe.o;yeT H3 pBC. 56, MOMeHT HHepD;HH II( KOpld CK8"lKOM
YMeHhmSeTC.R ua BeJIH'qHHY M.
78

H3 38KOHa COXp8.HeHHa. MOMeHTa HMUYJIbCa HMeeM
l..m o = (1 - MIt)co l :::::> M.. = I. WI - roo .
to.
(1)
BCJIe,n;CTBHe BHyrpeHHerO TpeHHSI npoHCXOJJ;HT BhIp8.BHHB8.HHe yr-
JIOBLIX CKOpoCTeii KOphI H .HJJ;p8 HeiiTpoHHOii 3BeaJJ;hI.
3anHmeM 38.KOH COXpaHeHH.H N:OMeHT& HMnYJILCa:
CO2 - COo
(I K + 1.><0 0 = (I K + I. - AI K ) m 2 => AIl{ = (I K + Is) co 2 ' (2)
r1I,e I. - MOMeHT HHepIPIH BJU>a 3Be3JJ;hI.
H3 BLlpWKeHHii (1) H (2) nOJIyqaeN:
Ill.
I K + I.
1
I.
I K + I,.
1. r:
Y"ITa., "ITO I K + I.. - r 6 ' UOJIyqaeM
r..=r 5 1
«(r)2 - roo )co. (WI - CO 2 )ro O
= r 5
(WI - COo )CO Z (to. - (00 )CO Z
7. = 0,98, r = 98 . 10 3 M.

XXII Me>KAYHapOAHali
cl»M3MLleCKaR OnMMnMaAG
1991 r. , Ky6a
3a.n;aQa 1
Ha pRC. 58 nOKa38H O,ttHOpO.n;HLIH map pa,ttIlYCOM R. BaaqaJIe D;eHTp
Macc mapa HaxOJlCS B COCTOSIHJUI nOKOH, a caM map Bp8II1;8JICH BO-
Kpyr ropn30HTaJIbHOH OCH, npoxo,ttJlIIl;eii Qepe3 ero D;eHTp, C yr JIOBOH
Cl\OpOCTbIO <0 0 . CaMaS! HH3KM TOQl\a mapa H8XO,AHJIaCb Ha BbICOTe h B
nOJIOM. TIoCJIe OTUYCKaHHH map IIa;:I;aeT no)]; BJIHHBlIeM CHJIbI TSIatecTH
Ha nOJI, Y,ttapHeTCH, 3aTeM nO,ttCKaKHBaeT )];0 H3BeCTHoii BbICOTbI ah.
BeIIl;eCTBa H3 KOTOpbIX COCTOSIT ITOJI H m8p TaKOBbI QTO MomBO He
yqHTbIBaTb ,tteij)opMaD;IIH npeTepneBaeMbIX 8TlIMH TeJI8MH BO BpeM.H
CTOJIKHOBeHH.H. K08ij)(}JnD;HeHT TpeHH.H CKOJILmeHH.H Mem.n;y mapOM H
nOJIOM , Macca mapa m. CqHTaiiTe, 'f:ITO map HftXO.n;HTCH B BaKYYMe H
BpeMS CTOJIKBOBeHHH OlleRb MaJIO (HO He paBHO HyJIIO). HCnOJIb3yihe
2
OPMYJIY I = '5 mR 2 ,ttJISJ. MOMeHTa HHepD;HH mapa OTHOCHTeJIbHO OCH,
npoxo.n;.HIII;eii Qepe3 ero D;eHTp.
IIpe,lJ,noJIaraH, t:lTO Cy:n:J;eCTByeT npOCK8JIb3LIBaHHe Mem,ny mapOM H
nOJIOM B Tet:leHlIe Bcero BpeMeHH y,lJ,a.pa, HaH.AHTe: TaureHc yrJIa OTCKO-
PRC. 58
80

K8 B; rOpH30HTaJIhHOe CMem;e-
BHe eHTp8 mapa Mem,ZJ;Y nep-
BlUM H BTOpHM y.n;ap8.MH 0 nOJl;
NHHHM8JlLHOe 3B8'f:1eHHe roo .n;JI.R
STOro CJI8.R.
BhInOJIHHTe npe.n;LI.n;ym;:ne
38J];8HH.R, npe,AIIOJI8r8JI, q-ro npo-
CK8Jlb3hIBaHHe npeKp8.IIl;aeTCH
1J,O HCTeqeBH.H BpeMeHH y,lJ,ap8.
IIocTpoHTe rpacpHKH 38BH-
CHMOCTH tg e OT roo.
c
PHC. 59
38aq:a 2
IIo HenpoBo.n;HrqeMY BHTKY B (}JopMe KBBJJ;P8T8 co CTOPOHOii L C 3a-
KpyrJIeHHhI14H yrJIaMH .n;BHmeTC.H 60JILmOe KOJI1fqeCTBO 3apH»ceHHhIX
mapoB MaJIhIX pa3MepoB (pHC. 59). CKOpOCTb Kam.n;oro mapa U, 3apH,IJ;
q, paCCTOHHHe Me)K.n;y mapaMH a. HenpoBo.n;.HII:J;He CTep.mHH, COCTaBJlH-
IOII:J;He BHTOK, 3ap.HmeHLI C 01J,HOpo.n;HOii nJIOTHOCTLIO 3apH1J,a (STOT 38-
pH1J. KOMlleHcHpyerr o6rq:nii 38pH,IJ; Bcex mapoB). Bce STO BepHO B CHCTe-
Me OTC'f:IeTa, CB.H38HHoii C BHTKOM.
PaCCMOTpHM cJIyq:aii Kor.n;a BHTOK J];BHmeTCH co CKOpOCTLJO V na-
paJIJIeJILHO CTopOHe AB B 06JI8CTH, rAe cym;ecTByeT O.n;Hopo,lJ,HOe SJIeKT-
plrfleCKoe nQJIe C HanpameHHOCThIO E , nepneHJUiKYJIHpaoii CKOpOCTH
BHTK8. Be BpeMa 1J,BHmeHHH BHTOK HaX01J,HTCJJ B nJIOCKOCTH, KOTOp8.H
06pa3YeT yrOJI e C aanpaSJIeHHeM BeKTopa H8np.RmeHHOCTH nOJIH. Y'f:IH-
TLIB8H peJI.HTHBHCTCKHe SlBJIeHHH, onpeA,eJlHTe CJIep;}71OII:J;He BeJInIJIHLI
B CHcTeMe OTCQeT8 HenOA,BHHCHOro Ha6JIIOA,8TeJIH, OTBOCHTeJILHO KOTO-
paro BHTOK J];BlDKeTCSI co CKOpOCTLIO V:
1) paCCTOHHHe Memp;y mapaMH Ha Ka.m.n;oH H3 CTOpOH BHTKa;
2) pe3YJIhTHPYIDJ::qHH 38PHJJ; (cTepmHH + maphI) Ha Ka»C,lJ,OH H3 CTO-
pOH BHTKa;
3) M014eHThI 8J1eKTpl{'qeCKHX CHJI, CTpeMHHXC.H Sp8II:J;aTL BHTOK
B14eCTe C map8MH;
4) 8HeprlllO B3aHMo.n;eiicTBHH BHTKa H mapoB C 9JIeKTpn'tleCKHM no-
JIeM.
n p H 14 eq a H He. 3JIeKTpH'IleCKHH 3apsr.n; He 3aBHCHT OT CHCTeMhI
OTCQeT8, B KOTopoA H314epHeTCH. Ha pHCyHKe nOK8.38HLI TOJILKO OT-
HOCHTeJILHLIe HanpaBJIeBH.H BeKTopOB. H3JIyq:eHHeM 14omHO npeHe-
6peQL.
81

H e"omOpbte ,p0pMgJl,b/, cnellUaJl.bHOU meopuu omnOCUme.nb1tOCmu
P8CCMOTpHM CHCTeMY OTCqeTa S', ABH)KY1OCSI CO CKOpoCThJO V
OTHOCHTenhHO .u;pyro:A CHCTeMhI OTCqeT8 S. OCR KOOPAHB8T napa.n:JIeJIb-
RhI, If B MOMenT BpeMeBH t:;; 0 uaqaJIa KOOp,lJ,HHaT COB81OT. CKOpOCTb
v HanpaBJIeH8 B;o;OJIL nOJI<»KHTeJIbHOro Hanp8.BJIeHHJI OCB X.
1. PeJUITHBHcTcKoe CJIO)ReUHe CKopocTeii. ECJlH qaCTH8 ABHa<eT-
C.H co CKOpoCThlO U' B Hanp8BJIeHHH OCH X' (H3MepeSHoii B CHCTeNe OT-
CQeT8 S'), TO CKOpoCTL "IaCTHD;h1 B CHCTeNe S pUBa
u'+v
u= .
1 + u'vfc 2
2. Pen:.HTBBHCTcKoe COKp8lIteHHe ;o;JIHHhI. Ec.nH B CHCTeMe S 06'beKT,
HMelOHii: pa3Mep Lo BAOJIb OCB X, nOKOHTC.H, TO B CHCTeMe S' (;o;BH-
3tym;eH cH CO CK OpOCTLIO V B HanpaBJIeHHH X') ero AJIHRa p8BHa
L:;; Lo .J 1- v 2 fc 2 .
3a.o;aQa 3
J];JIH AeTaJlhBOrO H3yq:eUH.R CBoiicTB OTAeJIbHLIX aTOMOB Heo6xo-
AHMO HMeTL 3TH 8TOMLI B COCTOSlHHH noqTH nORO.H B He60JIbmOH qaCTH
npOCTpBHCTB8 B TeqeHHe onpe)J;eJIeBHoro BpeNeHH. 'TITo6hI nOJIytlHTh
TaRHe aTONLI, HeABBHO CTaJIH HCnOJIh30B8TL NeTOp;, Ha3B8HHLIH Me-
TOP;OM OXJI8eIlJUI aTOMOB C nOMOhlO n8aepa. PaCCMOTpHN ero OC-
HOBbI.
ATOMhI 23Na, B03HHK8IOIqHe npH Hcn8peHHH R8TpH.JI npH TeMIIepa-
Type T = 10 3 K, C nOMOm;blO KOJIJIHMaTOpa 4x>PMHPYIOTCSI B Y3Kd ny-
1:I0K 1:1 Sanp8.BJI.HIOTC.H B B8.KyyMBYJO K8Mepy (puc. 60). IIo BceMY none-
pe'tlBOM:Y CeqeHHIO IJY1IOK aTOMOD OCBeaeTC.H BCTpe"IBLIM nyqROM CBe-
Ta MOHoro JIB3ep8. -qaCTOT8 JIa3epa Bhl6upaeTC.H TaKoH, 'lTo6LI
npOH30mJlO pe30HaHCIIOe nOrJIOm;eHHe tPoo'OHOB 8TOM8.MH, CKOpoCTb KO-
TOphIX p8.BHa v 0 H KOTopLIe H8XOASlTC.R B OCROBHOId COCTOHHHH.
nor JlolI(aa t}:JOTOH, aTOM nepeXO.D;HT Ba nepBhlH Boa6y:m,n;eIlHLIii YPO-
seUb C 3HeprHeii E H mHpHROH r (pHC. 61), B era CKOpoCTb MeHHeTC.H Ha
aeKoTopYI<> BeJIH'lHllY Av] ;:: v] - Vo . SaTeN B pe3YJIbTftTe CnOHTaHHOH
SMHCCUH aTON B3JIyqaeT CBeT H
I ===P B03BpaeTC.H B OCHOBRoe co-
4b cTofiHHe, CKOpOCTb ero npH
. aTOM H3NelllleTCII UB BeJIUQHHY
N a : w;  V; - ii. H COCTllJ!.JUleT yro.n
cp C nepBOHa"l8JIhBhIM Hanp8.BJIe.
PHC. 60 HneM (pHC. 62).
82

E + r 12
E - r /2
-
VI
E
Ns
o
PRc. 61
PRC. 62
STOT npol\ecc nOrJloeHH.H 8TOMaM:H <POTOHOB H nOCJIeAYI<UItero HX
H3JIyq:eHHH nOBTOp.HeTC.H NHorOKpSTHO 1I.0 Tex nop, nOKa CKOpOCTL aTO-
MOB He H3MeHHTC.R Ha BeJIHYlHHY fj. V Ii pe30HSHCHoe nOrJIOeHHe cseTa
IIPH )J;8HHoft 'laCTOTe CT8HeT ueB03MO)KHhIM. B .n;8JIbHeiimeM Heo6xo,lJ;n:-
MO H3MeHHTL acToTY H3JIyqeHH.H JI83epS, Tot5hI nOAAepmHBsTb pe30-
H8.HCHOe nOrJIOeHHe aa HOBOii CKOpOCTH H npo.n;OJImSTb TOpM03HTh
aToMhI, nORa HeKOTophIe H3 HRX He CHH3.HT CBOIO CKOpOCTb /1,0 .n;OCTa-
T01:IHO MaJIOI'O 3Ha'leHH.R.
]l.1ISI H81:I8JIhHOro H3yqeHH.H .HBJIeHH.H MomHO He IIpHHHMaTL BO BHH
MaHHe HHK8KHe .n;pyrue npoD;eccLI B3aHMo.n;eHCTBH.R aTONOD, HCKJllO.
"l8JI nOrJIOeHHe H cnoHT8HHyIO 8NHCCHIO CBeTa. BYAeM C1:IHTaTL, "ITO
HHTeHCHBHOCTh H3nyqeHHH JIaaepa HaCTOJlbKO BeJIHKa, "ITO aTOM B OC-
HOSBOM COCTOHHHB cpaay nOI'JIOSeT <pOTOH.
CtIHTaH, "ITO E:;; 3,36.10- 19 J];)K, r::;: 7,0.10- 27 J];m, CKOpOCTh
CBeTa c:;; 3,0 . 10 8 N/c, Macca npoToHa m = 1,67.10- 27 Kr, nOCTOSlH-
HaH lln8HKa h :;; 6,62.10- 34 J];:m. c, nOCTOHHHaa BOJlbD;Naaa k =
= 1,38.10- 23 ,IJ;m/K, OTBeTLTe Ha CJIeAYIOHe BonpOChI:
1. KaKOH .n;OJI:mHa 6hITL "IaCTOT8 JI83epHOI'0 H3J1yqeHluI ,lJ,JIH Toro,
TlT06hI 06eCne'lHTh pe30H8HcHoe nOrlIOeHHe CBeT8 8TON8MH, 3HeprHSI
KOTOpLIX pasHa cpeAHeA: KHHeTU"IeCKOii 8HeprUH aTONOB B 06118CTH .n;o
npoXO)KeHRSI K01l1lHM8Topa? Ha KaKoe 3H8"IeHUe fj.V] H3MeHHTCH CKO-
poeTL TaKOrO aTOM8 nOCJIe nepBoro nOrn0I:l\eHH.R «}>oToHa?
2. B KaKON HHTepB8JIe CKopocTeii fj.v o Haxo.n;SlTCSI aTOMLI. OTopLIe
MOryT UOI'JIOTHTL <pOTOH C 'IlaCTOTOH, BhI"IUCJIeHHOH B u. I?
3. Ha KaKOft MaKCHN8JILHLIH yrOJI Q) MO)KeT paCCeHThCSI aTOM nOCJIe
O,lJ,HOKpaTHoro HCnyCKaHHSI (}JoToHa?
4. IIpB KaKON MaKCHMaJILHOM yMeHLmeHHH CKOpOCTB fj.v ee 140-
)KeT nor JlOTHTLCH <pOTOH TOH :me "IaCToTLI?
5. OneHHTe1:IHCJIO 8KTOB nOI'JIOeHIUI-H3JIyqeHHSI, nOCJIe KOTOphIX
aTOM, .n;BHrasmHiic.H B;o;01lL OCH nyq:K8, 3aTopM03HTC.H OT CKOpOCTH V o
npaKTU"IeCRH.n;o HyJIesoii CKOpOCTH.
6. OD;eHHTe AJIHTeJILHOCTb up0Q.ecca a8Me;O;JIeHHH STOMa. KaKoe pac.
CTO.HHHe npoJIeTHT aTOM 3a aTO speMS?
83

PeweHMR
38)1;8'18 1
Hafi.AeM CKOpoCTh Henocpe,a;CTBeHHo nepe,u; y,a;apoM mapa 0 nOJI. .nJI.H
SToro npHp8.BHHeM nOTeHD;HaJILHyJO H KHHeTHQeCKyro 8HeprHIO:
mv. 2
mgh==T' (1)
oTKyna nOJI}"lHM
Vo :: .J2gh .
(2)
BLI'lHCJIHM. BepTHK8JlbHYID EOMnOHeHTY CKOpOCTH cpa3Y nOCJIe yna-
pa. IIYCTb v 2 % H v 211 - COOTBeTCTBeHHO ropH30HT8.JIhHWI H BepTHKftJIL-
BaSI npoeKIUlH CKOpOCTH D;eHTpa Macc CPS3Y me nocne y,u;apa. BhICOT8,
sa KOTOPYlO nO.D;HHMaeTC.R map nOCJIe y.n;apa, pUBa ah, H Tor,u;a
viII:: 2ga.h, (3)
oTKy,u;a BLIpa3HM. U2g qepea a (unn BBO.n;.R Kos4>(}JnD;nem BOCCT8HOBJIe-
HHSlC= ):
he. 63
v 211 = J2ga.h = CVO. (4)
3annmeM COOTHOmeHH.R
AJISI H3MeaeBHft HMnYJIbCa II
MOMeHTa HMuYJlbCa aa BpeMSI
coy,o;apeBHH, ytIHTLIBaSl, '1TO
N» mg(pHc. 63).
CqRT8.H qTO HMnyJILC eRn
f r pSBeH H3MeHeBHlO HM.UYJIbCa
TeJIS II BpaII{8TenhHhlH HM-
uynLc CHJI p8.BeH H3MeHeHHIO
ero MOMeHTS HMnYJlLCa, nony-
'IBM COOTHOmeUH.H:


If! = IN(t) dt = mv o + mv 2v = m (1 + c) ,j2gh ,
r.
(5)
"
1% = II,. (t) dt = mv 2 %,
"
(6)
'2 '2
Ie = fRt(t)dt = R ft,.(t)dt:;: [(roo -ro 2 ),
'1 '1
(7)
84

r.n;e I x' Ie H Ie - rOpH30HT8.JILHaJI H BepTHKaJILHa.R npoeKH HMnyJIb-
ca H Bpa.rqaTeJIhHhIH HMnYJlbC .n;eAcTByJOJ::qHX CHn; 0)2- yrJlOBaJI CKO-
pOCTL nOCJIe y.n;apa; t 1 H t 2 - COOTBeTCTBeHHO BpeMH BRTlaJIa H BpeMH
KOHD;a COYA8peHH.H.
B Ha'flaJIe coyp;apeHH.R map B JII060M cnyqae Bpam;aeTCSl nOCKOJIb-
KY OH HMeJl HeKOTOpylO yrJIOByIO CKOpOCTb (0)0). )J;8JIee B03MO.mHbI .n;S8
CJI}"lWl.
C 11 Y 'fl a H 1. HMelOIIterOc.R TpeHH.R Hep;OCT8TO'flHO, ,lJ,JI.R Toro 'flT06bI
3a BpeMSl coy.n;apeHHH p;BHmeBHe nepemJIO B 'flHCTOe KaqeHHe T .e. 'flTO-
6bI TO'l:lKa COnpHKOCHOBeHH.R mapa C nOJIOM 6hIJIa HenO,lJ,BH)RH8.
CJIYQ8H 2. BK8KOH-ToMoMeHT BpeMenR t E (t., t 2 )CKOpoCTbTOq-
KH conpHKOCHOBeHHH mapa c nOJIOM paBHa HYJI1O. C STorn MOMeHT8 CHJIB
TPeHH$I paSHa HYJIIO.
PaCCMOTpHM OT;o;eJIbHO 06a CJIyqaH.
CnyqaH 1
B TeqeHHe Bcero coy.n;apeHHH ecTb npocKaJIb3LIB8HHe Mem.AY ma-
paM H nOJIOM, H CHJI8 TpeHHH BbIpam8eTCH Qepe3 HOpM8.JIbHyro (BepTH-
K8JIbHYIO) cnJIY CJIe.nyroHM 06pa30M:
f r = J.Ifl(t). (8)
IIo.n;cTaBJIHH (8) B COOTHomeHHH (6) H (7) H HCnOJIb3YSl (5), nOJIyqa-
eM
'2
Ix ;;: ilK IN(t)dt;;: ilK Iv ;;;; Km(1+c) .J2gh = mx (9)
fa
H
'2
Ie ;;;; RK IN( t) dt = RKm(l + c) .J2gh ;;;; 1«(00 - (02)' (10)

OTKy.na MomHO HaRTH ropH30BTaJIbH}'IO npoeKI\H1O CKOpocTH
Vb ;;: J11< (1 + c ) 2gh
(11)
H KOHe'l:lByIO yr JlOByIO CKOpOCTb B BH)];e
_ J.1.mR(l + c) f2::Lh
£0 2 - (00 - vc.gn.
I
(12)
3.n;eCh Bce BeJIH'I:lHBbI BhIpameHbI Qepe3 ,lJ,8HHLIe YCJIOBHH.
Y cnOBHe npHMeBHMocTH pemeHHH MOJKBO BaATH 83 BhIpWKeHHH
(11) H (12). STO pemeHHe BepBO B TOM CJIyqae, Kor.n;a B KOHD;e coy.n;ape-
85

BHS! T01:lKa conpHKOCHOBeHHS! mapa C DOJIOY ABH>KeTCS! B CTOpoHY OT-
pHIt8TeJIhHhlX x. T8.KHM 06p830M,
ro > V 2x ;
roo - fla mB (! + c) .J2gh > JJ,1(1 + c) .J2gh ;
I R
1lx 2gh (l+c) ( mR2 1)
roo > -+ .
R I
(13)
PemeBHe aenpHMeHHMO, eCJ1H yrJIOBaH CKOpOCTb MeHLme 3Toro
3H8'qeHHSI.
CJIYTiaii 2
B STOM CJ1yqae B MOMeHT BpeMeHH t Mey t] H t 2 BaQUH8eTCS! 'lJHC-
Toe KaqeUHe. Tor,lJ,a KOHeqUM yrJIOBa.s CKOpOCTb H ropll30HTaJILHaa
npoeKu,HH CKOpOCTH v 2x CB.H38H1d COOTHOmellHeM
co::; Vax. (14)
nO.n;CT8BJI.HSI (14) H (6) B (7), nOJIyqaeM
mRv",= r( roo - j; ).
(15)
OTKY,u,a A1I.H KOBeHbIX 3BaeHHA 6YAeM HMeTb
fro o
V2% :=
mR+L
R
I moR 2
= mR2 + I = 'i(J}oR
(16)
H
lIDo 2
(02 = mB2 + 1 -7ro o .
BhI'tIHCJIHM T8BreUChI yrJIOB.
CJIY'Iaii 1
)J.JISI tg 9 nOJIytl8eM: 113 (4) H (11):
tg9 == Vb == J!1(1 + c) .J2iii =  (1 +c) ;
g cJ2gh c
(17)
tg8 = J.1. (l+c) ,
c
(18)
T .e. yrOJI He 3aBHCHT OT COO'
86

Cnyaii 2 tgS
113 cOOTHOmelmii (4) H (16) UOJIyqaeM tg 6:
tge = V 2x = [moR 1 = [rooR .
V2 I + mR 2 c .J2gh (I + mIf) c .J2gh 
CJ)Omln
COo
t e _ 2ro o R
g - ,
7 c.J2gh
H Tor/(a H3 cooTHomeHHii (18) H (19) nOJIyqaeM rp84>HK nOK83aHHhIH
H8 pRC. 64.
MLI BRJJ;HM, 'tlTO e He 3aBHCHT OT 0)0' eCJIH 0)0 > O)Omin' r.n;e
(19)
PIIc. 64
".(He) ,j2ih (1+)
roo min ::::: R
7JJI< (1 + c),f2ih
0)0 min = 2R .
(20)
BLIHCJIHM paCCTOHHHe Mey T01lKaMH YAapoB mapa 0 non.
CJIyqaft 1
BpeM.R, a8 KOTopoe map ,lJ;OCTHrHeT MaKCHMWILHO:ft BhICOTbI H yna-
,lJ,eT C Bee, paBHO
t = 2 V21/ = 2c,j2ih = 2c f2h .
g g Vg
Tor.n;s HCKOMoe paccToHHHe p8.BHO
(21)
ct. = ....t = 11.(1 +c) ,j2gh . 2c f! .
d 1 = 4J.11( (1 + c) ch.
OHO He aaBHCHT OT 0)0.
(22)
CJIyqa:ii 2
B 8TOM CJlyqae BpeMSl MmK.u;y coy,u;apeHHHMH COBU8,ASeT C t 83 (21).
IT03TOMY  QTo6L1 HaHTH ropH30HTWILHoe cMeIIl;eHHe, YMHOJRHM t Ha
CKOpOCTL v 2x (16), Torp;a
 = v 2 t = ImoR .2c [2h = 2000Rc [2h 
x mR 2 +I Vg 1+ Vg
2
4  h
 = -c -Bro o .
7 g
(23)
87

B 3TOM CJI}"lse rOpH30HTaJIbHoe CMeeHHe MemAY,D;BYMH yp;spaMH
BOapaCTSeT JIHHe.HHO B aaBHCHMOCTH OT <0 0 ,
3aAaa 2
1. llYCTL S - JIa60paTopHafl CHCTeM8 OTCqeTa CBH38HHaJI C neno.n;-
BHmHhIM Ha6JIlO,D;aTeJIeM OTHOCHTeJIbHO ROToporo BUTOR ABH)KeTCSl co
CKOpOCTbIO v, a S' - CHCTeMS OTCqeTa, CB.H3aHH8Sl C BHTKOM (OCb X'
HSnpSBJIeHS BAOJIb HSnpSBJIeHHSI BeKTopa V , Y' - B,lJ,OJIb CTOpoHhI DA,
8 OCL Z nepneKYJUlpHa nJIOCKOCTH BHTKS). Ocn CHCTeMhl S' napan-
JIeJIbHLI OCSlM CnCTeMhI S', H Haq8JIa KOOp,lJ,HHaT COBnaIOT B MOMeHT
BpeMeUH t = O.
C TOp 0 H a AB. S AB - CHCTeM8 OTCqeTa, B KOTOpOH mapLl, ,lJ,BH)KY-
IItHecSI B,lJ;OJIbAB, nOKOHTCSl. Ee OCH rrapSJIJIeJILHLI OC.HM CHCTeM: S H S',
H OHa ,lJ,BHmeTC.H co CKOpOCTblO U OTHOCHTeJILHO CHCTeMhI S' .
IIo CPOPMYJIaM JIopeHIl;eBa COKpam;eHHH AJIHHhI no.rryqaeM, 'IlTO paCCTO-
mwe a r Mem.ztY coce,n;HHMH map8MH lIB. cTopoueAB B CHCTeMe S AB paBHO
a
a r =  1- u 2 / c 2 .
(1)
(STOT pe3YJIbTaT MomHO HCnOJIb30BSTb AJISI COce.D;UHX mapOB ua JII060H
H3 CTOpOH, eCJIH TOJIbKO aT H3MepeHO B To:ft CHCTeMe OTC'tIeTa, rAe mapLI
nOKO.HTC.R. )
ITo OPMYJIe peJISlTHBHCTCKoro CJIOmeHHH CKopoCTeii nOJI}"lHM CKO-
pOCTb, C KOTOpOH maphI .D;BHmyrcH B.n;OJIbAB B CHCTeMe OTCqeTa S:
v+u
U AB = 1 + uv/c2 . (2)
3UaQHT, B CHCTeMe S paCCTO.KHHe Me>Kp;y mapaMH pSBHO
U AB = .... 1- uj (3)
(no cpopMYJIe COKpaII1;eHH.ft AJIHHhI).
llOAcTaBJIH.H(1)H(2)B(3),no JI}"lseM
.,fl - v./c'
a AB = a
1 + uvJ c' .
(4)
C TOp 0 H a CD. ):J;JISI Ha6JIIO,D;aTeJISI B CHCTeMe S CKOpoCTb mapOB
B,D;OJIb CD paBHS
v-u
U CD =-.
1- UV
c 2
(5)
88

no cl»opMYJIe JIOpeHneBa COlCpaII(eHHH ,D;JIHHhI,

U CD =arvI-&.
(6)
nO,D;CTaBJI.R.H (1) H (5) B (6), nOJI ytlaeM
Jl- V 2 /C 2
aCD=a / .
1 + uv c 2
(7)
C TOp 0 H a DA. llYCTb B CHCTeMe S' B MOMeHT BpeMeHH to' R8.Koii-
JIH60 map HMeeT ROOPAHH8TLI x'. = y'.= z'. = O. B TOT:me MOMeHT Bpe-
MeRR cocep;HHH C HRM map RMeeT ROOp)1;HHaThI x' 2 = 0, y' 2 = a, z' 2 = o.
IIPOCTp8.HCTBeHHO-BpeMeHHLle ROOp,D;HHaThI 3THX mapoB B CHCTeMe
OTCqeTa S MomBO nOJIyqHTb no (}JopMYJlaM npe06pS30B8.HHii JIopeJIu;a:
x' + vt'
X= g 2
1--
c 2
y = y', z = z' 
t' + x'v
c 2
t=  2 .
1--
c 2
(8)
COOTBeTCTBeBHO )1;JISI nepBoro mapa B CHCTeMe S HMeeM
vt'
Xl= g 2 ;
1--
C20
Y. = 0;  = 0;
= g '
v 2
1--
c 2 0
(9)
8 AlIlI BToporo -
vt' t'
%2 = g o ; Y2 = a; Z2:= 0; t 2 = g o ,
v 2 v 2
1-- 1--
c 2 c 2
(10)
nOCKOJIbRY t. = '2' paCCTOHBJle Mem,ltY mapaMH B CRCTeMe S Mom-
HO HaATH H3 COOTHOmeHH.R
a1>A = (X2 - %1)2 + (Y2 - y.)2. + (Z2 - 2 1 )2,
(11)
T.e.
a DA = a.
(12)
89

C TOp 0 H 8 BC. nOBTOpHB Te me paCCY)KJI;eHR,ft 'lJTO H .n;JIH CTOpoBhI
DA9 nOJIyqHM
a se = a.
(13)
2. 8ap,ft.D; Kam.n;oH K3 CTOpOH, 06pa3YIOHX BRTOR, B CHCTeMe OT-
C'lJeT8, CB,ft38.HHOii C BHTKOM, paBeH
L
Qnopollw = --q
a
(14)
L
(- - T:lHCJIO mapoB Ha Kam,n;oii H3 CTOpOH). nOCKOJIhKY aapJl)]; HHBapH-
a
auTeH, 38pg.D;hI cTepmHe:H 9 H3MepeHHhIe B JIa60paTopHoii CHCTeMe OT-
C"leT8 9 6y.n;yr T8.KHMH me.
CT 0 P OB 8 AB. 38PBA mapoB H8. CTopoae AB B JIa60p8TOpHOH CHC-
TeMe OTCqeTa
Lq Jl- 
QAB m = .
. a AB
STOT peaYJIhTaT MomBO nOJIyqHTh, nepeMBOmHB "lHCJIO mapOB Ha
8TOH CTopOHe II (HHBapH8HTHhIU) 3apx.a; Kft.jl(,AOro mapa.
ITO.n;CT8BJIH,ft BhIpameRHe (4) B (15), nOJIyqaeM
(15)
QAB = c 2 + uv Lq
.m c 2 a.
CKJIa.p;hIBBJI (14) H (16), nOJIyqaeM 1I0JIHhIii 3apx.a; 3TOii CTOpOHhI:
uvL
QAB = --q.
c 2 a
C TOp 0 H ft CD. ITOBTOPHB Te me paCCY'fC,lJ;eHHH, 'lITO H ,ltJISI CTOpOHhI
AB9 nOJI}"IHM
(16)
(17)
Lq Jl- 
'kD. m =
OeD
CKJI8J1.hIBBJI (14) u(18). HaxOAHM
uvL
tieD = ---q. ( 19)
c 2 a
C TOp 0 H LI BC II DA. AJIHRa 3THX CTOpOH B CHCTeMe OTCqeTft S pu-
Ba L, a paCCTO.JlHUe Me)KJI;Y mapaMH paBBO a, II08TOMY
QBC.m = Q&.m = Lq . (20)
a
( 1- uV ) Lq .
c l a
(18)
90

CKXBaa(14)H(20),nonyqaeM:
QBC = 0,
QDA = o.
3. B 8JIeI<TpH'fIeCKOM nOJIe H8 cTopoay AB )1;eHcTByeT CRJI8
- - uvL-
F AB = QABE = --qE
c 2 a
(21.1)
(21.2)
(22)
8 aa CTOPOHY CD - CHJI8
- - uv L -
FCD = QCDE = ---qE.
c 2 a
(23)
CHJIhI F CD H F AB C03P;81OT BP8B.IOH:H NOIdeHT (pRC. 65)
M= F AB L sin e.
OKOaTeJIhHO HMeeM
(24)
uv L2 11 .
M=2'"-qEsln6.
c a
(25)
4. nYCTb Q)AB H Q)CD - 3JIeKTpOCT8TH1JeCKHe nOTeHD;H8.JIhI B TO'flKaX
cTopoaAB H CD COOTBeTCTBeBBO. Tor,lJ,8
W = CPAB QAB+ Q)CD QeD. (26)
38cpHKcHpyeM TOKY BYJIeBOrO nOTeBItH8Jl8 (cp::: 0) B llJIOCKOCTH,
nepnea));HKYJI.HpaOH E (pRC. 66), H8 npOH3BOJI:bHOM paccToSlHHH ROT
CTOpoHbI AB.
TOr)1;8
W = -ERQAB - E(R + L cos e) QCD'
(27)
,
"
C.  E
Ie'
"e
I ,
I ...
J "
I "
I ... B
r---------
I
I R
I
I
...
"
"
,
C
" B
,
,
" ,'L sine
" ,
"(
"
"
"
\0=0
PRC. 65
PRe. 66
91

Ho QCD = -QAB' n03TOMY
W = ELQAB COS 8.
IIoAcTaBHH(I7)B(28),nonyqaeM
W uvL.2qE e
= .2 cos.
ca
(28)
(29)
3a)J;aa 3
1. CKOpOCTb aTOMOB U o MomBO onpe,AeJIHTb, TaR KaK H3BeCTHa KH-
BeTH'lJeCKa.H 3BeprH.H aTOMOB nOCJIe npoxom.n;eHH.H ROJIJIHMBTopa:
!mv: =!kt => Vo = lkT .
22m
r.n;e m = 23 mp (m p - Macea npoToHa). C eTOM "IJHCJIOBbIX 3H8't1e:mm
nOJIyqHM
3.1.38.10- 23 .10 3 / 1 04 10 3 I
V. = M CR="' M C.
o 23 . 167 . 10- 27 '
IIOCKOJIbKY 3T8 CKOpOCTh U8MHOrO MeHhme C (V o « C), peJUlTHBHCT-
CKHMH a4>cPeKTaMH MomHO npeHe6pe"lL.
qJOTOHhI HMeIOT 8BeprHlO hv H HMnYJIhC hv Ie. B JIa60paTopHOH CHC-
TeMe OTC'tJeT8 aaTI:umeM 38KOBhI coxpaHeHH.H 3HeprHH H HMny.m.C8 /l.JIH
npoIteCC8 nOrJIOeHH.H:
1 1  W
-mv + hv = -ml1. 2 + E; mv o - - = mV I ::::> AV 1 = VI - V o = --;
2 2 c me
-im(v: - v) = hv - E:=;) m(VI + vo)(v t - v o ) = hv - E;
hv« muo' n03TOUY VI R=" V o ' a 3Ha1:IHT, mv o Vl = hv - E (NhI JIH, '1TO
v. + V o  2v o ).
KOM6HHHpyeM 3TH BhIpameHHH:
E
v= H
1+ Vo
c
E 1
Au] :: ---.
me 1 + U o
e
IIO.n;CT8BHB 3USQeHHH BeJIHtIHH, nOJIyqHM
v R=" 5,0 . 10 14 r.q,t Av}  -30 . 10- 2 M/c.
92

ECJIH 61d MhI paCCM8TpHBanH npoD;ecc B CHCTeMe OTClIer8, ABHmy-
m;eiicJI OTHOCIITeJILHO JIS60paTopHOii CHCTeMhI co CKOpOCTblO V O ' Mid 6bI
nOJIytllIJIH
1 2 E h '
-VI + = v 
2
rJ\e v' = {I +  ) (v - qaCTOTa CBeTa B J1a6opaTopHOil CHCTeMe). IIpe-
uoopersJI lIJIeHOM tJ. v: ' MhI nOJIyq:seM Te me YP8BHeHH.H, lITO II BhIme.
IIpeHe6pemeHHSI onpaB.n;aHHhI, nOCROJILKY
ltJ.vll  10- 4 .
va
Tor.n;a VI + Va = 2v o - tJ.v)  2v o '
2. ,lJ;JIH €}>HKCHPOB8HHO:ft: lIaCTOThI v HMeeM
va = c (  -1).
ECJIH mnpHH8 3HepreTHQeCKOrO YPOBHH paBH8 r  TO mHpHH8 HHTepB8-
JIa CKopOCTeii paBHa
cr ( l + vo )
.6v o = cr = c 1:::: cr = 625 Mlc,
hv E E
n03ToMY €}>OTOHbI nOrJIOIItaIOTCH 8TOM8MH, CKOpOCTH KOTOphIX JIemaT B
HHTepB8J1e [v o - 6;0 ; V o + 6;0 ] .
3. Ha 38KOHOB coxp8HeHHH ::tHeprUH H HMnYJIbC8 nOJIyq:aeM:
1 2 E 1 ,2 h '
"2 mVI + = imVI + v
(v' - Q8CTOT8 HcnyeHHoro €}>OToua),
hv'
mV I = mv COS<&, + -cose
c
O ' . hv' . e
= mV 1 smcp --SIn.
c
YrOJI paCCeHHHJI aTOMa q> 6Y,lJ,eT HSH6oJIbmHM, eCJIU e = .!:. , Tor,lJ,a
2
, hv', . t hv'
mV I = mV I cosCPDWt; - = mV 1 SlnfPlJW(::;:> gfP rnax = -
c mc
E
a nOCKOJIbKY v' 1:::: v  tg fP max  -.
mVlc
93

OrclOa
E
qJmax = arctg- :::) <Pmax  2.8 .10-s paA.
mvle
4. TaR K8.K CROpOCTb aTOMOB yMeUbm8eTC.R, TO qaCTOT8, U8 KOTO-
poi: npOHCXO,D;HT pe30H8aCHoe nOrJIOeHHe yueJIH'lHB8eTCa corJI8CHO
3aBHCHMOCTH
E
v = ----1L- .
1 Vo
+-
C
Kor,D;8 CKOpOCTb paDHa v = V o - Av, B03MomHO nOrJIOm;eHHe B HHm-
Rei: QaCTH 3HepreTH'IlecKoro ypOBBH, ecJIH
r
E--
hv= 2
1 Vo - Av
+-
C
E cr ( v{) )
-Av=- 1+-;
1 + V o 2E c
c
Av = 3,12 III/C.
5. ECJIH 38 R8m,D;LI:A aRT nOrJIOeHH.R-H3JIyqeHH.R CKOpOCT"h MeU.R-
E
eTC.R a8 AVt s::::: -, TO, 'lT06LI CROpoCTb YMeHDWHJIaCb OT V o nOQTH ,D;O
me
UYJIH, Heo6xo,D;HMO N TaKHX C06LITHH, rJJ;e
Vo mevo
N = _ I I s::::: - :::) N s::::: 3,56.10 4 .
f1Vl E
6. ECJIH nornOm;eHHe npoHCXOJJ;HT NrHOBeHHO, ,lI;J1HTeJIbHOCTh K8m-
,lJ,oro 8RT8 norJIOm;eUH.R-H3JIyq:eHHJI onpe.n;e.na:eTC.H CnOHT8HHLIM H3JIY-
'qeUHeM:. ATOM H8XO,lJ,HTCJI B B036ym,o;eHHOM COCTO.HHHH B TeQeUHe npo-
MemYTK8 BpeMeHH or = !!. TorJJ;8
r
Nh nu:hu -3
At = N't = - = ----!!..  At j:tI 3,37 . 10 c.
r rE
P8CCTO.HHHe, npoH,D;eUHOe 8TOMOM 38 3TO BpeNg, paDRO 88 = v.p.t/2
(nOJIaraeM, 'IlTO 3aMeJIeHHe npoHcxoll;HT paBHoNepHo):
1 mehVA
A8= 2 rE A8s:::::1,75N.
94

XXIII Me>KAYHapOAHaR
.M3HlleCKali OnHMnHGAa
1992 r _, Cl)MHJ1I1HAMSI
3ap;a-Ia 1
BOKPyr 3eMJIH no KpyrOBOA op6HTe pa.zurycoM R, JIemaeA B nJIOC-
KOCTH 3KBaTopa, .D;BH)f(eTCSI cnyrHHK, COCTO.RHH H3 HeCKOJILKHX TeJI
(pRC. 67). HeBecoNoe eHTp8JIbHOe TeJIO P coe.o;HBeHO JIerKHNH Hepac-
TSI)f(HMhIMH HHTSlMH C 'IleThIpLMH CHMMeTpJrCIBO pacnOJIOmeHHhIMH Te-
JIaNK B, Macca K8.aC,ltoro 83 KOTOpLIX m. &rs CRCTeNS Bce BpeMH Haxo-
AHTCSl B nnOCKOCTH Op6HThI II BpamaeTca C 38.,ltSHHOA yrJIOBO'A CKOpoC-
TblO co (oTHOCHTeJILHO Heno.o;BH)KHbIX 3Be3,lJ,) BOKpyr eHTpsP . ,l];JIH Toro
'IlT06L1 B38HMHoe paCnOJIOmeHHe TeJI He MeHSlJIOCb 6ea Heo6xOAHNOCTH,
pa.zr;HaJILHLIe HHTH CKpenJIeUhI Nem)J;Y c06o'A ,lJ,OnOJIHHTeJIbHLIMH TOH-
KHMH HHTHMH T8.K, QTo6LI yrJILI Me)l(Jt;y HHNH COCT8BJISlJIH 90°, a BCH
CHCTeN8 Ten ABHrlUI8Ch K8.K OABO TBep.o;oe TeJIO.
............
--
'.....
.... .... B
.....
"-
"
PRe. 67
95

1. Onpe.n;eJIHTe CHny, C KOTOpoii npHKpenJIeBH8.JI K TeJIY HHTL JJ;eH-
CTByeT ua 8TO TeJIO. PaccNoTpHTe cnyqaH, KorAS BeKTophI Ii H r na-
paJIJIeJIbHhl, aHTHnap8JIJIeJILBhI, nepnes.,u;uKYJIHpUhI (B03NO)KHO, HMeH-
HO B 8THX nono>KeHII.HX CHJIhI CT8.HOB.HTC.H MRKCHMaJIbHhIMH IlJIH MH-
HHMaJIbHLlNH).
2. B Ka)K.ll;ON 113 TeJI B HaxO)1;HTCB MeXaHH3M:, nOJIyqmo 8Hep-
rHIO 0'1' COJIueqHHX 6aTapeH. STOT MeXftHH3M 6hICT}>O BTHrHBaeT HUTL
BHyrpL B TOT MOMeHT, Korp;a CHJIa aaTHmeHHH M8.KCHMaJIbHa, H BhInyc-
KseT B TOT MONeHT, KOr,lJ,a CHJI8 MHHHM.MbHa npHQeM H3MeueOHe .n;JIH-
HhI HUTH COCTftBJIJleT 1 % 0'1' ee AJIHHld - Cpe,lJ;Hero paCCTO.HHH.H Me)K.ll;Y
BliP. Bee OCTaJIbHOe BpeM.H JJ;JIHBa RUTH HeH3MeHHa.
PaCCQHTaliTe cpe.n;HlOlO MOID;aoCTb T8KOrO NeXaHH3Ma 311 BpeM.H O,lJ,-
HOro 060POTft'I'M B OTHOCHTeJILHO n;eHTpa P (MOID;HOCTL onpep;eJISeTC.R
OTHomeHueM C}TMMapHoit pa60TLl CHJIhI HaT.ReHHg C yqeToM 3H8.KOB
ICO BpeMeHH O)J;Boro 060poTa).
3. Onpe.o;eJIHTe, KaICHe H3MeHeHH.H B ,lJ,BHmeUHH cnyrHHKa BLl3LlBa-
lOTCSI YKa3B..BHhIMH MeX8..H1I3MaMH. KaK MeHHIOTC.R Op6HT8JILH8.JI CKO.
pOCTb, p8JJ;UYC op6HTLl yrnoBa.H CKOpOCTL II rpftBHTan;UOHBaH nOTeH-
n;HftJILHftSl8HeprH.H cnyrHHKft? MO)KHO JIH nepeMeCTHTh cnYTHHK P;8JIh-
me 38Jt8.HHOH op6UTLl? Ha o-qeHL 60JIhmOe paCCTO.HHHe?
YKft3ftUH.H. PaCCMOTpHTe 06a B03MO)KHhIX 3H8.Ka yrJIOBOH CKO-
pOCTH i.O. He npHHHM8HTe BO BHUMSHlle npUT$UKeHHe COJIHn;S II
JIYULl- P;OCT8TO'qHO paCCMOTpeTb B38.HMO,lJ,e'AcTBue cnyrHHKft H 3eM-
JIB. He CT8paHTecb nOJI}"IHTL cOBepmeuHo TO'qHLIe pemeHHSl - BnOJIHe
AOCTaTO"lHhI npH6JIH)KeHH.R KOTopLIe o6eCneQaT TOQUOCTL 50/0.
Pap;HYC Op6UTLI n;eHTpMLHoro TeJIa P R = Ra + 500 KM. )J;JIIIHa pa-
,l);HaJIbHhIX HHTeA r = 100 KM, T.e. AHftMeT}> Bcero cnyrHHKs 200 KM.
Macea Kamp;oro H3 Ten B paDHa m = 1000 Kr. H3HS-q8JILHO Tena B o6pa-
aIOTe.R OTHOCHTenhBO Tenft P (B CHCTeNe OTCqeTa, CB.R38.HROH C HenO,lJ,-
BH3KHhIMH 3se3p;aMH) C qaCTOTOH 1006/'1. MaccaMH RUTeA H n;eHTp8JIb-
HOro Tens P r.lOmHO npeHe6peQh.
3a,ZJ;a'la 2
B 3TOH 3MaQe paccMaTpHB8eTCSI oceBOe (npop;OJILHoe) ,lJ,BH3KeBHe 8'1'0-
MOB JIHUeHBo'A MoneKYJILI B.n;OJIL ee OCR, npH KOTOpoN Bee aTOMLI B npo-
Itecce ABHmeHH.R OCTWOTCJt Ha OAHOH npJtMOH. ITpe,lUlOJIaraeTCSl, 'tITO MO-
JIeKyn8COCTOHTH3NftTOMOBM8CCftMHm], m 2 , ma,".' mNCOOTBeTCTBeH-
HO H K8m,ALIii aTOM CB.H38a co CBOHMH (iJIHmBHmHNH COCeA.fIMH
XHMHTIecKHNH CB.fI3S1MH, KOTophIe MO,lJ,eJIHpyIOTC.R HeBecoMhIMH npY'KH-
HaMB, nOJJ;tlHHmoID;HMHC.H 38.KOHy I'yKa. meCTKOCTL KOTOphlX COOTBeT-
CTBeHHO paBlI8. k] k 2 , k3 ..., k N - 1 (puc. 68).
96

m l
m 2
m;
m N
PRC. 68
H3BeCTHO TiTO cBo6o,lJ;Hoe npo.n;OJIbHOe KOJIe6aTeJILHOe ,lJ,BH:meHlIe JIH-
HeHHon MOJIeI-CYJILI SlBJISleTCH cynepn03HU;lIen KOJIe6aTeJILHLIX ,JJ;BIDKeHHH
OT,l1;eJILHhIX aTOMOB. TaKHe ,l1;BH:meHH.H H83LIB81OT HOpMaJIbHbIMH KOJIe6a-
HH.RMH HJIII HOpMftJILHbIMlI MO,JJ;aMll II npe,JJ;CTaBJIHIOT co60ii ,lJ,BH)KeHHe
npH KOTOpOM Bce 8TOMbI cOBepmmoT rapMOHUqeCKHe KOJIe6B.HHH OP;HOH H
Toe :me qaCTOTLI npOXO,lJ,H npH aTOM O,lJ,HOBpeMeHHO 'tIepe3 CBOR nOJIo.me-
BHB paBHOBeCH.H.
1. IIycTb Xi - KOOPWIHB.TR CMe:m;eHufl i-ro 8TOM8 OT ero nOJIOmeHHH
PaBHOBecIIH. BLIpa3HTe CHJIy Fi .n;eHCTBYK>YK> Hft STOT aTOM. KaK 4JYHK-
D;HIO MrHOBeHHLIXCMe:m;eHHH Xl X 2 , X 3 , ... XNHJKeCTKOCTeiik l . k 2 . k3 ...
k N - I . Krome COOTHomeHHe lIMeeT MeCTO Me.H<,ZJ;Y CHJI8MH F 1  F 2' F 3' ..., F N?
HCnOJIb3YH aTO COOTHomeHHe YCTaHOBHTe 3B.BHCHMOCTb Me.H<,ZJ;Y CMee-
HHHMH Xl X2 x 3 ' ..., X N II ,lJ,aHTe eH cpH3U'tlecKyro lIHTepnpeTan;IIIO.
2. IIp08HaJIH3IIpYllTe ,lJ,BHmeHHe ,ZJ;ByxaToMHoH MOJIeKYJIbI AB
(pIiC. 69). IIoJIrIHTe BhIpameHH.fl ,lJ,JIH CHJI ,lJ,eiicTBYlOIIJ;HX a8 8TOMhI A
II B Tlepe3 cMeeHH.fl x A II x B II meCTROCTb k. OUIIIIIHTe B03MOJKHLIe
THULl )1;BlImeHHH aTOMOB B MOJIeKYJIe II onpe,lJ,eJIHTe COOTBCTCTByrom;He
qaCTOTbI KOJIe6aHIiH. 06"b.HCHIITe nOJI}"IeHHbIH peaYJILT8T H B qaCTHO-
CTII T.OT cpaKT, T{TO aTOMhI Moryr KOlIe6aTLC.H C O.ll;HHaROBOH 'tlaCToToft
XOTSI ilK MaCChI pa3JIH'tlHbI ('tITO II npe,lJ,CT8BJI.HeT C060H HOpMaJIbHYIO
MO,lJ,Y).
3. IIpo8H8J1H3HPYHTe B03MO)RHLIe )J;BlImeHHH Tpex8ToMHon MOJIe-
KYJIhIABA (pIIC. 70). BhIpaaHTe CHJIbI ,lJ;eHcTBYIOHe Ha Kam.n:hIH aTOM
MOJIeKYJIhI, Qepe3 CMe:m;eHH.H Xl' X 2 ' x 3 . OnpCAeJIHTe HOpM8.JILHhIe MO.n;hI
,lJ,8HHOH MOJIeKYJIhI II COOTBeTCTBYlO:m;ne 3H8'tleHH.H qaCTOT.
4. qacToTLI ABYX rrpO,lJ;OJILHhIX KOJIe6aTeJIbHLIX MOA MOJIeKYJIhI CO 2
p8BHhI 3,998. 10 13 II 7,042. 10 13 rD,. BhI"'InCJ1HTe eCTKOCTh ,lJ,JIH CBH3H
c-o. OeHHTe, H8CKOJIbKO XOpOIIIO HCUOJIL30BaHHoe rrpH6JIHmeHHe

m A m n
B
XA
X n
m A m B m/J,
- - -
Xl X 2 Xg
PHC. 70
97
-
-
Puc. 69
4 --:J7!J

c 3 h 3 u(T.f)
81th" Tot
1,6
1,2
/",
I \
I
/ \
/ \
I "
J '
/ '"
/  
0,8
04
o
2
4
6
8
10 11 = hf
kT
PRe. 71
(CIIJIhI ynpyrOCTH) corJIacye-rCH c npHBe,n;eHHbIMH 'IlHCJIOBhIMH ,u;aHHLI-
MH. Macca aTOMa yrJIepo,u;a p8BHB 12 a.e.M. MBccaaTOMa KHCJIOPO-
Aa - 16 B.e.M. (1 B.e.M. = 1660 .10- 27 Kr).
3a,a;a q;a 3
CnyrHHK npe,u;CTaBJISIeT c060H paBHOMepao HarpeTLm map pa.n;HYCOM
R = 1 M. BeH nOBepXHOCTb CnyTUHKa HMeeT O)1;HH8.KOBOe nOKphITHe.
Cnyrum< H8XO,l);HTCSI Ue,u;aJIeKO OT 3eNJIH, HO He B ee TeHH. TeMIlepaTypa
nosepxHOCTH COJIHna, paCCMaTpHBaeMoro K8.K a6COJIIOTHO qepHOe Te.n:O
T c = 6000 K, ero Pft,ZJ;HYC Rc = 6,96 · 10 8 M. PaccToSIHHe MeCOJIHneM H
3eM.JIeit L = 1 5 . 1011 M. CnyrHHK H8rpeB8eTC.H COJIHe'tIHhIM H3JIyqeuHeM
lI,O TeMIIepBTYPhI npH KOTOpoH MOn::pIOCT:b H3.rryqeHH.R cnyrHHKa (ero Mid
T8.Kate paCCMaTpHBBeM KBK a6COJIIOTHO qepHoeTeJIO) paBHa norJIOm;aeMOH
HN NOII{HOCTH. H3BeCTHO, 'tIT0 MOn::pIOCT:b, H3JIyqaeM8JI e.nmnm.eit nOBepx-
BOCTH a6COJIIOTHO qepaoro Tena, no,u;qHHSleTCSI 3aKOHY CTeaHa-
BOJI:bM8Ha:
P = aT 4 , rp,e 0" = 5,67.10- 8 BT. M- 2 . K-4.
B nepBoM npn6JIH)f(eUHH MO)f(HO c'tJHTaTb 'liTO ConH:Q,e H cnyrHHK
nOrJIOm;aIOT Bce na,zJ;wom;ee Ha HHX H3JIyqeHHe.
98

1. HWt.ztuTe Bldp8.)KeHUe ,AJlH TeMIIepaTyphI T cnyrHHKa H BhIllUC-
JlHTe ee 3HaqeHHe.
2. CneKTp H3.nyqeHHH a6COJllOTHO qepHoro TeJIa onHCIdBaeTC.R T8.K
Ha3LIBaeMOA CneKTp8.JIbHOii «l»YHKHeii u( T, f), 38,ltBHHOii B BH,Q;e
u(T,/)dl = 87tk"T" T) 3 dT) ,
c 3 h 3 e ll -l
r/1,e u(T. 1 ) dl - nJIOTHOCTb sHepruH 3JIeKTpONarHHTHoro H3JIyqeHHsr,
TIaCTOThI KOTOpOro JIemaT B HHTepB8JIe [1; 1 + df] h = 6,6.10- 34 )J;m. C -
nOCTO.HHHBH IIJI8.HKa, k = 1,4 .10- 23 A. K-l - nOCTOHHH8JI BOJIbI(,Ma-
hi
Ha, c = 3 . 10 8 M/ C - CKOpOCTb CBeTa, T) = hT '
ECJIH npoHHTerpHpOB8Tb c})YHKnHIO u(T, f} no BceM QaCTOT8.M H Bce-
B03NOa(HIdM Hanp8BJIeHH.HM, TO nOJIytlHTCH BhIpWKeHHe J];JI.H MOIqHOC-
TH H3.nyqeHHH C e,q;HHhI nOBepXHOCTH (3aKOH CTec})aua-BoJlbQMa-
Ha).
Ha pHC. 71 npHBe.u;eH TaR HaShIBaeMbIA HOpMHpOBftHHLIii cneKTp,
onHChIBaeMLlA «l»YHKeii
c 3 h 3 u(T,f>
87th" .
Ha upaKTHKe cTpeMHTcH nOAAepmHBaTb CnyTHHK npH B03:M0HO
60JIee HH3KOii Te:MnepaType. C 9TOii neJIhlO npHMeHHIOT cooTBeTcTBYIO-
rqee nOKpLITHe KOTopoe oTpamaeT H3JIyqeHHe Bcex qaCTOT, npeBhIma-
IOrqHX HeKoTopYJO rp8.HFlHYJO qaCToTY, HO nponycKaeT 6e3 oCJIa6J1e-
HHH H3JIytleHHe HB 60JIee HH3KHX QaCTOT8X.
IIycTh npe,q;eJIbHaH qaCTOTa YOBJIeTBOpHeT COOTHomeHHIO
hi = 1200K.
k
O!\eHIITe. KaKYJO TeMllepaTypy 6Y/1,eT HMeTh cnyrHHK B aTOM c.nyqae.
Y K BS a H H H. He CTapaihecb UOJIyqHTb COBepIIIeBBO TOqabIA: OTBeT,
BOCnOJIL3yjhecb npH6JmmeHHoA c}>oPMYJlo:i, cnpaBe.D;JIHBOii llpH Ma.JILIX x:
e r =: 1 + x.
3HalleHHeHHTerpaxa
oo J 113dl1 = n 4 .
o e'l -1 15
M8.KCHNYM c})YHKIJ;HH '11 3 / (e 11 -1) /1,OCTHraeTCH npH 11 = 2,82.
99

3. B peanhHOM cnYTHHRe 06LI'lHO YCTaHaBJIHBaeTCH 3JIeKTpOHHSH
annapaTYPa KOTopaSl nHTaeTCSI OT COJIHe'lHO:H 6sTapeH. By,u;eM C'lHTaTb,
QTO MOIIJ;HOCTb 1 KBT KOTOpyIO nOTpeMSleT annapaTypa, pacceHBaeT-
C.H BHYTPH cnYTHHKa B BH,u;e TenJIOTLI. K8.Ks.H TeMnepaTypa YCT8HOBHT-
C.H B 3TOM CJIytlae Ha nOBepXHOCTH cnyrHHKs?
4. BeRM cpHpMa peI<JIaMHpyeT CIIH8JIhHyro KPSCKY CJIe.nyroIIJ;HM 06-
pa30M: .ara KpacKs 6Y,lJ;eT O'l'pIDKaTb 60JIee 900/0 II8,D;aIOIIJ;ero H3.nyqeHH.H
(K8.K B BH,lI;HMO:H, TRK H B HHcppaKpSCHO:H 06JIacTH) KaR a6cOJIIOTHO 'lepHOO
TeJIO, H368BJISI.H T8.KHM 06pB.30M CnyTHHK OT JIHIIIHero TeIIJIS. HTftK RpaCKS
06ecneQHT oxnam.n;eHHe cnymHKS,u;o H8.H60JIee HH3KHX TeMnepaTYP.. Mo-
HCeT JIH Cym;ecTBOBaTb B npHpo,u;e TaKaH KpacKs? OrBeT 060cHyHTe.
5. KaKHMH CBO:HCTBaMH ,u;OJIHCHO 06JI8,lI;aTb nOKpLITHe ,u;JI.H Toro, QTO-
6LI no,u;HSlTb TeMIIepaTYPY ccpepH'leCKOrO TeJIa, no,u;06Horo paccMaTpH-
BseMOMY B 38,l1;a'le cnYTHHKY, BLIIIIe 3HSQeHHSI, BbIQHCJIeHHOrO B n. I?
PeweHMII
3aJ(a'la 1
II p e]J, B a p II T e JI b H hI e C B e]J, e H II .H. BeKTophI 0603HaQSIOTC.H KaR
r, R. Be3 BeKTopHoro CIIMBons 6YKBbI r II R 0603HS'lSIOT ]J,JIHHY BeKTO-
pOB. E,u;HHHqHble BeKTophI 0603HS'lSIOTC.H e (ih - B cK06Kax 3anHcaH
BeKTOp, B Hanp8BJIeHHH KOToporo 6epeTCSI e,u;HHHQH:bI:H BeKTOp e (H8-
npHMep e (R) - STO BeKTOp e,u;HHH'lHOH ,u;JIHHbI, HaIIpSBJleHHLI:H OT
eHTps 3eMJIH It TOQKe P). E):\HHH'tIHbIH BeKTOp B HsnpaBJIeHHH OT B
3 - (R - - ) - - (R - ) K KR
K eHTPY eMJIH paBeH -e + r , S a = e 2 = - ----a - 9TO rpa-
R R
BHTaHOHHoe YCKopeHHe B TO'lKe C pB.,ll;HycoM-BeKTopoM R, rp;e
K=GMa.
B II. 1 CJIOBO .nsp8JIJIeJIbHLI. 03HS'tIaeT, 'QTO nap8JIJleJILHLI BeKTO-
pLI R H r, T.e. BHeIIIHee Teno B HaxO,D;HTC.R HS M8.KCHM8JIbHOM y,u;aJIe-
HRH OT eHTps 3eMJIH. AHaJIOrHTlHO .aHTHnapBJIneJ1hHLn 03HaqSeT
qTO TeJIO B H8XO,u;HTCSI aa MHHHM&nbHOM paCCTOHHHH OT 3eMnH. B no-
cne,u;YIOIqHX nYUKTftX CJIOBa . nap8J1J1eJIbHbI I) H .aHTHnapanJIeJIbHbI I)
OTHOC.HTCH K PS3JIHQHLIM HanpaBJIeHHstM BpaIIJ;eHHSI cnYTHHI<a: eCJIH
BeKTopLI yrJIOBOH CKOpOCTH ffi H g napanJIeJILHLI (g - yrJIOBaH CKO-
pOCTh BpaIIJ;eHH.R P OTHOCHTeJIbHO eHTpa 3eMJIH), 3TO 03HaqaeT qTO
cnYTHHK BpaIIJ;aeTCH B HanpaBJIeHRH CBoero Op6HTaJIbHOrO ,u;BHmeHH.H.
100

1. IIpH onpeneneHHH4CHJI HaTSl)KeHHSl nOTpe6yeTC.H HcnOJI&30BaT&
SeKOTOphle npH6JIIUKeHHJI:
1) neHTp cnyrHIlKa ,D;BHmeTCJI no KpyroBOR KenJIepOBCKoii op6HTe,
T.e. n 2 R =..!i.-. To "IToop6HTaKpyroBaSl, cKa3aHoBYCJIOBHH.BnpHH-
R 2
nHne OHa Morna 6&IT& 3nJIHnTHTlecKoii;
2) co H n nOCTOJlHHhI;
r
3) r < < R, II03TOM)' BhIcmHMH CTeneHSlMH R MO)KHO npeHe6peqb;
4) H3 pHC. 67 BH.D;HO, 'ITO KOH4hI cB06o,D;HbIX yqaCTKOB HHTeR MOryT
CBo6oAHo KOJIe6aTbCH C pe3YJIbTHPYIOIItHM ycKopeHHeM TeJI B. KaK npa-
BWIO, CWla. ,D;eAcTByIOID;a.a Ha B, H8.IIpaBJIeH8 He K neHTPY P. B JIJ060M
c.nyqae cB06o,u;HDIe yqaCTKH HHTeii, CoeJUIHHIO P H B, HallpaBJIeBhI
B,D;OJIb neiiCTByIOID;eii CHJIhI. IIpHMeM, 'ITO AJIHHa 3THX CB06o,D;EbIX KOm::t0B
OqeHb M8.JIa 11 HX Kone6aHlfH He BJIHmoT 38.MeTHLIM 06pa30M Ea ABH)Ke-
HUe TeJI B, T.e. ro If r ,D;OCT8TOTUIbI ,D;JI.H OnHC8HHSI ABH)KeHWI B OTBOCIf-
TeJIhHO P;
5) 60KoBoe nonomeHHe B - 3TO nOJIOmeHHe, Kor.u;a BeKTopLI R H
r nepneH,D;HKYJI.HpHhI. P8CCTO.HBHe OT 3TOA TOQKH 1I.0 neBTpa 3eMJIH
paBHO .J R' + = R Jl +  .. 1,00011 R .. R + 0,008 r. TIIKHM 06paaOM,
,nOCT8T01JHO XOpoillHM npH6JIHmeHlfeM .HBJIJleTC.H TO'lKa, p8CCTO.HHHe OT
KOTOpOH ,lJ,O neHTpa 3eMJIH p8BHO R. Omlf6Ka OneBKH CH.1IhI pacC1JH-
TaHHOA AnSI npH6JIHmeHHOro, 8 He peaJIbHoro nOJIOmeHH.H, 6YAeT MeE&-
me 5 0/0.
IIpe,D;noJIo.meHH.H 1, 2, 3 II 5 HenpHeMJIeMbI, eCJIH r (pa,a;HYc cnyr-
HIIKa) COCTaBnS:leT ThlCH KIIJIOMeTpoB H cnYTHHK HaxO;ulTCSl He,D;a.ne-
KO OT 3eMJIH. H3 QHCJIOB&IX ,lJ,8HBhlX YCJIOBH.H 38,lJ,a'lH CJIe,D;yeT, "qTO
r
-  OOI45, n03TON:y yKa38HHLIe npH6JIHmeHHH Haxo,u;.HTCS:I B npe,lJ,e-
R
JI8X O)KHJ];aeMoH TOqHOCTH pemeHHSI. TaTeJILHhlA auaJIII3 3THX npH-
6JIHmeHHA nOTpe6oB8JI 6Ll 60JIhme YCHJIH:i 'leM Henocpep;CTBeHHO pe-
weHHe 38,AaQH. MhI ero 3p;eCb npHB01I.lITh He 6y,u;eM, If OT }"IaCTHHKOB
OH He Tpe6oBaJlC.H.
Pa,zJ;HYC-BeKTOp Ten8 B paBeH R + r. AJI.H CKOpocTH H YCKopeHH.H
Tena B nOJIyqaeM
V  [0, R] + [ro r]. a = _(0. 2 R + cl r).
101

C He4>OpM8JILHOH TOq:KH apeHHJI ,;o;BHmeHHe TeJI8 B npeJJ;CT8.BJIHeT
C060H cynepn03HI:tHIO yx BpeHHH: O,;o;HO - BOKpyr 3eMJIH, .n;PY-
roe - BOKpyr D;eHTpa cnyrHHKa. nOSTOMY' H YCKopefiHe TeJla B eCTL
CYMMa .n;BYX COCT8BJI.HIOIQHX: OABa HanpaBJIeHa K I:teHTpy BeNJIH H HMe-
eT MO,lJ,YJIb n 2 r, a .n;pyra.H HsnpaBneH8 K neHTpy cllyrHHKa H pasBa no
MOJJ;YJIIO ro 2 r.
[paBHT8ItHORHaH CHJI8, ,;o;eHCTByronJ;asr UB TeJIO B, 3BBHCHT OT pac-
CTO.HBH.H OT B ,;0;0 neHTpa 3eMJlH, T.e. OT Mo.n;ynSl BeKTopa R+r:
K
Frp ;:;; m I _ 2 ' H OHB n8npaBJIeHa K I:teHTpy 3eMJIH, T.e. npOTHBO-
R+r I
uon03CHO BeKTOpy R + r . B BeKTopBLIX 0603Ha-qeHH.HX:
F _ _ K; (R + r) _ _ K(R + f)
rp - m l.il+rr - m l.il+rl s . (1)
.z:tJI.H B}JqHCJIeHHH BTOpaJI (}>OpM8 38llHCH 'IlaCTO 60JIee y,;o;06Ha.
1I0JIBaJI CHJI8, eACTBYJOIQa.H HB Teno B, 3BnHChIBSeTC.H 'tIepe3 YC-
KopeHHe cne,;o;yJOnJ;HN 06pBaoM: F ==  + F rp ::: ma = m( _0 2 .il - (02;:),
oTKy,;o;a
Fa = -o.2.il _ CJ)2;: + K(El + ) .
m l.il+
STO TO"lHhIA pe3YJlLTST. H33Toro BhIp8)KeHH.R yace Moryr 6ldTL no
JIyq:eHId 'IlHCJIOBlde 3HBqeHHH (npHMep npHBe,D;eH fia c. 112). O.n;H8KO
(2)
(}>OPMYJIB .n;JIH F H CTBueT fioJIee narJISI,ZJ;HoA, eCJIH MLI paCCMOTpHM ee B
npeHe6pe)l(eHHH BldcmHMH CTeneHSlMH NaJILIX BeJIHtlHH, npSKTH'Ilec.
KE He BJIH.H1OIQHMH HB pe3YJIbTaT.
nOJIOmeHHe Tena B MO)I(eT C5LITh YK838.HO c nOMOnJ;LIO pBCCTO.HHHSI
OT B ,;0;0 I:teHTpS 3eNJIH:
IR+;I = R+p, re - r ce; p ce; r.
Bldpa3HM 3HBNeB8TeJ.IL B (}>opNYJIe (2) q:epe3 £.. :
R
(R+pr s =n-S[l-  +6( ) "J
(3)
102

C 4»U3lAecKO:A TO'({KH apeHH.H aTO npH6JnDKeHHe 03Ha-qaeT, 'ITO MbI
cq}fTaeM 38.BHCHMOCT& H3MeHeHHH rpaBHT8IUlOHHOro yCKOpeHKSI OT H3Ne-
HeBH.H pa,D;HYca JIHBeitHOH. IIOACT8BJI.RSI Bhlp8}KeHHe (3) B (}>oPNYJIy AJI.H
 , nonyqaeM AJlJI npoH3BOJIhHOI'O nOJIo»teHHSI B8IOm;eI'OCSl CnyTHllK8
_ _ (11+ T) [ 1- 3 : + 6( : ) 2 ]
F. :;;: -02R-m 2 i + K =
m R 
[ - - ( ) 2 - ]
2 - 2- 2 R i R p R
=-OR-ror+OR -+--3p-+6 - - =
R R R 2 R 2 R
::-ro 2 r+0 2 r-302p R .
R
(4)
HaH,n;eM nOCJIe.D;Bee CJIaraeNOe B BhIpa)KeHHH ( 4) ,n;JIa pa3HHX nOJlo.
)KeuRA B H P. Teno B 3B.HHMaeT BepXHee nonomeHHe (R H r nap8JI.
JIenLBhl), HH)KHee nOnO)KeHUe (R H r aHTHnapaJIJIeJIhHhl) II 6oKoBhIe
UOJlOmeHH.fI (.D;JI.H 6oKoBoro nOJIO)l(eHHSI B03hNeM He onpe,n;eJIeHHe, a
npH6JIHmeHHe KOTopoe COOTBeTcTByeT 3HaQeHHIO p = 0):
- -R
- BepXHee nOJI02KeBHe: p=+r, R = r-, OTKY,lJ,a pR=rR, H no-
r
CJIep;Bee CJIaraeMOe paSHO - 30 2 r ;
- -R -
- HH3RHee nOJI02KeUHe: p= -r, R=-r-. oTKy,n;a pR=rR, H no-
r
CJIe,D;uee CJIaraeMOe paBHO - 30 2 ;= ;
- 60KOBoe nOJIO)KeHHe: p = O. H nOCJIe,n;Hee CJIaraeMOe paBHO O.
1I0,n;CT8.BJI.fI.H Bce 3TH 3HaQeHHSI nOCJIe,n;Hero cnaraeMoro B 4»oPNYJIY
(4), no.naeM CHJlY HaT.H3ReHHSI B Tpex CJIyq8.Slx:
P min = - (m 2 - d) i7n ecJlH p = 0;
Fmax = _«(02 - 2( 2 )im, eCJlH p = -:f:r;
(5)
M':;: Fmax - F min ::;: a0 2 rm.
Bo Beex cn)7'I8.RX CHJIa napaJIJIeJ1&Ha HJIH aHTHnap8.JIJIeJIbHa BeK-
TOPy r. BeJlH'({HH8 (co 2 + 2( 2 ) BCer.D;a nOJIO)KHTeJl&Ha nOTOMY' CHJIa
103

FmaJ( Hanp8.BJIeH8 npOTHBOnOJIOmHO r, T.e. HHTh y,n;epmHBaeT TeJIO B
B BepXHeM II HH)KHeM nOJIOmeHH.JIX. A BeJIJP.IHHa «(02 - ( 2 ) CTaHOBHTC.H
OTpHI{.8TeJIhHOA, ecJIH CO < O. STO )];OJIJKHO 6LIJlO 6LI 03Ea'qaTb, q:TO F min
COHanpaBJIeHa C r , T.e. HHTL OTT8JIKHBaeT TeJIO B, EO aTO HeB03Mom-
HO H npHBeJlO fiLl K pa3pymeHHIO cnyrHHKa, T8K RaI< TOHICue HHTH MO.
ryr 06ecneQHTh TOJIhKO y,n;epmHBaIOIIJ;y1O, HO He OTT8JIKHBaroym CHJIY.
II03TONY B nocJIe,n;ymm;eM 8.HaJIH3e npUHHMBeTC.R co > 0 H BhIpameHHe
,D;1IB AF a&nHC8HO B 3TOM npe,lJ,nOJIOmeBHH.
2. CHJIa, ,n;eAcTBymm;a.R Ha R8KOe-JIH60 H3 TeJI B, npHHHMaeT M8K-
CHI\II8.JIhHOe 3HaQeHHe )];B8JK.ALI 38 OAHB 060POT - B BepXHe:M II HHJKHeM
nOJIOmeHH.RX, TaR me K8K H MHHHM8.JILHOe 3HaqeHHe - B ,n;BYX 60KO-
BLIX nOJIO)l(eHHHX. MraOBeHH8JI OCb BpaII(eHH.R cnyrHHKa caMa Bpam;a-
eTCH C Op6HT8JILHoA yrJIOBoA CKOpOCThlO O. ECJIU cnyrHHK Bp8IIJ;aeTC.H
B HanpUJleHHU CBoero op6HT8JILHOrO ABHmeuuSi (ro H n nap8.JIJIeJIL-
BbI), TO, QT06L1 COBepmHTh O,n;HH nOJIHLIA 060POT OTHOCHTeJILHO MrHO-
BeHHoA OCH Bpam;eHHSI, B CHCTeMe HerrOp;BH>KHLlX 3Be3A OH AOJI)I(eH no-
BepHYThCSI T.IYTL 60JILme, qeM Ha 360°, 3Ha'llHT, ero yrJIOBaJI CKOpOCTh
OTHOCHTeJILHO M'rHOBeHHoA OCR Bp8nteHHH PUH8 0) - Q. B TOM cnyqae,
Kor,n;a aanp8BJIeHHSI co6cTBeHHoro H Op6HT8JIbHOro BpaIIJ;eHd npoTH-
BOnOJlO)l(HLI, yrnoBaSI CKOpoCTh cnyrHHKa OTHOCHTenLHO MrHOBeHHOH
OCB Bpan:teHHH PUH8 (0 + .0. B BeKTopHLIX 0603Hsq:eHH.HX B 060HX CJlY-
qa.JIX OTHOCHTeJILH8S1 yrJIOBaJI CKOpOCTL 38llHCLlBaeTC.H KaR ro - 0.
MeXB..HH3M COBepmaeT )];Ba pa60QHX I{.HKna 3a O,n;HH OTHOCHTeJIbHbIH
060POT (O,n;HH IJ;HKJI: BTSlrHBaHHe HHTH + OTnYCK8BHe HHTH). TorAa pa-
60Ta, COBepmaeMaSl CHJIaMH, ,n;eACTBYJOHMH H8 TeJlO B, 38 OAHH 060-
poT cnyrHHK8 p8.BH8
AE = 2.Ar (F max - F min) = 6ml1rr 0 2 = O,06mrQ2.
IIepHo,u; o,n;aoro OTHOCHTeJlhHOrO 060pOTa paBeH
AT =.
c.o:l::Q
r.u;e3H8.K .+. cooTBeTcTByeTCJ1yqalO, Kor,u;a co II n 8HTHnap8JIJIeJIL.
HLI.
MOIQHOCTL NeX8HH3M8
p = &E = 2Ar(F max - F m1n ) = Ar (F max - F min )«(1) :t 0) .
AT 2n 
ID:tO
104

qUCJLenHbte pe3YJLbmambt.
K
H3 BhIpameHH.R 2;;;;; 0. 2 R nonyqaeM K R = 2,54. 10 21 M 4 / C 2 H
R
o.;;;;; .JKR- 3 = 0,001106 pap./c. Op6HT8JJbHhIii nepHOp; paBeH 5678 c.
YrJIOB8S1 CKOpOCTb cnyrHHK8 paBH8 m;;;;; 21t/(360 c) = 0,01745 p8.JI,/c.
OrHocHTMbHhIe yrJIOBLIe CKOpOCTH paBHL! co - 0: = 0,01634 P8Jl./ C (eCJIH
ro H 0 nap8JIJIeJIbHb!) H 0 + co = O01856 p8Jl./c (eCJIH ro II 0 8HTHll8-
paJInenLHh[). Cnep;OB8TeJIbHO,
F min = (co 2 - ( 2 ) rm = 30 339 H;
Fmax =«(02+ 2 ( 2 ) rm = 30706 H;
Frnax - F m1n = 3rm n 2 ;;; 367 H;
P = r (F roax - F roin) (co ::t:: {2)/1t;
Pn =2168BT, PM =1909BT.
3TH 3HaQeHH.R p;onmHLI 6LITL O:KpyrJIeHb! AO P;BYX 3Ha'q8m;HX rU'l4»p,
nOCKOnLKY B pemeHHH 6LIJIH HCnOJlb30B8HhI npH6nHmeHHB
P =2200BT P tt =1900BT.
n
AHWIH3 CHJI p;eiiCTBytOID;HX Hft TenO B B 60KOBOM nOJIOmeHHH (CM.
pHC. 67):
r
yrOJI a H8XO"nHM H3 COOTHoweHHH tga = -;
R
CHJIa HaTSI.meHHSI P; = sina  F,.tga = F g !:.....
R
3. B npHH:Q;nne a8 C'tIeT ,lJ,eiiCTBH.R MeX8HH3MOB Op6HT8 cnyrBHK8
MomeT CTaTL :JJIJIHIITH'tIeCKoii. Ho nOCKonbKY MeXRHH3M cpa6aTbIB8eT
QeTbIpe p83a 38 OP;HH 060POT cnyrHHKa, npOH3Bo,lJ;HMbIH s4>4>eKT P8C-
npep;eluIeTcH no op6HTe Gonee HJIH MeBee cHMMeTpIPIHo. BnarO,lJ,apH
KpyrOBOii cHMMeTpHH MbI H8BepHBK8 Mo»ceM CqHTaTb '1TO op6HTa oc
T8eTCH KpyrOBoii, p;8me Korp;8 pa60TalOT MexaHH3MbI. IIoSTOMY Op6H-
T8 H3MeHHeTCSI C OP;HO:i KpyroBOH :KennepoBCKoii Op6HTLI HR .ApyrYIO.
MOJKHO 6hIJIO 6b! p8CC'lHT8Tb H3MeHeHHe Op6HTbI, 8H8JIH3HPY.H peaYJI&-
THPYIOm;y1O rp8BHTaI{.HOHHLIX CRn, .n;eACTBytOID;HX H8 'leThIpe TeJIa B.
O,u;H8KO aTO CnOJKHbIH H Tpyp;oeMKHH cnoco6. IIoJIHhIii 8U8JIH3 38,l1;aTlH
B STOM cnyqae 6b!JI 6b! OqeHb TPY,u;HLIM. (3TO 6bIJI 6hI e.n;HHCTBeHHO B03-
NOJKHhI:i nYTb pemeHHB 38,AS-qH, ecJIH 6b! MeX8HH3N pa60T8JI He nOCTO-
BHHO, a cpa68TLIBaJI 6Ll anH30p;H"tIeCKH H BCJIeACTBHe :JToro op6HTa H3-
MeH.HJl8Cb 6Ll Ha SJIJIHnTHqeCKYIO.)
B .n;aBHOM CJI}"I8e STO He HBJI.HeTC.H He06xop;HM:LIN. ]1.JIH MHorHX 4»H-
3HQeCKHX 38P;8Q HCnOJIL30BaBHe 38KOHOB coxpaueHH.R .RBJI.ReTC.R HftH-
60JIee npocTLIM cnoc06oM pemeHHH, eCJIn HaHAeTCH nOCT8T0'1HOe 'lHC-
105

JIO COXPaB.HIOlQIIXCH BeJIHqHB, 38,A8TJ8 CBO)J;HTCH K pemeHHIO ypSBHe-
Hllii COXpaHeHHH.
KorAa pSCCN8TpHBaeTca Bp8TeJIbBOe )J;BHmeBHe, MomBO, KaK
np8BHJIO, HcnOJIb30B8Th 38.KOHbI cOXpaHeHH.R SHeprHH H MOMeHTa HM-
nyJIbca. TpYAHOCTb qaCTO COCTOHT B TOM, QT06bI yqecTh Bce DHAbI 3Hep-
nm: (TaR KaK COXpaH.HeTCH nOJIBaJI SHeprH.R CHCTeMhI) II Bce COCT8BJI.f[-
IOHe NOMeUTa HMIIYJIhCa.
CU8"18.JI8 paCCMOTpHM MOMem HMnYJIhCa. HMnYJIbC cnYTHHK8 npn
BpeHHH ero BOKpyr 3eMJIH He coXpaU.HeTC.H, a CYMMapHLIii MOMeHT
HMUYJIbCa 1 0 0THocHTeJILHo neUTp8 8eMJIH COXpaH.HeTCSl nOCKOJILKY
ep;HHCTBeHHM BUemWl.R CDJIa, Ae:AcTBYJO8.Ji H8 cnyrHHK (cHJIa rpa-
BHTanHOHHOrO npHT.HmeHH.H), Hanp8.BJIeH8 K .q;eHTpy 3eNJIH (STO 6&1.110
6hI HeBepHo, ecJIH 6hI nJIOCKOCTb Op6HTLI npoxoa TIepe3 nOJIIDC H ecJIH
6hI npHHHM8JIaCL B paC":leT 8CHMl\leTpHH 4>OpMLI nOBepXHOCTH 3eMJIH).
I D CKJI8,lJ;bIB8eTC.R H3 )J;BYX KOMnOHeHT: BuyTpeHHero MOMeUTa HMnYJIL-
ca, 06YCJIOBJIeBHOrO co6cTBeuHhIM Bpam;euHeN cnyrHHKa Hero op6n-
TaJILHOrO MOM:eHT8 HNIIYJIhC8, 06YCJIOBJIeHHOrO ,D;BHmeuUeM IteHTpa
Mace cnyrHHK8 BOKpyr 8eMJIH.
pyroA COXpaH.HIOm;eHCa BeJIH'IHBOH .RBJI.HeTC.R 3HeprH.H; Touee,
nOJIDaR 3UeprH.R Ell cnyrHHKa B03paCTBeor 38 CqeT PS60ThI, COBepmae-
MoA CHJISMH B MeX8.HH3Ne. IIOJIB8J1 3BeprH.R CJIaraeTC.R H3 CJIe)J;YIOHX
aHeprHii:
8ueprHH Bp8IIteHHa CTIYTHHKa, p8BHOH !.. 4mro 2 r 2 ;
2
Op6HT8JIbHOH KHHeT}[tICCKOA 3ueprn:H, T .e. KHHeT}fqeCKOH sHeprHH
neHTps Macc, p8BHoii !.. 4m!i'R 2 ;
2
nOTeBD;H8JlLHOH 3HepmH CnyrBHKS B nOJIe T.RrOTeHH.H 3eMJIH paB-
BoA -4m, K .
R
MbI B Hame.M pemeHHH Be3)J;e npeHe6peraeM TIJIeH8MH BLIme nepBo-
1'0 nopII,ltK8 .M8JlOCTH, n08TO.MY paCC":lHTLIBSeM nOTeH:QHaJILHyJO 3uep.
I'HIO cnyruHK8 T8.K )Ke KftK eCJlH fiLl ero :Macca fiLIJIa cocpeAOTO'leHa B
eHTpe P.
JIfi nOJIaoA 8HeprHH nOJIy'lae.M BhlpameHHe
K 1 2 1 2 2
Ii' = 4m { --+-!1 rt +-(0 r )
 R 2 2 '
HJIH C yqeTOM COOTHomeHufi KeuJIepa
ED;:;: 2m (- (22 R2 + ro 2 -,2 ).
106

TpeThe yp8BHeHHe CHCTeMhI nOJlyq:aeM H3 38.KOH8 Ken.nepa, CB.R3LI-
B81Om;ero P8JJ;HYC Op6HThI cnyrHHKa II ero Op6HT8JIhHyJO CKOpOCTh. IIo-
JIyq:aeM CHCTeMY 113 Tpex ypaBHeHHR:
K 2
2;;: 0 R (38.1<OU KenJIepa),
R
In= 4m (ror  (}.R2) =const,
E 2 - El = AE,
r11.e El H E 2 - :3'1'0 nOJIH8H 3HeprH.H CnyTHHK8.11.0 II nOCJIe TOro, K81( co-
sepmeH8 pa60Ta M. Bepxmlli 3uaK COOTBeTCTsyeT CJIyq8lO, KOr11.a ro H
n nap8.JIJIeJIhHhI: cnyrHHK Bpam;aeTC.R B UanpaBJIeHHH Op6HT8JIhHOrO
,lJ,BHHCeHH.R, a IUUKHHH 3HB.K COOTBeTCTByeT CJIyq8lO, KOr11.a (j) II n 8.HTH-
n8paJIJIeJIbHhI, T.e. HSnp8.BJIeHH.H Bpa.m;eHHH npOTHBOnOJIO)KHhI.
8TH TPH ue3aBHCHMhIX ypaBHeHHSI ,lJ,OCTaTo-quhI 11.JIH onpe11.eJIeHH.H
Tpex HeH3BeCTHhIX - 0, co II R. 3anHC8HHhIe B TaKON BHJJ;e yp8BHeHHSI
He MOryl 6hITL HarJISI)I;HOA HJIJIIOCTpHeA npOHCX011..RID;nX H3MeueUHR.
MOMeHT HMnYJIbC8 38BHCHT 0'1' Tpex nepeMeHHLIX, K8m,u;8.JI H3 KOTOpLIX
MeH.ReTC.R npII H3MeHeHHH Op6HThI. AHaJIH3 yp8BHeuHit JIyqme Ha'lH-
HaTh C H8xo:m,n;eHHH 38BHCHMOCTH MOMeUTa HMnyJILCa 0'1' R C nOMOII(blO
ypaBueHH.H KenJIepa:
10 =4 =4 mn2 =4 m.JKR .
TaKIIM o6pa30M, Op6HTaJILUhIA: MOMeHT YBeJIH'tIUBaeTCH npH YBe-
JIH'tIeUHH R. )J;JI.H nOJIHoro MOMeUTa HMnYJIbCa nOJIyqaeM
1(1 = 4m (0),-2 :f:  KR ) = canst.
IIOCKOJIhKY r II K nOCTOSIHHhI, aTO ypSBHeHHe CBH3hlB8eT co II R.
ECJIH ro H 0 nap8.JIJleJILHLI TO npu YBeJIHQeHHH (0 p8JJ;uyc R lJ.oJI-
)KeH yMeuLmaTLC.R, a npn YMeHLmeHHH co - YBeJIH'IUBaTLCH.
ECJlH (j) II n aUTHnap8.JIJIeJIhHhl, TO, 'IT06LI 10 COXp8HHJIC.H, npH
YBeJIINeUHH (0 PHYC R lJ.OJImeH YBeJIH't{UB8ThCS, 8 npH YMeHLmeUHH
(0 - yMeHLmaTLC.H.
HH'1'YHTHBHO K8meTCH, "tlTO YBeJIu'tleuue DOJIHOR aueprnH CnyTUH-
Ka (38 c't[eT nOJIOmHTeJIbHoi\ pa60ThI, COBepmaeMoA CHJIaMH B MeXaHH3-
Me) 11.0JI)KHO npHBecTH K YBeJIHqeUHIO co. JIH nOJJ;'l'Bep:m,lJ,eHH.H SToro
npe,lJ,nOJIOmeHH8 HY.)KRO npo8.H8JIH3HpOBaTh ypaBHeHHe lJ.118 3HeprHH.
SueprH.Jl Op6HTaJIbHoro .ltBH)KeHHH KB.K 4JYHKItH.H 0'1' R paBHa
Eo K 1 2 2 1 K
=--+-O: R =---
4m R 2 2R'
107

a nOJIHaH 3HeprHH
E K 2
-1L;;: -_ + ro 2 r .
2m R
K8.K nOR838.H0 BhIme. B TOM CJIyqae. Kor,o;a ro H 6: aHTllnSpSJIJIeJIb-
HhI 3aKOH COXp8.HeHH.H MOMeHTa IIMUYJILCa Tpe6yeT op;HOBpeMeHHoro
YBeJIl{qeHH.H HJIII o,o;HOBpeMeHHoro YMeHbmeHH.H m II R. IIOJIHSH aHep.
rHJI CIIyTHIIKa YBeJII{qIlBaeTC.H nOaTOMY' pe8JIH3YeTCH rrepBLIH caH,
TaR R8K YBeJIHQIIB81OTC.H 06a CJlsraeMhIX B BbIpameHHH P;JIH SHeprHH.
B TOM cJIyqae, eCJIH co II 0 napaJIJIeJI:hHLI, Tpe6yeTcH 60JIee Trqa-
TeJIbHhIH 8HaJIH3. 3anHmeM ,u;H4J4>epeHItH8JI In
2 1 ( ) _!
dmr +-K KR 2dR;;: 0
2
H nOp;CT8BHM 3TO COOTHomeHHe B BldpameHHe ,D;JISI ,D;u4JcPepeHIJ;H8JIS
nOJIHoii 3HeprHH:
dEn = d ( - K + m 2 r 2 ) = dR+ 2m dill. r 2 :;; dR - 2m !.K(KRrdR =
2m R R 2 R 2 2
=KdR [ -ro(KRri]=KdR(  -  ) = KdR. R- 2 (1- J.
IIOCKOJIbKY CD > 0:, YBeJIHQeHlle nOJIROH SHeprllH COOTBeTcTByeT
YMeHbweHlllO R II YBeJIlPleHHIO roo &ro II nOJ];TBepaeT C,lJ,e.rraHHoe BHa-
"laJIe npep;nOJIOmeHHe 0 TOM, llTO ro YBeJIHqHBaeTC.a: B 060HX cJIyqa.a:x.
Pa.u;HYC Op6HTId R YMeHLmaeTcSI B TOM CJIyqae, RorAa ro II n na-
paJIJIeJIbHLI II yseJIHqHBaeTC.fI B TOM CJIyqae, Ror,u;a OHH aHTHnapaJI-
JIeJIbHId .
H3MeHeHHe Op6HTRJILHOH CKOpOCTH npoTHBOIIOJIOmHO H3MeHeHHIO
pa.zr;Hyca R: Op6HTaJIbH8SI CKOpOCTL YBeJIHBaeTCSI B TOM CJIyqae, Kor-
.na co H (} napaJIJIeJlbHbI, II YMeHbmaeTCH B TOM CJIyqae, Kor,u;a OHI1 aH-
THIIap8JIJIeJILHhI.
YrJIOBaJI CKOpoCTL 0) YBeJIUqHBaeTC.R.
IIOTeB8JILHaSi 8ReprHH YBeJIH'lHBaeTC.R npn YBeJIUlleHHH R. T .e.
YBeJIHtlHBaeTCH ecJIH ro II Q 8HTlInapaJIJIeJIbHLI, II }TMeHhmaeTC.H, eCJIII
OHII nSpaJIJIeJILHbI.
KaK BH.D;HO H3 3THX pe3YJIhTaTOB P8AIIYc Op6HTId cnyrHHKB MomeT
YBeJIIftIHB8TbC.H (T.e. CnyTHHK nepexO,D;HT HS 60JIee BhICOKyro Op6HTY).
aTO II npoHCXO,lJ,UT B CJIrIae, Kor,u;a ro II 0 8HTHnapaJIJIeJIbHhI.
108

IIpH OTBeTe HB nOCJIe,AHHH BOnpOC npe,lJ,nOJl8raJlOCb 'tITO 6YAYT pac-
CMOTpeHbl pa3JIBHSI MeX8.HH3Ma Ha cIIyrHHKe II MeX8HH3M8 YCKope-
HH.R 06blQHOH paKeThI: paKeTa MomeT npaKTHeCKH IIOKHHyrb 06JIaCTb
rpaBHTan;HOHHoro npHTHmeHH.H 3eMJIH, 3BTpaTHB HeKYIO KOHeHYIO
SHeprHIO. HanpOTHB, cnYTHHKY IIoTpe6yeTC.R 6eCKOUe'tlHOe KOJIHQeCTBO
SHeprHII ,lJ,JUI Toro, 'tIT06bI pa.u;HyC R BoapaCT8JI HeorpSHWleBHo: H3 ypaB-
HeHII.R ,lJ,JI.R In CJIeAyeT, 1lTO co HeOrpaHH1leHHO B03paCT8eT nponopn;HO-
HaJI:bHO KBa.ApaTHOMY KOpHIO H3 R. IIo3TOMY 6Y,lJ,eT cyecTBOBBT:b MaK-
CHM8JIbHhIH pa,z:J;uyc Op6UThI Rmax. IIPH KOTOpOM cnYTHHK psapymHTC.R,
TaK KaR nopBYTC.H BIITH.
3aAa'la 2
CyecTByeT HeCKOJlbKO npHHD;UIIU8JIbHO 0)];HH8KOBbIX peIIIeHlIit.
3a.n;sQs Cq.OPMYJIHpOBaUS TaKHM 06paaOM, 'tITO npHMeHeHHe TeopHH
MSTpnn" HCnOJIL3YIOI.II;eHC.R B 3a,n;aQ8X SToro THIIa Heo6.R38TeJIbHO. Ho
3neraHTHoe pemeHHe C HCnOJIL30B8HHeM TeopHH MSTpHn; 6Y,lJ,eT ,lJ,aHO
UHme.
1. CHJIa, AeHCTBYIOII(8.H H8 i-ji 8TOM MomeT 6hITb BbIQHCJIeHa C no-
MObIO pHC. 72.
i-I k j - 1
«
I
L-...-
X i -l
i + 1
9
I

X i + 1
i k i
9
I

Xi
PHC. 72
IIpH nOJIOJfCHTeJILHOM CMeeHHH X;-1 (i-1)-ro aTOManpY>KHH8, me-
CTKOCTL KOTOpoA k i - 1  cmHMaeTC.R Ii ,lJ,eHcTByeT Ha i-it aTOM C CIIJIOH
k i - 1 Xl-I' HanpSBJIeHHOH BnpaBO. COOTBeTCTBeHHO eCJIH CMem;eHHe i-ro
aTOMapaBHO Xi' as aTOM i AeHcTByeT CliJIa -h'-l X i - 1 - k j Xi HanpaBJIeH-
HaB BJIeBO. HaKOHen;, 38 cqeT CMeeHH.R: X i - I (i + l)-ro 8TOMa HR i-H
aTOM ,lJ,eHcTByeT CHJIa hi+1 X H1 , HanpaBJIeHHa.H BnpaBO.
CYMMapHaH CRJIa, AeAcTB}T1OIItaH ua i-j:i aTOM, paBHa
F i = - h'-l (Xi - Xi-I) - hi (x , - x , +1)' (1)
IIpHHHMM BO BHHMaHHe, qTO HeT aTOMOB JIeBee nepBoro H HeT aTO-
MOB npaBee N -ro, MomBO 3annCRTb:
FI = -hI (Xl - x 2 ),
F 2 = -hI (X 2 - XI) - k 2 (X 2 - x 3 ),
F, = - k t - I (Xi - Xi-I) - hi (Xi - X i + I ),
F N = - k N - 1 (X N - X N - 1 ).
(2)
109

m A k m B
'i' '(
I I
 
XA X B
PIle. 73
CJIOmHB Bce CHJILI, nOJI}"IHM nOJIH}TJO CHJIY F, J];eHCTByrorqylO aa MO-
JIeKY.ny:
F=F] + F 2 + ... + F H = O.
(3)
COrJIaCHO BTOpoNY 3aKOHY HLIOTOH8, 3Tft CHJIS pSBHa npOH3Be,lJ,e-
HHIO MaCChI MOJIeKYJILI Hft YCKopeHHe ee eHTpa Mace:
F=Ma=O.
(4)
K8.)I(J];8JI H3 BHYTpeHHHX C:HJI paBHa npoH3Be,lJ,eHHIO MaCCLI aTOMa Hft
ero YCKopeHHe:
Fj = mia i .
(5)
H3BHpameHHH(3)H(5)noJIyqaeM
m]a l + m2 +... + m'N =0.
H3 (6) nOJIyqaeM
(6)
m l VI + m2 v 2 + ... + mNv N ::;: Mv o ::;: const,
(7)
r,n;e V o - CKOpOCTh J];BHmeHHSI eHTpa Macc. ECJIH Mhl paCCMaTpHBaeM
MoneKYJIY B CHCTeMe OTC'tleTa, CBH3aHHOH C ee eHTpOM Macc, 3Ta CKO-
pOCTh pUHa HYJIIO.
Tor,n;a 1I.JISI cMeeHHH 8TOMOB MLI nOJI}"IfteM CJIe,n;yroIIJ;ee COOT Home-
RHe:
mlx l + m2 x 2 +... + mNx N = Mx o :;; const.
(8)
STy KOHCT8HTY MO)KHO npHHSlTb paBHoii HYJIIO, ecJIH CQHT8Th, 'tITO
HaqUO OTC'tleT8 CBH38.HO C eHTpOM Macc H 'tlTO BHyrpeHHHe CHJI:&I MO-
neKY JI:&I He BJIHJlIOT aa ,lJ,BHmeHHe n;eHTpa Macc.
2. MOJIeKyJI8 II HcnOJIh30B8HHhIe 0603HftQeHH.R IIOKa38.HhI H8 pHC. 73.
CUJIbI, ,lJ,eHcTByrorqHe Ha ftTOMhl, 38llHChIBB.1OTCH B BE,lJ,e
FA ::;: - k (X A - X B ) = m A a AI
F B =- k(x B - xA)=m B a B -
(9)
OnHTL
FA +F B ::;;: mAaA+mBaB=O.
(10)
110

B CHCTeMe eHTpa Macc
mAx A + mBx B =0
(11)
8, 3HaQHT,
m A
XB=--X A ,
m B
(12)
COOTHomeBHH (9) MOryT 6LITb nepenHC8.HLI TaR:
F k ( mA J k mA + m B
A :; - X A + -X A = - X A .
 IDS
(13)
"Ii' k( mB J mA + ms
L"B = - X B +-X A :; -k X B .
m A m A
COrJIaCHO YCJIOBHIO 38JJ;a'tlH, ,AeHCTB}T1OIIJ;8.JI Ha KamJJ;hIH aTOM CHJla
npOnOpItIlOH8.JIbHa ero CMeeHHJO, T .e.
FA = -rAx A ,
FB=-rB%B.
(14)
Koacp(}>HI{HeHThI nponOPItHOHaJIbHOCTH r A II r B MOryT 6hITb nOJIY-
'leHLI H3 BhIpameHHH (13) II (14):
_ k mA + m B . k mA + m n
r A - , Tn:; .
m n m A
(15)
PemaIOHM B 8.HaJIlI3e 3a,n;aTIH HBJI.HeTCH IICnOJIb30B8HHe ,A8.HHOrO
:B YCJIOBHH c}>8KTa, qTO aTOMhI KOJIe6JIIOTCSI C O,AIIH8KOBO:H 'tIaCTOTOH
co.. = J T.. = k m .. +m B COB"
m A mAmB
(16)
,npyroe pemeHlle, CJIe,AYIDII(ee 113 BhIpameHHH (9) II (11), TPHBH8.nb-
80 II cooTBeTcTBYeT
X A = X B .
(17)
OTKy.u;8 ill = 0, 'tIT0 COOTBeTCTByer OTCYTCTBlllO )1;e4JOPMBItHH MOJIeKYJILI
(Toe. OTCyrCTBllJO KOJIe6aTeJIbHOrO ,ABH»CeHHH) HJIH, B CHCTeMe IteHTp8
Mace. nOKOeHc.H MOJIeKYJIe.
38.,ZJ;aQY MomHO TftKme pemHTb B npe,lJ,nOJlOmeHHH. QTO X A II X B npo-
nopJ:{.HOHaJIbHLI (aTO CJIe,AyeT 113 pemeHHH n. 1) T.e. MhI nOJI8raeM
X B =CX A .
(18)
111

TIO.n;CT8BJI.fI.H COOTHomeHHe (18) B ypaBHeHHe (13), nOJIytlaeM
FA = - k (X A - ex A) = - k (1 - c) x A = - TAXA'
(19)
F B = -k(;XB - X B ) = -k( -1)XB = -TBX B .
Kpyron8..a: QaCTOT8 KOJIe6aHHH
"'A = ,--;:;: =  ="'B rr; = .
vtn; V vTn; 
(20)
Perna.H nOJIyqHBrneec.H ypanHeHHe BTOpOM CTeneHH OTHOCHTeJILHO
c, IIpHXO,lJ,HM K TeM .»ce peaYJILT8T8M:
1 - _ m A
C 1 =, C z - -.
m B
(21)
PemeHHe C 1 = 1 AaeT FA = F B = 0 6ea AonOJIHHTeJI:bH:bIX YCJIOBHH Ha
X A Ii X B' PemeHHe C 2 = - m A COOTBeTcTByeT HCTHHHOMY :KOJIe6aTeJIL-
m B
.}10M)' 1];BJI.»ceHHJO.
TpeTHH cnoco6 pemeHH.H - 3TO KOHeqHO, pemeHHe ypaBHeHHH ,D;BH-
.»ceHIIH:
FA =m A X =-k(x A - x B ),
F B =m B x; =-k(x B - X A ).
3aIIHmeM rapMOHHqeCKHe pemeHH.H B BRAe
X A = xAOe tmt; X B = xBOe imt.
(22)
(23)
IIop;cTaHoBKa BhIpa.»ceHHH (23) B ypaBHeHH.H (22) npHBO,D;HT:K CHC-
TeMe JIHHe:HHLIX ypanHeHHH:
(k - m A (t)2)X AO - kx BO = 0,
-kx AO + (k -m B (02)x BO = O. (24)
Cym;ecTByeT H p;pyroe, HeTpHBH8.JlbHOe, pemeHHe: AeTepMHH8HT
MaTpHD;hI H3 :Kosc}>4>HD;HeHTOB npn HeH3BeCTHbIX p;OJImeH 6hITL paBeH
HY JIIO:
I k - m A (02
-k
-k I
=0.
k - mBol
(25)
112

A k
m", 9
I

Xl
B k
mB 9
I
L-..-
X 2
A
mA 9
I

X3
Puc. 74
PemeHHe 3Toro ypsBHeHH.fl COBnaeT c nOJIyqeHHhIMH paHee pe-
aYJIhTaTaMn:
WI = 0; (02
k m A + m B
mAm B
(26)
C TaKHMH me SMIIJIHTy,o;SMn x A H X B'
3. MOJIeKYJIa, KOTOPYlO HymHo HCCJIe,lJ,OBftTh, H HCnOJIb30B8HHhIe
0603H8'1.:IeHH.R npe.u;CT8BJIeHhI aa pllC. 74.
,D;eHcTBYlOrqHe ua aTOMLI CHJ1hI paBHhI COOTBeTCTBeHHO:
FI =-k (x. - %2);
F 2 =-k (X 2  X.) - k (X 2 - X 3 ) = -k (-x. + 2X 2 - x 3 ); (27)
Fa = -k (x 3 - x 2 ).
MbI On.HTh MomeM C'tJHTaTh, 'tITO CMeeHliSl nponOpI1;HOHWIhHbI, no-
CKOJIhRY CMeeHHe eHTpa Mace nOCTOHHHO:
X 2 = C 2 X.;
%3 = C 3 X l'
(28)
r,Ae C 2 Ii C 3 - 3TO Te ROHCT8HTbI KOTOpbIe H8)XO onpep;eJIHTb. COrJIaCHO
YCJIOBHlO aa,nsqn, CILTIY, ,o;e:HCTBYJOIIJ;YJO ua K8.jKJ];bIH aTOM, Momao BhIpa-
3HTb 'Iepe3 cMerqeHne SToro aTOMa. TIO,lJ,CTftBJIHSI (28) B (27), no.nyqaeM
Fl =-k (1- c 2 ) Xl = - rlx l 
F. =-k( - c: +2-"" )x. =-r.x..
F. =-k(l- ::)x . =-r.x."
(29)
KOHCTaHThI c 2 n c 3 Momao onpep;e.n:nTh H3 Taro YCJIOBHH, 'tlTO HTOMM
KOJIe6JlIOTC.R C O,AHHHKOBhIMH KpyroBhIMH qaCTOTaMH:
 = J r 1 (iJ.= J r2 (iJ.= J r. "
m A m B m A
(30)
113

B03BO,lJ,.s: B KBa,z:J;paT H HCnOJI&3YH (29), nonaeN
-+2- C3 1- c 2
1- c 2 = c2 c 2 _ c 3
m A mB m A
(31)
STH YCJIOBHH ,AOJDKHbI BbITIOJIHHT&CSI O,AHOBpeMeHRO, n03TOMY MhZ
nOJIaeM CHCTeMY
I C2
l-c 2 =1--,
c3
l-c 2 1 1
-= [ 2-(1+C 3 )- J -.
m A c2 m B
(32)
(33)
IIepBoe H3 3THX ypaBHeHHA HMeeT ,ABa pa3.J1HYlHhIX pemeHHSI:
1)c 2 :;: 0 H c 3 '* 0,
2) c g = 1 H c 2 :;t; O.
(34)
TIepBoe pemeHHe npH nO,lJ,CTaHOBKe B ypaBHeHHe (33) p;am
= ( 2_ 1+C3 ) .
m A C2 roB
(35)
ECJIH Tenep& nop;CT8BHTh C 2 = 0 B npaByro Yl8CT& yp8BHeHH.H (33), TO
1
ORO He 6Y,AeT HNeTIJ CMhICJI8. t.{T06bI 3Toro He npOH30mJIO -qJIen c ;-
2
,lJ,OJImeH OTCYTCTBOB8TL, T. e
C 3 = -1.
(36)
OTcro,lJ,a nOJIa.eM
X 2 = 0, X 3 = -Xl-
(37)
H3BhIp8meHHH(29)H(37)noJIaeM
Tl  k, ""-  J k .
m A
(3B)
KpyroBhIe qaCTOTbI (01 H (03 p8BHLI, nOCKOJILKY pemeHHe 6hIJIO no.ny-
qeHO Bo06II:{e I'OBOpH, npH aTOM YCJIOBHH. 3,Aec& B03HHKaeT ,nonOJIHH-
TeJILH8JI CJIOmHOCTb, nOCKOJILKY (02oKa3h1B8eTCH Heonpe,D;e.JIeHHo:A - npH
3TOB RORKPeTHO:i KOJIe6aTeJILHOH MO,D;e aTOM 2 He )J;BH>KeTCSI Boo6IQe.
114

BTopoe H3 ,D;BYX pemeHHli (34), T.e. c 3 = 1 H c 2 *" 0, .o;aeT npn nO,lJ;CTa-
ROBKe B ypaBHeHHe (33)
l-c 2 =2 ( 1- ) ,
m A C2 m B
oTKy.o;a ,lJ;JI.H c 2 nOJIyqaeTc.H ypaBHeHHe BTopon CTeneHH:
2 ( 2m A 1) 2mA 0
c 2 + -- c 2 --= .
m B m B
(g9)
(40)
Ero KOpHR
- 1 . - 2m A
C 2 . 1 -  C 2 . 2 - --.
m B
(41)
IIepBoe pemeHHe cooTBeTcTByeT O,lJ,HHaKOBhIM a:MnJIHTy.o;8:M .o;JUI
Bcex Tpex aTONOB, T.e. CJIyqalO, KorAa npymHHLI HeCmaTLI R KOJIOOa-
TeJIbHOrO ABHmeHHSI He npoHCXOAHT. )J;JISI BToporo KOpH.H nOJIaeM
( 2m A )
1;. = -k 1 + m B
= -'iXx,
(42)
H COOTBeTCTBYIOIItaSl KpyroBa.H qaCTOTa
co. =  = k (  + -.!.... ) .
 m B m A
(43)
RaK H BO BTOpOH qaCTH 38.)];a'tlH, pemeHHe Morno 6bITb nonyqeHo Ii C
nOMoID;blO onpe,lJ,eJIHTeJIH M8TpHqbI, COCT8BJIeHHOH n3 K03<p<pnIJ;neHToB
B ypaBHeHHHx ,lJ,BH:meHRH. 3anHmeM HX:
1\ = mAX;' = -k(Xt -X 2 ).
.F2 = "'-B' = -k (-Xt + 2 - Xa),
(44)
 = mAx;'= -k(X 3 -x 2 ).
3anHChIB8.H pemeHH.H B BH,ZJ;e KO:MUJIeKCHOH 3KcnOHeHT:bI
x, = xwe toot, (45)
no.o;CT8BJISleM HX B ypaBHeHHH H, COKpam;8.H Ha 3KcnOHeHTY. nOJIyqaeM
cHcTeMY JIHHe:HHhIX YP8BHeHHH:
(k - m A (02) x lO - kx 20 = 0,
-kx 20 + (2k - m n w 2 ) x 20 - kx 30 = 0,
-kx 20 + (k - m A (02) x 30 = O.
(46)
115

Y CJIOBHe .n;JI.fI cyecTBoBaUHJI UeTpliBHaJILHOrO pemeHH.fI:
k - m 0)2 -k 0
A
-k 2k - m B 0)2 -k =0 (47)
0 -k k - m 0)2
A
KOpHH 3Toro ypasneHH.fI
"" = 0; "', = J k ; m. = k (  +  )
m A m n m A
(48)
COBn8,L{8IOT C 1I0JIyqeHHhIMH paHee pe3YJIbTaTaMH. AMnJIHTYjJ;hI 1I0JIY-
'IaIOTCa npH nOjJ;CTaHOBKe KopHeH B CHCTeMY ypaBHeHHH.
4. MomBo 38MeTHTb, 'IlTO B npHH.fITOH HaMil MOjJ;eJIH ffi3> CO2' 1I03TO-
MY JIOmtIHO npe.zr;nOJIOJKHTL, 'tlT0 66JILW8JI qaCTOT8, T. e. 7,042. 10 13 rn;,
cooTBeTcTByerr ffi s ' a MeHLma.R - 3,998.10 13 rn; - ffi 2 . CHaqa..rra 3aIIH-
meM CB.R3L Mem.n;y qaCTOTOH II KpyrOBOH qaCToTo:fi:
ffi= 21tv,
(49)
OTKY,lJ,a
<02 = 21tv 2 , ffi3 = 21tv a .
(50)
Tor.a;a 1I0JIyqaeM cJIeyrom;lie On;eHKH AJIH k:
k;. :; m A ro == 1670 HIM,
ka = ( mAmn ) ro = 1420 HIM.
2m A + m n
(51)
Pe3YJILTaTbI 60JIee IlJIH MeHee cOrJIacyIOTcH.
3aJ{aa 3
1. MOID;HOCTL H3JIyqaeMaa co Beefi: 1I0BepxBocTH COJIHn;a, paBHa
4 cng . Be.R 3Ta MOntBOCTL paCllpe.n;eJI.fIeTC.H no ec}>epH't{eCKOH no-
BepXHOCTH pa,lJ,HYCOM R (pa.n;HYC Op6HTLI 3eMJIII), II, 3Ba'IHT IIBTeH-
CHBHOCTJ, H3JIeH:H.H Ha 3TOM paCCTO.RHHIi pUBa 4 aT . CnYT-
47tR 2
116

HRK IIOrJIOm;aeT, KaK Kpyr, H IIOrJl0ntaeMaH MOn::tHOCTI» pftBHa
2 4 (JT
1tT.
en 41tR 2 '
a H3JlyqaeT co Bceii (c4»epuQecKoii) nOBepXHOCTH MOHOCTb 41tre (j.
IIpHp8.BHHBa.H H3JlyqaeMyro Ii IIOrJIOm;aeMyro MOntHOCTH, IIOJIyqaeM
Tm = Tc  . T.e. T <u = 289 K = 16 'C.
2. BLrtIHCJlRM, K8.KaR qaCTb ll np IIornOm;aeMoH MOIIl;HOCTH H3JIyqa-
eTCH yqacTKoM CIIeKTpa C TeMnepaTypOH < 1200 K:
1200 K
T)np :;:::-=0,2«1-
6000 K
Tor,na OTRomeHHe MOntHoCTeH
'1"" 3d "I.., 3
f  I f}2dTJ 1lup
e'1 - 1 3
0' = 0 .L...-:;::: _ = 4,1.10- 4 .
Il.. f TJ3d1l 1[4 7t 4
e'1 -1 15 15
o
TeMIIepaTypa CIIYTHHK8 M8.JIa no CpB.BHeHHIO C 1200 K, II09TOMY MbI
MomeM npeHe6peQb TeM, qTO qaCTb U3JIyqeHHSI CIIYTRHKa KaK TBep,[l;O-
ro TeJIa 6y.neT y.u:epm8.Ba. BaJlBRc 38IIHmeTCH B BH,l(e
2 4 2 47tR; (JT
41tra> UD = a1tr... 2 '
47tR
TaKHM 06pa30M, BLI'tlHCJleHHa.H paRee TeMnepaTypa H3MeRHeTC.H B
0'1/4 PB3, T.e.
Ten = (4,1-10-4)1/4 .289 K = 41 K.
3. IIoJIH8..H IIOrJI0ntaeMa.H MontHOCTb p8.BHa
2 47tR aT
anr.,. = 2 = 05 BT I
47tR
'tITO O'1eRL MaJIO 110 cpSBHeHHlO C MOID;HOCThlO P = 1 KBT.
117

B8JI8.HC MOm;HOCTH MomBO' 3anHCaTh B BH.zr;e
p:::;: 41trc aTc' oTKy.zr;a Ten :::;: 274 Ie (11 = 4,38) .
3 aM e q 8 H II e . CTporo roBOpH aTO' O''leHh rpy68S1 OneHRR, T8.R Kllli
330/0 3HeprHH H3J1eHHSI qepnoro Tena C TeMnep8TypO:H 274 It npHxo-
AHTCH Ha AH8n8S0H T> 1200 Ie. A 3HaQHT, CIIYTBHK .zr;OJIJKeH 6hITh 60-
JIee HarpeTbIM, TfT06hI aa TfaCTOTax BHme KpHTIIQeCKO:H H3JI}"I8JIaCh
MOm;HOCTh 1 KBT. Pe3YJIhTHpyrom;ee ypaBHeHHe 3allHChIB8eTCSI B BlI)1;e
'1 TJ3dTJ
( TJ ) 4 l;ct
4,38 = 7t 4 /15 
KOTopO'e MO)KHO pemHTh QHCJIeUHO' nO'.u;6opO'M. Ero pemeHHe -11 = 3,80,
a COOTBeTCTByrom;8JI TeMnepaTypa - 316 Ie.
4. IepacK8 He MO»CeT cym;ecTBoBaTh, nO'CKOJIhKY T8KOH MeX8BII3M
npOTHBOpeqHT BTOpO'My Haqa..rry TepMo,znmaMHKH. OO':LHCBeHHe OCHO-
B8RO BR npHH1J;Hne ,neTaJIhHOrO paBBOBeCHH; aTO 03HaqaeT, QTO .n;JIH cy-
m;eCTBOBaHHH p8BHO'BeCH.H H3JIyqeHHe II nOTJIO'III;eHHe B A8HBO'M HRTep-
B8JIe qRCTOT .n;OJI)KHhI TO''tlHO KOMneHcnpOBaThCH. 5To 0.zr;H8KO, He npo-
THBOpe'tlHT TOMY, 'liTO OTpaJKeHHe H nOTJIOm;eHHe MO'rYT 6bITh
COBepmeHHO pR3BhIMH. ECJIH 6bI KpaCKa cym;ecTBOBRJIa, MOJKHO 6hIJIO'
6hI CO'3.u;aTh B 38MKHYTOH cnCTeMe pR3HOCTh TeMnepaTyp a 3aR'tlHT, H
Be'tlHhIfi: .u;BHraTeJIb.
5. IIoKphlTHe .zr;OJImHO 6hITL np03p8'tIHhIM AJI.H BhICOKHX qRCTOT
(B O'6JI8CTH MRKCHMYM8 nnH 3a M8KCHNYMOM COJIHeqaOrO H3JlyqeHH.H),
HO' OTpamaTL HH3KHe qaCTOTbI.

XXIV Me>KAYHapOAHaR
.H3H'IeCKGII OnHMnHaAG
1993 r. , CWA
3a.o;a'la 1
C T01:IKH 3peHHJI 3JIeKTpOCT8THKH IIOBepXHOCT:& 3eMJIH MomHO pac-
CM8TpHBaT:& KaK xopomHH npOBO,lJ,HHK. DHa HMeeT IIOJIHhIH 3ap.H,u; Qo H
cpe.zr;HIOIO nOBepXHOCTHYJO nJIOTHOCTb 38p.H.zr;8 0 0 .
1. IIpH 6JIftrOIIpH.HTHhIX noro.n;HhIX YCJIOBH.HX Hanp8BJIeHH8.H BHH3
HanpsmeHHOCTb 3JIeKTpeCKoro nOJI.H Eo Ha IIOBepXHOCTH 3eMJIH P8.B-
Ha no MO.zr;YJIIO npH6JIH3HTeJIbHO 150 B/M. Haii.zr;HTe nOBepXHOCTHy1O
nJIOTHOCT:& 3ap.R.zr;a H nOJIHWH 3ap.H,lJ, nOBepXHOCTH 3eMnH.
2. Hanp.HmeHHocT:& 3JIeKTpIfqeCKOrO nOJIJl 3eMJIH yMeH:&maeTC.H C
BbICOTO:H n paBBa npH6JIH3HTeJI:&Ho 100 B/M Ha BbICOTe h = 100 M. BhI-
QHCJIHTe cpe,lJ,HIOIO 06'heMHy1O nJIOTHOCTL 3apa B CJIoe aTMoccl>ephI
BblCOTOH 100 M.
3. IIJIOTHOCTL 3ap.R.ZJ;a KOTOPYJO BLI BhIqHCJIHJIH B n. 2, 06ycnOBJIe-
aa B ,neHcTBHTeJI:&HOCTH H8.JIIPlHeM nO'1lTH pftBHbIX KOJIlPIeCTB nOJIO>KH-
TeJI:&HhIX H OTpHn;aTeJIbHhIX ep;HHHqBO 38pJDKeHHLIX HOHOD, KOHeHT-
paH.H KOTOpLIX COOTBeTCTBeHHO p8BHa n+ H n_. B6JIH3H nOBepXHOCTH
npH 6JIarOnpH.RTHbIX norO,lJ,HLIX YCJlOBH.RX n+  n_  6 .10 8 M- 3 . 8TH HORbI
,nBHmyrC.R no,lJ, .n;e:HCTBHeM BepTHK8JILHOH 3JIeKTplPIecKOH CH.TILI. Hx
CKOpOCTL nponopJJ;HOH8JI:&Ba nanpxmeHHOCTH IIOJIH: v  1,5. 10- 4 E
(v - CKOpOCTL, M{C, E - HanpmKeHHOCTb, B/M).
CKOJILKO BpeMeHH IIoTpe6yeTC.H, QT06LI nO.TIOBHHa IIoBepxBocTHO-
ro 3ap.H.n;a 3eMJIH 6bIJIa HeHTpaJIH30B8Ha BCJIe.v;cTBHe ,nBHmeHIIJI aTMO-
ccl>epHbIx nOBOB, eCJIH HeT .n;pyrHX npOJJ;eCCOB (BanpHMep, MOJIBHH),
nOMep.mHBalOII(1IX 3UaQeHHe 3Toro 3ap.R.ZJ;a?
4. Hanp.H>KeHHocT:& ftTMoccl>epHoro 3JIeKTpHQeCKoro IIOJI.H H CJIe-
.n;OBaTeJIbHO ll.1IOTHOCT:& U o MomHO H3MepHTL C IIOMOIIl;blO CHCTeMbI, no-
K838HBOH Hft pHC. 75, a. IIapa MeT8.J'IJIH'tIeCKHX KB8.Z\P8HTOB, H30JIHpO-
B8.HHbIX OT 3eMJIH, no coe.n;HHeHHhIX O.n;HH c p;pyrHM, pacnOJIOmeHbI no)];
383eMJIeHHbIM Bpam;aIOIII;HMC.H J];HCKOM C P;ByMJl OTBepCTHJlMH B .popMe
ICBa.a;panTOB. (Ha pHC. 75, a paCCTOJIHHe Me.mp;y KBB.ltp8.HTaMH H .zr;nc-
KOM.zr;n.JI aarJI.H,u;HoCTHyBeJIlNeBo.)AB8m)];hI npH Kam,AOM 060poTeH30-
119

-
Eo
Bp a 14 alO -
14UUCSl ouc"
Keaopanmbt
Y CUJLUmeJlb
<0
8.
a
6
PRC. 75
JIHpOBaHHhIe KBs.zr;paHThI nODM81OT B 3JIeKTpH'tJeCKoe I10JIe, a nOTOM
(qepea 1/4 nepHO,lJ,a) I10JIHOCTLIO SKpSHHpyroTC.H OT Hero.
nYCTL T - nepHOp; BpeHHH, a BHyrpeBHRfi: H HapymHhI:H pa.ttHY-
ChI H30JIHpoB8HHhIX KB8.,ZJ;p8HTOB paBHhI r. R r 2 (pRC. 75,6).
nYCTL t = 0 - MOMeHT BpeMeHR, Kor.n;a H30JIHpOB8HHhIe KBMpan-
ThI IIOJIHOCTblO 3aSKpSHHpOB8.HhI. IIOJIyqHTe BLIpWKeHHe .D;JI.H IIOJIHOro
3apH,lJ,a q(t) HHHpOBB.HHOrO a8 BepxHe:ft IIoBepxHocTH H30JIHpOBaH-
HlaIX KB8.,ZJ;paHTOB 38 nepByro 1I0JIOBHHY lIepnop;a BpSeHIi.R ,ltHCKa.
HapHcyihe rpacl>HK SToff 3SBHCHMoCTH.
Scl>cl>eKToM OT 1I0TOK8 aTMoccl>epHhIx HOHOB B 3TOH CHTyan;HH Mom-
BO npeHe6pe1:Ih.
5. CHcTeM8 OIIHCSHH8H B n. 4, CBH3aHa C YCHJIHTeJIeM, BXO,ltHOH
KOHTYP KOToporo 3KBHBMeHTeH COe,lJ,HHeHHLIM nap8JlJIeJIbHO KOH}I;eH-
C8TOpy eMKOCThlO C Ii pe3HcToPY conpOTHBJIeHHeM R (pRC. 76). (BLl
MomeTe npe.n;nOJIOmHTb 'lITO eMKOCTbJO CHCTeMbI RBa,l{p8HTOB MomHO
npeHe6peqb 110 CpaBHeHHIO C C.)
M
':'
R
N
BxoiJ YCUJlUmeJlSl
Pac. 76
120

HaplWYHTe rpacl>HK pa3HOCTII nOTeHIUiSJIOB U Mem,lJ,Y TOllK8MH
M II N K8K cl>YHKl{UH BpeMeHH t B TeqeHHe OJ];Horo 060POT8 AIICKS
cpa3Y rrOCJIe HallaJIa BpaeH:H.H C rrepHO,lJ,OM T  eCJIu: 8) T:;::: Ta « CR;
6} T = Tcs» CR. (IIpe.n;rroJIomHTe, ':ITO C H R HMelOT cl>HKCHpoB8HHLIe
3UaQeHIUI, TOJIJ>KO neplio,nLI T OTJIU'l8IOTCSI B CJIyq8..HX a Ii 6.)
IIoJIyqHTe npH6JIH3IiTeJIJ>Hoe BLIp8meHHe P.JISJ. OTHomeHHH U a IU 6 -
B8n60JIbIDIIX 3HaqeHHH U(t} B CJIyq8HX a II 6.
6. IIycTJ> Eo = 150 B/M, r 1 = 1 CM, r 2 = 7 CM C = 0.01 MKtI>,
R = 20 MOM II nYCTb '1'l8CTOT8 BpaeHH.R ,lJ,HCK8 50 o6/c. 'tleMY paDHO
(npH6JIH3liTeJIJ>Ho) B8n60JIhmee 3H81leHHe U max B TeqeHlle O.ZJ;Horo 060-
poTB B STOM CJIyqae?
3aJJ;8'18 2
IIpeJIoMJUISJ.ch, MOHhIH JIaaepHLIft JIyq Mome'!' BLI3B8Tb nO.HBneHHe
3HaQHTeJIJ>HLIX CHJI, .ZJ;e:HCTBYIOHX HB MaJILIe rrp03paTIHhIe 06'LeKThI.
'tlT06hI y6e,lJ,IiThCH B STOM p8CCMOTpHM MBJIeHhKYIO CTeKJIHHHYIO Tpe-
yrOJIhH}'lO npll3MY C yrnoM IIpH BepmHHe A = 1t - 2a, .D;JIIIHO:H OCHOBa-
BHSl2h II nmpHHoA b (pHC. 77). npll3Ma H3roTOBJIeHa H3 BeD1ecTB8, HMe-
JO:r:qero nOK838Tenh npenOMJIeHIiH n Ii IIJIOTHOCTh p. IIpH3Ma paCIIOJIO-
meHa Ha nYTH JI83epaOro JIyqa,
H,[Q'III;ero ropH30BT8JIhHO BJ];OJIh y
OCH X. (IIpe,nnoJI8raeTc.H, "tITO
npH3M8 He IIOBOpa'IHBaeTCH
T.e. ee BepmHua Bcer,na pacno-
JIOmeH8 H8SCTpeQy B8npaBJIe-
BHIO JIytl8, OCHOB8HHe napaJI-
neJILHO rrJIOCKOCTH YZ a Tpe-
yrOJIhHhIe rpaHIi napaJIJIeJIhHhI
nJIOCKOCTH XY.) IIYCTh RoS«l>-
cl>HHeHT npeJIOMJIeHIUI B03JJ;Y-
xa peeu e,nHHHJJ;e.
IIpeJJ;IIOJIOmHM, qTO ua rpa- PHC. 77
HII npH3NhI H8.HeCeHO aHTHOT-
pa)f(.aIOIqee nOKphITHe II OTpa-
)KeHH.R BeT. IIHTeHCIiBHOCTb JI83epHOro JI}"I8 0,lJ,HH8KOB8. no ero illlipH-
He B H8IIpWmeHBH OCB Z, HO y6hIB8eT JIHHeHHO C paCCTOSlHHeM Y OT OCII
X (pRC. 78) TBR qTO HHTeHCHBHOCTL p;OCTHraeT CBoero MaKCHMB.JILHO-
ro 3uaQeHual o npH Y = 0 II naeT.ZJ;O HynH npH Y = % 4h. (HHTeHcHB-
HOCTh '1'lHCJIeHHo paBH8 MOHOCTH, npHxOA.HD1eHCH H8 eJJ;UHHD;Y nJIO-
8JUI; OBa MomeT 6LITh BhIpameHa, H8rrpHMep, B B8TTax HB KBpaT-
HhIii MeTp.)
121

...
..
.-------: -- r --- a
..
.
.
.
: J_
z
x
y
2h
r
Yo
..
.
.
Puc. 78
1. HannmHTe ypaBHeHUe, H3 KOToporo yron e (pRC. 79) MO)KeT 6hITb
onpe.Q;e.neH Qepe3 BeJIH'IlUHhI a H n B TOM CJIyqae, Korp;a JIa3epHhIA JlytI
n8,l{aeT Ha sepXHlO1O rp8.Hb npR3MbI.
2. BLIpa3HTe npoeKItRH F % R F II CUJIhI, eiiCTBYI<>eH HB npH3My
npH JIB3epHOM 06JIyqeHRR, Qepe31 0 ' e, h, b II Yo (paccTomme OT OCR X),
r.Q;e I Yo I "3h (CM. pRC. 78). IIocTpoHTe rpacl>RKH 3aBRCRMOCTH BepTH-
KBJIbHOit H ropR30HTaJIhHo:A npoeKHit CRJIL! OT BepTRKBJILHOro CMe-
eHH.R Yo OCR nyqKa CBeTB OT BepmRHbI npU3:MLl.
3. llYCTb mnpRHB JIa3epHOro JIyqa B HanpaBJIeHHH OCR Z paSHa
1 MM, a B HanpOJIeHHR OCH Y - 80 MKM. IIpR3Ma RMeeT T8.KRe napa-
MeTphI: a = 30°, h = 10 MKM, n;: 15, b = 1 14M H P = 2,5 r/cM 3 .
KaKoA MOJ::q;HOCThlO ,nOJI>KHO 06JI8,l{aTh JI83epHOe R3Jlyq:eHl:le, QTo6hI
yp;ep>KHBaTh llpH3Ny B p8..BHoBecHH B none CRJIhI T.HI'OTeHHH (8 Hanp8.B-
neHHH OCR Y), eCJIR BepWHHa npH3MhI p8CnOJIOmeH8 HH>Ke OCR n83ep-
h
Horo JIyq:a npH Yo ="2:; -5 MKM?
4. llYCTh STOT sKcnepHMeHT npoBoJJ;HT-
CS:( B COCTOJrHHH HeBeCOMOCTH C TeMB me
npR3MoH II JI83epHLIM JI}"lOM, HMeJOMH
TBKHe)Ke napaMeTpLI, qTO H B n. 3, HO C 10 =
"= 10 8 BT 1M 2 . KaREN 6YJJ;eT nepHo,q; B03HBRa-
IOHX ROJIe6amd:t, eCJIH npH3M8 cMeJ::q;eua II
h
OTnym;eHa Ha pacCTOS:(Hlle y = 20 OT OCR JIa-
3epHoro JIytla?
Puc. 79
122

38JJ;8'18 3


Y CKOpSUOI:Il;ee HanpSI*eHHe U 0 C03,lJ,aeT O.v;HOpO
HLIA n8pamIe.JIhHbIH

OK 9JIeKTpoHOB C BhICOKOH 9Heprneit. 3JIeKTpOBhI npoxo.D;SIT B6JIH-
3U ,lI;JIHHHOH, TOHKOii, nOJIOmHTeJIhHO 3apSl:meHBoH Me,lJ,BOA npoBOJIO-
KU, pacnOJIOmeUHOH no,ll; npHMhIM yrJIOM K nepBoaa'tJaJILHOMY HanpaB-
JIeHHIO nyqKa, KftK nOK838HO aa pne. 80 (b - paCCTO.HHne OT npoBOJIO-
.KH
 HS KOTOpoM npome.n 61d 3JIeKTpoH, eCJIH 6bI npoBOJIOKa He 6hIJI8
3apHmeHa). IIocJIe 3Toro 3JIeKTpOH non8):1;aeT aft 3KpaB (nnocKocTh Ha-
6JIIO,neHH.H)
 pacnOJIO)KeHHhIH Ha paCCTO.HHHH L » b 38 npOBOJIOKofi.
nepBOHaTlaJILHO n
OK RMeeT pa3MepLI ::I::b max OTHOCHTeJIhHO OCR npo-
BOJIOKH. K8.K mHpHHY n
Ka, TaR Ii AJIIIHY npOBOJIOKH MomBO C"In-
T8Th 6eCKOHe'tJHhIMH B aanpasJIeHHH, nepneHJ];H.KYJUlpBOM nJIOCKOCTH
pHCYUKa. 'tJHCJIOBLIe 3HaQeHHSI napSMeTpOB TaKOBLI: p8,L{UYC npOBOJIO-
KH TO:::;; 10- 6 M; b max = 10- 4 M; JlHHeHH8JI nJIOTHOCTh 3ap.H.D;a npOBOJIOKH
q,= 4,4 .10- 11 KJI/M; YCKOp.HIOI:II;ee Hanp.HmeHHe U o = 2.10 4 B; paccToH-
HHe OT npoBOJIOKH 11.0 3KpSH8 L = O
3 M.
1. BhITlHCJIHTe aanp.H)KeHHOCTL E 9JIeKTplPleCKOrO nOJIS, C03,lJ,asa-
eMoro npoBOJIOKOii. HapHCyHTe rpScl>HK 38BHCHMOCTH Hanp.HmeHHOC-
TH E aT paCCTORHHJI T 11.0 OCII npOBOJIOKH.
2.lIcnOJIh3ya3sKoHLI KJIacc:n'tlecKOH cl>H3HKH, BhI'IHCJIHTe yrJIOBOe
OTKJIOHeUHe 9JIeKTpOHa OT npaNOJIHHeHUOH TpaeKToplIH (yrOJI e Me)K-
1l.Y nepBOHa'l8JIbHOH CKOpocThIO 9JIeKTp0H8 II ero CKOpOCTblO, KOr)];8 OB
na,n;aeT Ha 9Kpau) .D;JISI TaRIIX 3HSQeHHA napaMeTpOB b, npH KOTOpl:dX
9JIeKTpoH He CTWIKHBaeTC.H C npoBonoKo:A.
3. BhItIHCJIHTe II H306pa3HTe rpat})H"IeCKH 38BHCHMOCTb HHTeHCHB-
HOCTH paCCeJlHHOrO n
Ka 3JIeKTpOHOB OT Koop
HHaT B UJIOCKOCTH 3K-
paB8 (
HarpaMMa paCCemIHH), KOTOpyro npe,nCK83LIBaeT KJI8CCHtIeC:KM
cl>H3HKa.
4. KBaHTOB8H 4>H3HKa npe
CKa3bIBaeT c
eCTBeHHoe OTJIHQHe B
pscnpe,AeneBHH HHTeHCHBHOCTH
 KOTopoe ,naeT KJIaCC}[tIeCK8S1 clJH3HKa.
H306pa3HTe rpa4>WlecKH KB8HTOBLI:ft pe3YJILT8T H cll.eJIaiiTe KOJIlf'qe

CTBeHHLIe OD;eHKH.


CD

c -r:- 8Kpan



 _ _j
:-: ____n!

O


 ___ _ _ _ __ __


 ---- I L
t::


;:;



PRe. 80


123




PeWeHH!A


3a,zJ;8 '18 1


1. TaK R8K BeRTOp Eo Hanp8.BJIeu BeprHK8JIhHO BHH3 (pRC. 81, a)

TO
 no TeopeMe raycc8, a = -EoE{):
(J = -8,85 . 1 0
12 (I) 1M · 150 B/M
 -1,3. 10- 9 KJI/M 2 ;
Q = 41(R2 a = 41(. (6,4.10 6 )2 M 2 . (-1,3.10- 9 ) KJI/M 2 = -6,7.10 5 KJI.


2. PaccMoTpHM IUUIHH,lI;P C nJIOm;a.n;hIO nonepeTIHoro ce'tJeHHSI S II oc-
HOB8HHSlMIi
 pacnOJIOJKeHHLIMH us BhICOTRX 0 Ii h =100 M (pRC. 81,6).
no TeopeMe raycca,


Eo (O)S - E(h)S = q"" = Pc phs ;
Eo Eo
= EO [E(O) - E(h)] = 8,85.10- 12 .pIN' 50B/M
 4 4 .10-12KJx /
.

 h 1
 '


3. ECJIU KOHneHTpa]J;IUI 38pSl,lJ;OB B npOBo,nHHRe paBH8 n, 3H8qeHHe
3ap.a.zr;a q Ii 3apXJJ;LI nepe,lIBHraJOTCH co CKOpOCThIO V, TO nJIOTHOCTh
TORa j onpeJJ;eJISleTCH BLIpameHHeM j = nqv.
3,lJ,eCb HMelOTCH R8K nOJIOmHTeJIhHLIe, TaK H OTpllI!;8TeJIhHLIe 3apH-
,lIlt! (:t: e). IIoHHTHO
 '1TO B 3JIeKTpINecKOM nOJIe, Hanp.RmeHHOCTL KOTO-
poro HanpaBJIea8 DUH3, nOJIOJKHTeJIhHhIe 3apH.ZJ;hI JJ;BHmYTCH BHH3, a OT-
pHn;8TeJIbHLIe - BBepx. TaKHM o6paaoM, B OUHC8HHOfi: cHTya]J;HH TOJIb-
RO nOJIOmHTeJILHLIe 3apH)J;bI OTBeTCTBeHHbI 38 HeATp8JIH3an;HIO
nOBepXHOCTHoro 3ap.H,Zt8 3eMJIH. no STOH npH'lHHe npHMeM HaUpaBJIe-


E(h)


-I:


h=OM


h = 100 M


a


PRC. 81


124




Hue BHH3 3R IIOJIOmHTeJIbHoe. Tor.n;a IIJIOTHOCTb TOKR (n e.n;l:IHUn;aX CM)
6y,neT paBHa
j = n+ev
 (6 . 108). (1
6 .10- 19 ). (1
5 .10- 4 E) = 1,44 .10- 14 E. (1)


3,lJ,eCh j - CKOpOCTb H3MeHeHH.R (da / dt) IIoBepxHocTHOH IIJlOTHOCTH 3a-


cr
PMR cr
 a E =
 (eCJIH IIOJIOml:lTeJIhHbIM BbI6paHo HanpaBJIeHHe BHH3).
TI03TOMY BbIpameHl:Ie (1) MomBO nepenllcaTh (B e.n;HHHn;ax Gll) B BH.n;e
da _ 1 44 10 -14 cr - 1,44.10- 14 - 163 10 -3 1
(it - -
.
 - - 8
85 .10- 12 cr
 -,. er
 - 600 0'.
3TO BbIpameHHe IIMeeT TOT me BHA, qTO H ypaBHeHHe pa.n;lloaKTHBHoro
pacna.n;a. Ero pemeHHe - sKCnOHeHIJ,HaJIbHhIH cna.n; er CO BpeMeHcM:


a( t) = er o e- t / t , r.n;e "[::=: 600 c.


a
IIOJIO:mllB a( t) = -f
 nOJIyqaeM t = "[ In2 = 0
693. 600 C = 415 C ::=:

 7 MllH. BOJIee npocTYIO o
eHKY MomHO IIOJIyqllTb, npeAIIOJIo.>KHB
 -qTO
nJIOTHOCTb TORa OCTaeTCSI nOCTO.HHHOH II paBHoH cBoeMY Ha-qaJILHOMY
3Ha-qeHHJO io:
io = 1
44. 10- 14 Eo = 1,44. 10- 14 . 150 A/M2
 2,15.10- 12 A/M 2 .
IIo.n;cT8BJIHH I a o 1= 1,3.10- 9 KJI/M 2 II3 II. 1
 nOJIyqaeM
1 0 01 = io t, oTKy.n;a t = (0,65.10- 9 ) / (2
5 .10- 12 ) c
 300 C
 5 MHH.
2


4. ECJIH npHH.HTh 38 t = 0 MOMeHT, Kor.n;a U30JIHpOBaHHLIe KB
paH-
ThI nOJIHOCThlO 3aSI<pRHHpOBRHhI. MLI IIOJIyqaeM CJIe.n;YJOIIJ;He COOTHO-
meHH.H:


IIPII 0
 t <
 q(t):=: -2n (r; - 712) EoEo
 ;


npH : < t < i q(t) = -n(.... - Ij2) EoE1> (1-
 ).


COOTBeTCTByIOIll,He n3MeHeHHSl npOlIcxo,n;SlT KR:m,lJ,hIe CJIe.n;YIOID;ne IIOJI-
nepHo,n;a (.n;Be qCTBepTH nep:uo,n;a) (pHC. 82).
MaKCIIMaJIhHblH (oTpHIJ;aTeJILH'hIH) 3ap.H,lJ, paBeH


- it ( 2 2 ) E
qrnax - -"2 7 2 -r 1 Eo o.


5. STOT Bonpoc MomeT 6bITL paCCMOTpeH 6e3 p;eTRJIbHOrO 8HaJIH38
n;enH. Hy:mHO TOJILKO 3aMeTHTb, -qTO TOK 'lepea YCHJIIITeJIb paa,lJ,eJI.ReT-


125




q
T/4 T/2
0 t
qmax


C.R na TOR 3apH,lJ;KH KOH,lI;eHCaTOpa ( - C

 )
H TOK Qepe3 pe3HCTOp (
 ). Tor.zr;a, B 3RBH-
CHMOCTH OT COOTHOmeHHH 38px.n;a, nOTep.RH-
HOrO 3a CqeT yre"lKH 38 "leTBepTL nepHO.n;a,
H 3apH,lI;a CIf, B03MomHLI ,lI;Ba KpaHHHX CJIY-
"la.R:


PHC. 82


uT
a) ECIIH Cu» R 4' T.e. T = T.« CR



TO 38 BpeMSI T Qepe3 pe3HCTOp npOXO.,[J;HT OqeHb M8JILIfi: 3apHJ];. B 3TOM
4


CJIyqae, KOr,lJ,S H30JIHpoBaHHDIe KBMpaHThI sapameHhI OTpHnSTeJILHO
38 CqeT HHJJ;yKD;HH, as KOH.l(eHC8TOpe HMeeTCSI nO'lTH paBHblH no MO,ll;y-
JIJO nOJIOmHTeJIbHhI:H 3ap.R,ZJ;. Tor.n;a u(t) B03pacT8eT CO BpeNeHeM nOqTH
JrnHeHno OT t = O.n;o t = I.... H nOqTH JIHUeHHO y6LIB8eT npH t, H3MeHHIO-
4
T T 0 Iqmaxl
:r:q;eMCSIOT 4" /1.0"2 (pRC. 8u). BaTOMCJIyqae U max = U o =:I-c (3H8Tie-
HHe qmax nOJIyqeHo B ll. 4).
II p H N e q a H He. B ROHD;e KOUD;OB (eCJIH BeJIHTiHB8 CR He 6ecKo-
ReqUa) rpa4»HK u(t) CTB.HeT .nHJIoih, JIeaca.m.;efi: Ne)K)l;Y 3Ha"leHHSIMH
BanpSmeHH.R, COOTBeTCTBeHHO p8BHbIMH _ Iqrnax I H + Jqrnax I . IIoCTaHOB-
2C 2C
Eft 38)J;aTiH n03BOJI.ReT H36e:m8TL 3Toro OCJIO>KHeHHH, TaK RaK yrO"lHX-
eTC.R, "lTO U H3MepHeTC.fI CPS3Y nOCJIe Haq8JI8 BpaIqeHHSI.
6) ECJIH me T = To» CR, T.e. CR« To, TO 60JILma.R qaCTL 3apH,lJ;8
6hICTpO CTeKaeT Qepe3 pe3HCTOp. Kor.zr;a 3HaqeHHe 3apH.l(S q YBeJIH'lHB8-


u


U rnax


o


T/2


T


t


PHC. 83


126




o


po-
T/2 T t
-----


U
U b


-Ub


PUC. 84


eTCR
 Qepe3 pe3HCTOp Te1JeT nOCTOSIHHI:J:H nOJIOJRUTeJIbHhIH TOR, 8 Kor-
Aft 3SpHP; q yNeHbmSeTC.H - p8.BHhlH eMY no MO,lJ,YJIIO OTpHIJ;BTeJILHhIH
TOK. CHJIR TOKR npH6JIH3HTeJIhHO paBHB I qmax 4 ' . PeaYJIhTHPYJOII:I;ee aa-
Tf,/
npHmeHHe aB pe3HCTOpe npU6JIH3HTeJIbHO nOCTOHHHO B TeqeUHe KSac-
,D;OH qeTBepTH nepHo,n8 H JIH60 nOJIOHCHTeJILHO, JIu60 OTpHItSTeJILHO
(puc. 84). B STOM CJI}"IBe
1 4 Qrnui R -
Urnax==u,,1II:::
 .


U. T"
06'bep;HHH.H peaYJIbT8ThI, nOJIyqaeM U 6
 4CR .
6. }fMeeMCR== lO-8<P.2.10 7 0M:;;:O,2 CH T==0
02c. IT03TOMyCR==
== lOT H YP;OBJleTBOpHeTCH KpHTepnH CR » T. 3HaTlUT, MhI MomeM HC-
nOJIb30BaTb pemeHHe 5
 8.


Torp;a Smu. == ; (7 2 - 12) = 75 CM 2 = 7
5 . 10- 3 N 2

Eo == 150 B/M:::) <r = toEo III::: -1,33.10- 9 KJI/M 2 (KRR H B n. 1),
qmax = -1,33 . 10- 9 Kn/M2 . 7,5.10- 3 M 2 ::t: -1,0. 10- 11 KJI
_ Iqmaxl _ 1,0 .10- 11 KJI _ -3 _
H,3HSQHT, Umu. - -C - 1,0 .IO-8
 -10 B-1 MB.


38.n;a'la 2
1. 8To I'eOMeTpH'tlecK8.H aa.n;aqa 118. 38KOH CHeJIJIHyca (pHC. 85).
YroJI nap;eHH.R a 3 :;;: a, nOCKOJlbKY a 1 :;;: a H a 1 + a 2 == a 2 + a 3 ;;: 90° .
YroJI 13 aaA:.zteN H3 38.KOH8. CaeJIJIHYca: sin a ;;: nsin p . YroJI na,n;eHHSI aB


127





I a2 .-
X I al
I
I
I
,
TL
as
i
PHC. 85
OCHOB8BHe npH3Ml>l paBeII  - [ " - n - (  -I! ) ] = n -I!. CnI\AOBaTeJIhHO,
sin6= n sin (n -I!)... -. 6 = arcsi n [ nsm( n = arcsin m;:n ) l
2. CUJI8, ,JJ;eiicTBYIOaSl Ha npH3MY, paDRa no MO,AYJIIO H npOTHBO-
nOJIO>KH8 no Hanp8BJIeHHIO CKOpOCTH H3MeHeHHSl HMUYJIbCa JI83epHOro
H3JIyqeHHH, UpOXO,lJ,Hm;erO 'tlepe3 Hee.
lIYCTh Ha BepxHIOIO nOJIOBHHY npH3MhI napaJIJIeJIbHO OCH X nap;aeT
B ceKYH,l(Y r B cl>OTOHOB JI83epaoro H3JIrIeHHSI. ECJIH 3HeprHSI cl>oToaa
p8BHa E, TO ero HMUYJIbC paBeR Pi = E i II HMIIYJIbC cl>oToaR, BhIXO)J;H-
c
IQero H3 npH3MJd no,lJ; yrJIOM e K OCR X, 6Y,IJ;eT OTJIIPI8T&CH 0'1' HMUYJIhCa
nap;alOero cl>OTOH8 H8 BeJIH'qHHY
s:.- E ( e 1) -:" E. e -:
up =: - cos - ,,- -Sin '1,
c c
r.ne i H J - e}J;HHH'tlHhIe BeKTophl (CM. pRC. 85).
CKOpOCTL H3MeHeHHH HMUYJIbCa T8KHX cl>OTOHOB paBRa
- r. E [ - - J
11. = rBop =...!.- (cose -1) i-sinS j .
c
ITpoH3Be,lJ,eHHe rBE npe,ncT8BJIBeor C060H MOID;HOCTb P B H3JIyqeHHH
nap;alOero ua BepXHJOlO 'tl8CTL UpH3MhI, CJIe.nOB8TeJILHO, CHJI80T,lJ,a-
qn, 06YCJIOBJIeHH8a npeJIOMJIeBHeM CBeTa B BepxHefi 'tl8CTH npH3MbI,
puua
- E. [ ) - - J
Ii;. =....!... (cosO - 1 i-sine j .
c
128

AH8JIOrH'tJHO nOJIyqae.M CHJIY, ,lteHCTByrorq,yro Ba HHmHlO1O nOJIOBH-
BY npH3MhI:
Fa = PH [(I-cos8)i +sinSJ].
c
H3 3THX ,ltBYX BLIpBmeHHA HaH.n;eM nOJIHYIO CIDIY  ,lJ,eitcTBYIOIII;YJO
BB npH3MY:
F = ![(P H + P.)(I- coss}]i + ![(p. - Pa)sine]J.
C C
YrOJlS MOJKeT 6hITb BhIpameH Qepe3 a (CM. n. I). QT06LI HaHTH Be-
JIHqHHhI PHil P R paCCqHTaeM Cpe,lJ,HIOIO HHTeHCHBHOCTb I B H I R H3JIY-
qeHHSI, nMBIOIII;ero Ha K8)K,l{yro H3 nOJIOBHHOK npH3MbI, H YMHOJKHM
HB hb - nJIOIIl;B,lJ,b npoeKItHH nOJIOBHHOK npH3MhI aa nJIOCKOCTb, nep-
neHKYJIHPJIYIO J1B3epHOMY JIY'lY. nOCKOJIhKYHHTeHcHBHoCTbI(y) ecTb
JIHHeitH8.H «l>YHKHSI OT y, JIerno HaRTH cpe,AH101O HHTeHCHBHOCTb. Co-
rnaCHo YCJIOBHIO
I(Y) = 10(1- :h ) IIpII 0 < Y < -4h,
l(y) = 10(1+ :h ) IIpH -4h < Y < o.
lIpeJ];noJlo2KHM Tenepb, QTO npH3M8 pacnOJIOJKeH8 aa paCCTOJlHHH
Yo OT OCB X (Yo> 0). Bo3MomHbI .D;Ba pa3JIHtlHhIX CJIaJl.
a) ECJ1H h <: Yo <: 3h, TO npH3M8 pBCnOJIOmeHa enHKOM B BepXHeH
qaCTB CBeToBoro nKa.
I
10
o
Yo -h
Yo
Yo +h
y
PIIc. 86
5-379
129

KaK IIOKa3:&IBaeT pRC. 86, Cpe.n;H.R,R HHTeHCHBUOCT:b paSHa HHTeH-
h
CHBUOCTH Hft ypOBBe :qeHTp8 K8JK,ltOH CTOpOHhI, T.e. npn Yo +"2 .D;JIH
BepXHeii qaCTH H Yo -  JI,JIH HH)KHeit. OrclO,lta nOJl}"l8eM
[ h ]
Yo+-
- 2 7 Yo
I. = I. 1- = 1.(8- 410).
[ h ]
- Yo -2 9 Yo
I. =1. 1- =10(8- 4h}
H3 3Toro CJIeAYeT, 'tlTO
F. = 2h:l o (1-  ) (1- cose
hb10 . e
1;, = -""""4C" SIn .
6) ECJIH 0 < Yo < h, TO qaCTL BHmUe:H nOJIOBHHKH npH3MLI pacno-
Jlomeua B HH)KHeii: qaCTH CBeTOBOro nyqK8, K8K nOK8a8HO Hft pRC. 87.
TorAa nJIOIIl;8,ZJ;:b TOft qaCTH npH3MLI, KOTOp8JI JIemHT NemAY 0 H Yo' OT-
Yo
BOCHTCSI K o6m;eA nJIOm;8,ZJ;H HH)KHeH 1IaCTH K8K h' 8 CpeP;HSISI HUTeH-
CHBHOCTL na..D;alOm;ero Uft Hee
H3JI}"ieHH.H pUBa
I.. = I( ,;) = /.(1- }
OrHomeBHe fiJIOI:Q;a..D;R 118-
CTH, JIemam;eft Me)K)l;Y 0 H Yo -
h, K o6II1;eii: nJIOm;a..D;H paDUO
I
10
Yo-h 0 Yo
Yo+h
1 - Yo
h ' a cooTBeTCTByJOm;8JI
cpeJ];H.fUI HHTeHCHDHOCTL PU-
Y Uft
1.. = l( hYO ) = lo(i+ : }
PHc. 87
130

06'Le,nHHSISI 3TH peaYJIhTSTLI, nOJIytlSeM
p. == hb Yo I + hb ( 1 - Yo ) 1 = hbZ ( ! + Yo - y ) .
· h" h  0 8 4h 4h 2
Cpe.ZJ;RBSI HHTenCIIBHOCTL n8,D;slOII(ero HS BepxHJOJO 'l{aCTb 1I3JIyq:e-
BUB 38.BHCHT OT Yo TaR e, K8K II B nepBOM CJIyqae. l103TOMY P B =
= MI. (i - :;, ) . K8K H paBMlle. OIi'heJIHHHJI peaym.TRTLI. Uo.J1y'I8eJII
_ ( 7 Y )
 +  - hb10 - +  
4 4h
P. - P. == -hbL o Yo ( 1- Yo ) .
· H 2h 2h
OrCloa cJIe,nyer, 1lTO
F. == hb10 ( 1. - y )( 1- cose )
r C 4 4h 2 '
F. == - hb10  ( I-1&. ) Sine
11 c 2h 2h .
lIOCKOJILKY pacnpep;eJIeHHe HHTeBCHBHOCTU CHMMeTplI'IHO OTHOCH-
TeJIbHO OCH JIaaepHOrO JIyqs, pemeBHS B CJIyqse Yo < 0 6y.ZJ;yr 3epKaJIL-
HLIM oTo6pWKeHHeM pemeHHA npH Yo> O. rpacl>HKH F" H F1IK8K cl>YHK-
H nOKa38.HhI Ha pHC. 88, a, 6.
3. YpB.BHeHHe H rpacl>uK FIJ (Yo) IIOK8aLIB81OT, 'l{TO ,nJIH Toro, 'l{To6LI
FII 6hI.JIB npoTHBOnOJIOmn8 no HanpSBJIeHHIO CIIJIe TS)KecTH, Yo.ZJ;OJIeH
6HTb MeHbme HYJIS. Tor,na, '1To6LI BaHTH CHJ1Y, KOMneHCHpyIDyro
CHJlY T.HmecTH, HymHo CHaQaJI8 HaHTH Macey npH3MLI, aaTeM npHpu-
HHTb BePTHK8.JIbHYIO COCTaBJIHIOyro CHJIhI, AeHCTBYIOeA co CTOpoHhI
H3.rryq:eHHS K CRJIe T.HmecTH npH3MhI H .D;8.JIee HaHTH HHTeHCHBHOCTL 10
npH ,lJ,8HHLIX napSMeTpax. HCnOJIb3yeM 3TH pe3YJIbT8TLl, '1To6bI HaATH
IIOJIHyID MOOCTb H3JIyqeHHB. ,lJ;JIH SToro RabeM cpe.n;Hee 3R8-qeBue
I H8 Bceii: nJIO8,lJ;H nOnepeqBOrO CeqeHIUI CBeTOBOro nyqKS.
'tlTo6bI BaHTH Maccy npH3NLI, uaft:.ZJ;eM CnSq8JI8 ee 06'LeM (OB p8.BeB
h 2 b . tga ),Tor,na
m == }a (10-3t. 0,1 CM 3 .2,5 r/cM s = 1,44.10- 7 r == 1,44.10- 10 Kr,
mg = 1,42 -10- 9 H.
131

Fr
7/4 _______
3/2 . - - - - - -
I 1
I
1/2- ----r-
-3h -2h -h 0 h 2h 3h Yo
a
F,I
-3h -2h -h :
I
I
I
I h
Yo = -2"
6
1 10 bh . e
"4 ----c- SIn
Yo
PRC. 88
B pemeHHH TI. 2 npeJJ;TIOJI&rB.JIOCh CNem;eHHe B TIOJIOmHTeJIhHOM HanpaB-
JIeHRH OCR Y, HO 3Ma'I8 CHMMeTpH'IH8, n09TOMy MhI NomeN HCnOJIh30-
B8.Th STO pemeHHe. HaM uY)KHO TaKoe 3H8qeHHe 1 0 , :KOTOpOe YJJ;OBJIeT-
Bopsrno 61d YCJIOBHIO
lohb  ( 1 - Yo ) Sine ;::: mg ;::: 1,42 .10- 9 H
C 2h 2h '
rJJ;e e = 15,9°, Yo = i. h;::: 1.10-5 M. b = 10- 3 M;
l _ 3. 10 8 . 1,42 .10-9 BT . M -2 _ 8 3 1 1\8 B -2
0- - , . U 1l'.:M.
105 .10- 3 .0 274- 
 16
132

- 1
MOIIJ;HOCT& P paBHR npoIl3BeeHH1O cpe.n;HeH HHTenCHBHOCTH I = .JL
2
HB nJIOIIJ;M& nOnepeqHOrO ceG:eHHSI CBeTOBoro nyqKB T.e.
P =!. 83 .10 8 .10- 3 .80 .10- 6 BT = 332BT.
2
4. IIpH CMem;eHHH Ha 2 OTHomeHHe  == 0,05« 1, n03TOMY
B xopomeM npH6JIHmeHHII MomHO CTlHTaTL 'lTO BepTHKftJIbHaH KOMno-
lob sin e ..
HeHTa CHJIhI FII = 2c y. STO eCTh ypaBHeHHe KOJIe6aHHH rapMo-
HHqeCKOrO OCD;HJIJIHTOpa C KpyroBOH G:acToToH
10 sinS
2cph 2 tga .
IIO.n;cT8BJIHeM B 9TO BbIpa:m:enHe qnCJIOBhIe ,lJ,aHHbIe:
21t 2 - 3.10 8 . 25 .10 3 .1011 .1/ J3
T = - = 21t C = 112 MC.
ro 10 8 .0,274
3ap;aQa 3
1. 3JIeKTpHG:ecKoe nOJle pacnpe,lJ,eJIeHO cHMMeTpH'lHO, H JIHHHH Ha-
npameHHocTH HanpaBJIeBhI pa,a;H8JIhHO OT npOBOJIOKH, a 3H8G:eHHe Ha-
npJlmeHHOCTH aaBHCHT TOJIhKO OT p8.AHyca r (B D;HJIHnApHqecKHX KO-
OP.AHHRTax) (pHC. 89). lloMecTHM Bo06pa.maeMbIH D;IIJIIIHJJ;P BOKpyr npo-
BOJIOKH H aanHmeM TeopeMY raycca:
27trE(r) =!k.
Eo
(.n;JIH D;HJIHH,ltpa pa,u;nYCOM E ( r )
r;>- ro). IIOOTOMY B e
r
CII E(r) = ql
2n:r Eo
O791
npH r ;>-ro. llpM r(:; roHarrpH-
meHHOCTb 3JIeKTpnqecKoro
nOJIH paBHa HynlO, nOCKOJIh-
Ky BnYTpH npOBO,ltHHK8
(MeAnOH npOBOJIOKH) nOJIe
OTCYTCTBYeor .
o
r
r
PUC. 89
133

bJ
----  _.
.
I
PRe. 90
2. lIo ycJIOBHIO 3a;:J;a'IJH yrJIOBOe OTKJIOHeHHe MaJIO. OeHHM HCEO.
MJdA yroJI OTKJIOueUHSI e H3 cne.ztY]OII:tero COOTBomeUHSI: e  Ip.ll , r)];e
mv o
IdP.l.1 - H3MeueUHe HMuYJIhCa B HanpBBJIeHlIH, nepneH)];HKYJI.HpHOM
H8."IJ8.JIhHOA CKOpOCTH, mvo - H8'IJaJIhHLI:A HMUYJI&C. lIepBOHa"IJ8JlbHaH
On;eUK8 nepneuAHKYJISlpHO:n COCT8BJUIJOeA HMUYJIbC8 MomeT 6bITh
. c,u;eJIaH8 CJIe.n;yroHM 06p830M.
lIepneH,AHJ(YJI.RpH8.SI COCTaBJI.RIOJ:Q;aJI CHJILI (TaM, rJ];e ORB. 3H8."IHTeJIh-
eq,
ua) HMeeT nop.RJJ;OK BeJI}fqHHbI 27tE o b .
TOJIhKO B TeqeHHe Toro npoMe:mYTKa BpeMeHH, 38 KOTOpLlii :':tJIeKT-
paB npoJIeT8e-r paCCTOHHHe uopa 2b, T.e. B Te'IJeHHe npoMemYTKa
BpeMeHH UOp.H,ll;K8 2b lu o , Ha SJIeKTpou ,lJ,eACTSYeor 3HSqHTeJIhH8S1 nep-
neHJJ;HKYJIllpH8.SI COCT8BJUIIOJ:Q;aJI CHJIhI.
OeHHM nepneH,ZI;UKYJI.HPHYJO KOMIIOHeHTY HMnYJIhC8 KBI< npoH3-
Be.n;eUHe MO.n;YJIH CHJILI Ha npOMe3(yroK, B TeqeHHe KOTOpOro 3Ta CHJIa
.n;eACTBYeT:
I A - I - 2beq, _ eq,
L}.P..l - -,
27tE o bv o 7tE o V o
Tor,u;a
e = eq, 2 _ q, 3,96.10- 5 p
7t6 o mV o 2U () XEo
1
(3.n;ech HCnOJIh30BaH 38KOB COXpaHeHHSI suepnm 2mv = eU o ).
3aMeTHM, "ITO OTKJlOueBHe O'IJeUh MaJIO H He S8BHCHT OT nap8MeT-
pa b. IIoCKOJIhKY CHJIa, ,lJ,e:iicTBYJOaa ua SJIeltTpoH co CTOpOUhI nOJIO-
3aITeJIhHO aSpJl)KeUHo:n npoBOJIOKH, - 3TO CHJIa npHTSlmeUH.R, TpaeK-
TOpH.R 6YAeT HCKpHBJleHB no uanpaBJIeHllIO It npoBOJIOKe, XOTH II co-
BCeN: BeMBoro (pRC. 90).
MO;)KHO CAeJIaTh ooJ1ee aKKypSTHYJO On;eHKY, ec.nR BBeCTB SJIeMeH-
Tspaoe HHTerpHpoB8BHe AJISI H8Xom,u;eHHSI 11\ jJ..11 . ,D;JI.R 3Toro 38MeHHM
134

V o !J:.t
L
PHC. 91
(B nepBOM npn6JIHac.eHHu) pe8.JIbHyro TpaeRTopHIO np.HMOH JIHHHeH, KO-
TOpa.H npoxo,nHT H8 paCCTOSlHHH bOT npOBOJIOKH (K8K Ha pHC. 91).
Torp;a
IFLI = ..!!!Lcos cp ,
27tE o r
r!J.cp
!J:.t = , TO
V o eoscp
I FJ.. I !J:.t;:;: COSQ> r!J.cp = -!!!!L6.fP.
21t£ o r Vo coscp 27tEovo
'It 'It
CRJI8.,D;hISaeM Bee npHp8.IIJ.;eHHSl Ha HHTepB8.JIe OT - 2 P.O "2 H nony-
vo!J:.teosQ> ;:;: r.
T8K KaK
qaeN l.6.p.L1 = . ST8 oeHKa OTJIH'IaeTCSI OT nepBoA Deem JIHmh Ha
2E o V o
'It
l'tIHO)KHTeJIb 2 .
OKOHQaTeJlLHO nOJlyqaeM e = eq, 2 =  ;:;: 6,21.10- 5 pa,zJ;.
2Eomvo 4£oU o
3. Hu60JThmee HCKpRBJIeHHe TPaeKTOpHH npOHCXO,ltHT aa paCCTO-
.fIBH.HX OT npoBOJIORH IIOp.R,D;Ka b. no Cp8.BHeHHlO c L BeJIJlllHBa b eii-
CTBHTeJlbHO oqeHb 148.JIa. IIOSTONY' B npH6JIHmeHHH MomHO H306pa3HTb
TpaeKTopHIO B BlI.lte P;BYX np.HMhIX JIlmBit C E3JIOMOM B paiioHe npOBO-
9L -ro
b max -eL =
-5
=8,1.10 N
PRe. 92
135

,
c
T
C - HtJnpaBAeHUe na
Ma"CUMYM unmep-
tj)epenl.lUU
T - Hanpa8Aenue Ha
MUItUMYM unmep-
tj)epeHijUU
a
PRC. 93
JIOKH. TorAa nepneH)J;HKYJI.RpHOe CMeIJ:{eHRe x B nlIOCKOCTH Ba6JIIO.Ae-
BRB 6Y.Q;eT paSHO
r=9L=6,21. 10- 5 .0,3 N= 1,86. 10- 5 NSI:$ 19 ro» roe
(HapHC. 92 nOKa38HaBeJIIAHUar- ro= 18 ro.)
II03ToNY .ABe 'tJaCTH nyqK8 nepeKphIB8lOTC.R B HeROTopoH 06n8CTH
aa npoBOJlOKo:A:, RaK nOKaa8HO Bft pRC. 92. IIoJIHaJI mHpHHa 06JIaCTH
nepeKphIB8.IIH.R d = 2 (eL - ro> = 36 ro == 3,6 . 10- 5 N.
IIHTeHcHBuocTL pacceHHBoro nyqKa nOCTO.RHHa B npep;eJI8X Kam-
.AoA 06JIftCTH, H ee 3H8qeBHe B 2 p8aa 60JILme HHTeHCHBUOCTH B o6JIac-
TR nepeKphIBaHH.R.
4. 8JIeKTpoHHoNY D}"IKY COOTBeTCTByeT BOJIBR ,lJ,e BpoAJI.H C .AJIH-
BoA BOJIHhI A ;:;;  == h - 8,68. 10- 12 M. ):(J1HHa BOJIHLI p;e BPOH-
mv o .J 2meUIJo
JISI HaCTOJILRO NeBLme mHpRBhI nytrKa 2b max , 'tJTO ,lJ,H(j;)p8.K:Q;HeH nyqKR
NO)l(BO npeBe6peQh. OP;UaKO Cnp8BR OT npDBOJIORH .Q;Be DJIOCKHe Bon-
ULI, pacnpOCTp8H.RIOm;Hec.a no.Q; (j;)UI<CIIpOB8HHLlN yrJION p;pyr K .Apyry,
nepeKphtB8.IOTC.R H HHTeP4>epJIP)'1OT. B TOH 06J1aCTH, rp;e B KJIaCC}fqec-
KON cJIY'Iae ABe nOJIOBHHhI HallaJILHOrO n}"lK8 npOCTO nepeKpLIBalOT-
C.R, 6yp;yr Ha6JIIO.Q;8TLC.R N8KCHNyMLI If NHHHMyN:LI HHTepc}>epeH:Q;HH.
113 pRC. 93 HaXO,lI;HN, 'tJTO p8CCTOHHRe a Nem,u;y coce.u;HHNH MaKCH.
MYNOM H MHHHMYMOM HHTep<l)epeHq;HH pasHO
A. ).. ! . 868 . 10- 12
a =   .£ SI:$ 2 M = 7,00 .10-8 N .
sine e 6,21-10- 5
nOCKOJIhKY nOJIB8JI mHpHHa 06J18CTH nepeKphIBBHH.R COCTaBJI.ReT
3,6. 10- 5 N, 6yp;er OKOJIO 500 N8.KCRMyN:OB HHTepc}>epeBIUlR. 38MeTHM,
'tJTO HHTepB8Jl NemP;Y COCep;HHMH NaKCHMYM8NH He 38BBCHT UK OT b,
HK OT b max ('tJTO COBepmeHHO He COOTBeTCTSYeT CJI}":I8IO HHTepclJepeU:Q;KK
OT ,lJ,BYX m;eJIeiI).
136

xxv Me>KAYHapOAHaR
tM3MlleCKali OnMMnMGAa
1994 r., KItTGM
3ap;aa 1
B CneD;HaJILHo:A: TeopllH OTBOCHTeJILHOCTH CB.H3L Memp;y 3HepI'lleit
E H HMnYJILCOM p .Q;JI.H cB06op;aofi Q8CTHD;LI onpep;eJIHeTC.H COOTHome-
HHeM
E = ,J p 2 C 2 + C4 = mc 2 .
Kor,u;a TaK8.fI QRCTII:q,a HftXOP;HTC.R nop; .Q;eHCTBHeM KO HcepB8THBB OH
CHJILI, ee nOJIB8.fI 3HeprHSI, HBn.moIIJ;8.fICSI CYNMOit E = ,J p 2 C 2 + mc4 II
nOTeBD;H8.JIhBOH 3HeprHH, COXpaHHeTC.H. ECJIII3HeprH.R qaCTH:q,LI O'TleHL
BeJIHKa, TO 8HeprHen nOKOSI NO)l(HO npeHe6peQL (TaKa.R -qaCTH:q,a Ha3LI-
BaeTCSI YJILTpapeJI.HTHBHCTcKoii:).
1. PaccMoTpHM O,lJ,BO:MepHOe ,ABHmeHHe -qaCTH:qLI C oqeHb BLIcOKO:A:
3HeprHefi, ua KOTOPYJO .n;eitCTByeT npHTHrHBaIOIIJ;8.H D;eHTpaJILHaJI CIIJIa,
Nop;ynh KOTOPOit DOCTO.RBeH II paBeH f BHe Ha-q8.JIa KOOPP;HH8T II HYJIIO B
TOqKe, COOTBeTCTBytOJIJ,;eH Ha't:laJIY KOOP.Q;HH8T. lIpH p;BHmeHIIH TaKOH
qaCTH:q,LI B HeKOTopLIe KopOTKHe npoNemyrKH BpeMeHII qaCTa ne-
peCTaeT 6LITh YJlbTpapeJI.RTHBHCTCKoA. ITpH pemeBHH P;aHHOH 3a.u;aQIl
STHN .HIlJIeHHeN CJIep;ye-r npeHe6peQ:b H C'IHTaTh 'IaCTIID;Y YJIbTpapeJISI-
THBHCTCKOfi Ha Bcex 3Tanax ee p;BH»ceHH.R (KpoMe u. 4).
.l(onYCTHN, qTO B Hall8JIhHLIfi MOMeHT t = 0 3Ta 'IBCTHD;a HaxO.Q;HTC.R
B Haqane KOOPAHHRT (B CHJIOBOM D;eBTpe) H HMee-r HMUYJILC Po. Hao6pa-
3HTe rpa4>JPIeCKII p;BHmeHHe qaCTHD;LI C nONOIIJ;hlO rpacIJHKoB 3aBHCH-
NOCTH HNnYJ1LCap OT Koopp;HBaTLI x H KOOp,lJ,HHaTLI x OT BpeMeHH t .n;JISI
OP;HOro nepHop;a p;BH»ceHHH. Onpep;eJIHTe KOOPAHHRTbI TOqeK nOBOpo-
Ta qaCTHD;1d qepea napaMeTpLI Po H f II yxa»CHTe CTpeJiKaMH HanpBBJle-
BHe ,ABHateHH.R aa JJ;H8rpa:MMe P, x.
2. Me30H JIBJI.ReTCSI qacTHD;eA, CocTOHJIJ,;eH H3 .u;BYX KBapKOB. MaccR
nOKO.H M MeaOHR onpe.u;eJI.HeTCa OTBomeHHeM cYMNapHoit 3HepI'HH CII-
CTeMhlII3.u;BYX KBapKOB K c 2 .
137

P8CCNOTpHM O,ABOMepH)11O MO.l(eJIb nOKO.HIl\erocH NeaOHa, B KOTO-
poi: npeAIIOJlar8erCB, 'liTO 068 KBapKa BH)KYTCB BOJlL OCB X B npHT.R-
rBB810TCB APyr R Jq>yry C nOCTOSlHBOH no NO,D;YJlIO CHJIO:A: ,. IIpenoJI8-
r8eTC$I,  KsapKB MOryI' CB060,D;HO npoXO,D;HTL Jq>yr CKB03L .u;pyra. lIpH
HCCJIe)I;OB8.HHH .u;BHmeUHSI KBapKOB C BLICOKOfi 3HeprHefi HX MaCCOH
nOKOSI MO.HtHO npeHe6pe1:JL.
ITYCTh npH t = 0 06a KBapKa HaXO,D;.HTCSI B T01:JKe C KOOp.n;UH8TOH x =
= O. Hao6pB3BTe ,D;BHmeHHe 060HX KB8.pKOB (no OT,D;eJIhHOCTH) C nONO-
IIt;LIO rpal}>HKoB rl(t), x 2 (t), a TaKme C nONOIItLIO AHarpaNMPl' (x) R P2'
(x). OnpeAeJIHTe KOOpHB8TLI TOTleR nOBOpoTa 1:Jepea BeJIH1:JHHhI M H ,.
YK8JKHTe Hanp8BJIeHHe npoq;eccoB H8 HarpaMMax p, x H onpeeJIHTe
N8KCHN8JILBOe paCCTOJIHHe d Me)K,U;y .u;BYM.R KBapK8MH.
3. CHcTeNY OTCqeTa, HCnOJIh30B8.HH)11O B n. 2, 6Y,D;eM 0603HaqaTh
1:Jepe3 S. IIycTh Jq>yr8JI CHCTeNa S' (HB30BeN ee JIa60paTopHoA) .D;BHmer-
CII C nOCTO.RBHo:A: CKOpoCTLIO v = 0,6 C B 07pH:q,8TeJIbHOM Hanp8BJIeHHH
OCR X. KOOp.D;HHaTLI B 3THX ABYX cHcTeNax BhI6Hp8lOTCSI TaR, 1:JTO T01:J-
Ka x :;: 0 B CHCTeMe S COBna.u;aeT C T01:JKoi: x' = 0 B CHCTeMe S' B MOMenT
BpeMeHH t = I' = O. H306pa3HTe ABHmeBHe 060HX KBapKoB B CHCTeMe S'
C nONOIIt;hlO rpal}>HRoB xi(t') H x;l.t'). OnpeeJIHTe KOOp,D;HHaThI TOqeK
nOBopoT8 B CHCTeNe S' Qepe3 M, f H C H HHTe NBKCHMWIhHoe pac-
CTOJIHHe d ' Ne)K I<BapKBNH, Ha6JIJOaeMhIMH B JIa60paTopuoi: CHCTe-
NeS'.
B COOTBeTCTBHH C npeo6paaOBaHHSlMH JIopeHq;a CBSl3L Mem.n;y KO-
Op,D;HHaTaMH x H x' H BpeMeHeM t H t' B CHCTeM8X OTCQeTB S H S' HMeeT
BHJJ;
x' = y(x+pct), t' = y(t+ :).
R V 1
re tJ = -, y:;: r;t:;2'
C v l - 13 2
4.l1.JI.R MeaOHa C aHepI'Heii nOKoa M c 2 =140 MaB, ,ltBHmytI(erOCSI co
CI<OPOCTLIO v = 0,60 C OTHOCHTeJIDHO JIa60p8TopBOA CHCTeMhl S', onpe-
eJIHTe ero 3ueprHIO E' B 3TO:A: CHCTeMe.
3a1Ja 2
CBepxnpoBo,D;JUIJ;He N&rHHTLI mupoKo npHMeU.RIOTCa B J1a60paTo-
pHIIX. H8H60JIee pacnpocTp8.HeHHoit l}>opMoA cBepxnpoBOgm;ero MBr-
HHTa .HBJIaeTC.R COneHOHJJ; H3 cBepxnpoBoASlIIte:A npOBOJIOKH. YAHBH-
TeJIbHLIM cBOAcTBOM CBepxnpOBoero NarHHTB .JIBJUleTCSI TO, "lTO OB
C03,D;ae-r CBJILHLIe MBl'BHTHhIe nOJISlOO3 ;D;HCCHDH (paccelIHua) 3Hep-
rHH, 06YCJIOBJIeHHo:A: A)KoYJIeBLINH nOTepJll'tUl, T8K KBI( 3JIeKTpH1lecl<Oe
conpOTHBJIeUHe CBepXnpOBO,lJ,.RIItefi npOBOJIOKH CTaBOBHTCH pBBHLIM
138

I
t
T
I
-L-
r-- ---
1 2
I
I
I
I
I
I
I
L____
PUC. 94
HYJIIO, Korp;s MRrHHT norpymalOT B )KH.D;KHH reJIHH npH TeM1IepSType
4,2 K.
06hI'lHO MarHHT cHa63<aeTCB Cnen;H8JIhHLIM CBepxnpOBo.u;.HHM
KJIIO'tIOM. RaR nOKa3aBO H8 pHC. 94. COnpoTHBJIeHHe r RJIIO-qS MOmeT
H3MeHHTLCH - OHO JlH60 paBHO HYJIIO, Kor.u;a RJIlOq HaxoAHTcH B cBepx-
npOBo;D;eM COCTO.HHHH, JIH60 p8.BHO r n' KOr,lJ,S KJIlOll HaxO;D;HTC.H B HOP-
M8JILHOM COCTOHHHH.
Kor.u;8 RJIlOq HftXOP;HTCB B CBepxnpoBO;D;.HIIJ;eM COCTO.RHHH MRrHHT
MomeT pa60TaTh B YCTO:WlHBOM peJKHMe. B STON CJlyqae TOK D;HPKYJIH-
Pye-r qepe3 M8.I'HHT R CBepxnpoBo.u;HI::qHH KJI10ll HeOrpSHH'leHHO .u;OJIro.
y CTOHBhI:A pe)KHM n03BOJIBeT C03.u;8B8TL M8.I'HHTHOe nOJIe, KOTopoe
,AJIHTeJILHOe BpeM.R: nOAAep)KHB8eTC.R: neH3MeHHhIM npH OTKJIIOQeHHOM
HCTO'tJB.HKe TOKa.
]:(eTaJIH cBepxDpoBop;ero 1<Jl101l8He nOK838BhI as pRC_ 94. 06LrQ-
HO OH H3rOTaBJIHBSeTCH H3 RYCKS cBepxnpoBO.R:e:H npoBOJlOKH, Ha-
MOT8.HHOH BMeCTe C npOBOJIOQHhIN HarpeB8TeJleM, TenJIOH30JlHpOB8.H-
BhIN 0'1' B8HHhI C )KH,ZtKHMreJlHeN. ITpH HarpeBaHHH TeMIIepSTypa CBepx-
npOBO.HeH npOBOJIOKH YBeJIH'lHBSeTC.R: H OHa nepexo.u;HT B
HOpM8JIhHOe COCTOJnlHe. TUnKtlHoe 3HalleHHe r n COCT8BJI.R:eT HeCKOJIh-
KO OM (npHNeM B p;aJILaeHmeM r n = 5 OM). HH,lI;YKTHBHOCTL CBepx-
npOBO.D;HJ:IJ;ero MBrHHTa 3aBHCHT 0'1' ero pa3MepOB (npHNeM ee pRBHOH
L = 10 rH). ITOJIHblH TOK B n;enH (CM. pHC. 94) NO)KeT 6hITL H3NeHeH
nyreM perYJIllpOBRH conpoTHBJIeHHH R nepeNeHHOrO pe3HcTopa.
CTpeJlKH Ha pRC. 94 0603H81l8IOT nOJIomHTeJIhHhIe HanpB.BJleHHH
TOKOB I, 11 H 1 2 - 06Be.u;eHH8.H UynKTHpHOH JIHHHeH "laCTh YCTftBOBRH
norpymeHs B mHJJ;KHA reJIHH npH TeMnepRTfl)e 4,2 K.
139

fo 
O,510 -V
° t 1 t 2 t 3 t 4
a
.
n L.=.I=r.=
o t 1 t 2 t 3 t 4 t
6
: :  -- -----------
0,51 0 -----------
I I I .
o t 1 t 2 t a '4 t
6
::  -- -----------
0,51 0 -----------
I I I ..
o t 1 t 2 t 3 t. I
Z
PRe. 95
t
o,51 t- -----------
L I I ..
o 1 2 3 t, MHH
a
ntJ:=r.==
o 1 2 3 t. MHH
6
0,5  t -------------
- I I I )I
o 1 2 3 t MHH
8
0, 51 t --------- ----
- I I I ....
o 1 2 3 t, blHR
2
PRe. 96
1. ECJIH npe.n;nOJIo»tHTb, 'lTO nOJIHhIH TOR I H conpo'1'HBJIeHHe r
cBepxnpoBop;am;ero KJIIO'l8 H3MeHSlIOTCH co BpeMeHeM TaR, R8.K nOK83a-
HO Ha pHC. 95, a, tJ COOTBeTCTBeHHO, H HB HHTepB8.JIe BpeMeHH 0 - t 1 TORH
II H 1 2 'IIepea M8rHHT H KJIIO'l O;l];HHftKOBhI H ueH3MeHHLI (puc. 95, e, z),
TO RaIC OHR 6YAyr H3MeH.fITbC.H Uft HUTepBftJIe BpeMeHH 0'1' t 1 AO t 2 ? Ha06-
paauTe pe3YJIbTaT pemeBH.H rpa4>IDIecKH Uft pHC. 95,8, z.
2. ITpeAnOJIOmHM, 'liTO KJIlOll K, nOAKJIIOTl81Om;HH MarHHT R HCTOq-
BHRY, 6h1JI BKJIIOQeH npu t = 0, Kor1J;8 r= 0, II = 0 H R = 7,5 OM, H nOJI-
HhlH TOR OKB38JIC.fI paBHhlM I ;:::; 0,5 A (pRC. 96, a). ITYCTb KJIIOlI OCTaeT-
C.H 38.MRU)'ThlM, a COnpOTHBJIeUHe r CBepxnpOBop;am;ero KJIIOqft R3N:e-
H.HeTC.H TaK, RaR nOK83aHO Uft pUC. 96, tJ. KaK 6yp;yr H3MeHXTbCH co
BpeMeHe1'd TORR I, I] H 1 2 ? BhlilOJIUHTe paClleThI H H306pa3HTe peaYJIb-
TBTLI rpa4>lAeCKR ua pRC. 96, a, 8, z.
3. TOJILKO NftJIhle TORR, MeaLmRe 0,5 A, pa3pemfteTCH nponYCKaTb
llepea CBepxnpoBO,D;.s:m;uii KJIIOq, KOr,Aa OB HaxO,AHTCa B UOpM8JlLHON
COCTO.HHHH. ITpH 60JlLmUX TOK8X KJIIO'l neperopHT. AonYCTRN, 'lJTO
CBepXnpoBO.HH:ft MarBUT pa60TSeT B YCTOitclHBOM pemHNe, KorAa
I:;:: 0 (pUC. 97, a), II:;:: i l (HanpH1'dep, 20 A) R 1 2 = - ii' RaK nOKa3aHO Uft
pHC. 97, 8, Z ,D;JIH HHTepB8Jl8 BpeMeHH 0'1' t = 0 1J.O t = 3 NUB.
140

I
20 A t - - - - - - - - - - - - -
_ f I I I ..
o 1 6 9 12 t. MIIH
a
n t - - - - - - - - - - - - -
- J 1 1 , .
o 1 6 9 12 t MilE
(5
20  t - - - - - - - - - --
- I 1 1 ..
o 1 6 9 12tMIIH
8
I
30A t -------------
20 A - - - - - - - - - - - - -
J 1 1 1 '"
o 3 6 9 12 15t.MIIH
a
n t --- --- -------
- '" .-
o 3 6 9 12 15 t. MIIH
t5
30k  -- -----------
20 A - - - - - - - - - - -
1 . 1 I_
o 3 6 9 12 15 t. MIIH
6
1 2
30A
20A
o
6 9 12 t. MIlH -20 A
----------- -30A
z
1 2
20A
o
PRC. 97
3 6 9 12 15t.MIIH
-----------
z
PHC. 98
ECJIH SKcnepHMeHT Heo6xo,ll;HMO OCTSHOBHTb n}'TeM YMeabmeaHH
CHJILI TOKR qepea MarHHT AO HYJlH. TO K8K BhI aTO 6YAeTe .u;eJI8Tb? 3TO
.u;OJI.>KHO 6hITb C,lJ,eJI8HO B aeCKOJIbKO STRnOB (onepau,HOHHLIX marOB).
Hao6pS3HTe rp8cPINeCKH aa pHc.97, a-z COOTBeTCTByroJ:q;He H3MeHe-
HHft I, r, I] H 1 2 ,
4. .l(OIlYCTHM, qTO MarHHT pa60TfteT B HHTepB8JIe BpeMeHI:I OT t = 0
AO t = 3 MHH B YCTOWlHBOM peatHMe npH TOKe II = -1 2 = 20 A (pRC. 98,8, z).
KaKH1\II o6pa30M MO.>KHO nepeBecTH MarBHT B YCTO:AqRBLIH pe.>KH1\II C CH-
noA TOKa I] :::; -1 2 = 30 A, He llpeBhIm8JI1\lIaKCHM8JIbHO p;onYCTIIMoro
aaftqeHHH CHJlhI TOKa qepea cBepxnpoBOAHIIJ;HH KJIIOq B HOpM8JlbBOM
COCTOHHHH (CM. II. 3)1 H306pa3HTe aft pRC. 98, a-z COOTBeTCTBYIOIItHe
R31\1IeaeHHH CHJlhI TOKa II conpoTHBJIeHHH.
3ap;aWla 3
O.u;aopoJJ;HLIH JJ;HJIHH)J;pl{qeCKHft AHCK A M8CCOH m II PMHYCOM R A
ABHmeTC.H nocTynsTeJILHO co CKOpOCTLIO V no rJI8,.IJ;KOH ropH30aTaJIb-
BoA nJIOCKOCTH XY B Banp8BneHHH OCR X aft paCCTO.HHHH bOT 8TOR OCR
(pRC. 99). OR CT8JIKHBSeTCB C APyrHM HenO)J;BHmHbIM O)J;HOPOAHLIM
141

ABCROM B TOft me MaCCbI H p8,ll;HY-
COM R B , IteHTp KOToporo HaxOAHT-
C.H B aft'llaJIe KOOPAHHaT. Bo BpeMJI
CTOJlKBOBeBHH Me WlCK8..MH B03-
HHK8eT B TO'llKe conpHROCHOBeHH.R
,AOCTaTO'llaO 6oJILmoe TpeHHe. ITpe.o.-
nOJlaraeTC.H CJIeP;YIOJ:IJ;8.a NO,AeJIb
B3aHMo.u;eiicTBHa. IToCJIe CTOJIKHO-
BeHH.R COCTaBJI.fIIOIItHe CKopOCTeH
X
,AHCKOB, nepneHAHKYJIHpHLIe JIH-
BRB HX D;.eBTpOB, B T01:JKe conpHKOC-
BOBeBU.H paBHLI no MO,AyJIIO p;pyr
APyry, 8 OTHOCHTeJIhHLIe CKOpOCTH
AHCKOB BOJIb JIltBHH IteHTPoB Op;H.
H8KOBLI AO H nOCJIe cOYAapeHHSI.
B paMK8X STOR MOJJ;eJIH onpeJJ;eJIHTe:
a) npOeKD;.HH cKopoCTeA. IteBTpOB AHCKOB nOCJIe CTOJIKBOBeHII.H BB
KooPAHH8THLIe ocn OX H OY, T.e. BLIpa3HTe v.:u, V, V Bx H V Bg 'lepea
n8p&MeTpLI mt RAt R B , V H b;
6) RHHeTH'lecKHe sHeprHH E;" H E  060HX .ltHCKOB nOCJIe HX CTOJIK-
ROBeHH.H 'IIepea napaMeTphI m, R A , R Bt v H b.
y
Puc. 99
PeweHMIl
3ap;aQa 1
1. ITpHMeM, 'lITO '118CTHa H8XOA.RIIt8.HC.H B Ha'laJIe KOOPP;HH8T, HMe-
eT nOTeH8JIbHyJO 8Heprmo, paBHYJO IIYJIIO.
lIoTeIlI\H8JIhH811 8HeprHJI U(x) Q8CTH:Q,hI npH ee op;noMepHoM ABH-
meHKH BOJIL OCB OX p8.BHa
U(x) = f I x I , (1)
a ee 06.rqall 8HeprH.R W pURa
W = .J p 2 C 2 + mlc" + flxl,
.l(JIH YJILTpapeJI.RTHBHCTCKoft TlaCTUI\LI
w= IPx1c+f Ixl.
(2)
COrJIftCHO 38KOHY COXp8.HeHHH SHeprHH, a8llHmeM:
I p x Ie + f I x I = Po c.
142
(3)

PaCl<pOeM 3H8RH a6COJIJOTHOH BeJIIPIHHbI:
I pc + Ix = POC eCJIH x > O.
-pc + Ix = POC ecJIn: x > O
pc - Ix = POC, eCJIH x < 0,
-pc - Ix = POCo eCJIH x < 0,
Px > O.
Px < o
Px > 0,
Px < O.
(4)
MaKCHMaJILHhIM OTKJIOHeHHe L Q8CTH:Q;L1 OT H8'l8JIa KOOp,lJ,HHaT 6y-
POC
JJ.eT, eCJIHp = o T.e. L = ,..
H3 BIJp8meHHH (3) II TpeTbero 38KOH8 HLJOToHa CJIeyeT:
dp { -t npH x > O
-=F=
dt f npH X < o.
MhI NomeM HaHTH CKOpOCTL qaCTH:U;hI:
I : I  71 :1  c.
MhI npllillJIH K BhIBOy, 'qTO qaCTHnLI KOTopLIe )J;BHmYTCH C OqeHL
60JILIIIOH 3HeprHeH Bcer.n;a BllmYTCH co CKOpOCTLJO c, KpOMe Tex MO-
MeHTOB. Kor,lJ,a QaCTH:Q;8 H8XO,lJ;HTC.R 6JIH3 3KCTpeMaJlbHbIX TOQeK C KO-
op.n;HHaTaMJI x = ::!:.L.
BpeM.R ,lJ,BHH<eHHH qaCTHQbI OT Ha'tJaJIa Koop)J;HHaT ,lJ,O TO'lKH C KOOp-
.o;HHaTOH x = L paBHO
=  = o .
TaKHM 06pa30M 'tJaCTHna 6y.o;eT P;BHI'8TbCH Mem.n;y TO.I1,BMH C KO-
Op,AHHaTRMH x = L x = - L co C:KOpoCTLJO C H C nepHo.n;OM
4po
T=4t=-.
I
CB.R3b Mem.u;y x H t npe.n;CT8BJIeH8 CHCTeMOH ypaBHeHHH:
{ X = ct npH 0  t  ;
x = 2L - ct npH t < t  3'[;
x = ct - 4L npH 3 < t (;; 4'[.
MCKOMbIH OTBeT OTpameH Ha rpa.pHK8x (puc. 100, 101).
143

x
A
o
-L
C
JJ.
1"=
f
L = Poc
f
Poc.lOO
P
Po
-L
L
x
L = Poc
f
PRc.IOI

Xto X 2
L
A E
----------..,
/.'
/ . ,
.
o
,
-L
B
T= Me
2[
L= Me 2
2f
D
PHc.l02
PI
Po
P 2
Po
-L
-L
L X2
-Po
Me 2
L=-
2f
Me
Po=T
-Po
a
6
PRC. 103

2. 06IIJ;YIO 3HeprHJO ,nBYXKB8pKOBOH CHCTeMhI MomHO BhIp83HTb B
q>opMe
Me 2 = I PI Ie + P2 Ie + f I Xl - x 2 1,
(5)
r,lJ,e Xl H X2 - KOOp,lJ,HHaTbI RBapKOB, PI II P2 - HX HMnYJIbCbI.
,l];JIH CS060,lJ,HOrO MeaOHa 06HH HMnYJIbC .u;syx KBapKOB paBeH
HYJIIO, H, TaR R8R .D:Ba RBspRa .D;BHmYTC.JI CHMMeTPUqHO B npOTHBono-
JIO)fCHLIX HanpaBJleHHSlX, MomHO a8nHC8TL
Px =Pb + P2x= 0, Ph: = -P2x' Xl = -X 2 .
(6)
0603H8QHM HMnYJIbC KB8PK8 1 B MOMeHT, Kor.u;a X = 0, qepea Po'
Tor.u;8 HMeeM
M 2 2 Me
e = PoC HJIH Po =-.
2
(7)
113 BbIpWKeHUH: (5) - (7) HCHO QTO nOJIOBHH8 nOJIHOH 3HeprHH MO)fCeT
6bITb BbIp8.meHa Qepe3 BeJIHG:HIThI PI II X 1  OTHOCHHeCg K KsapKY 1:
POC = I PI x I e + f I x 1 I,
(8)
qTO COBna.n;aeT C BLIp8meHHeM (3) .u;JIH O.u;HOH qaCTH:Q;LI C HaTUUILHhIM
Me
HMUYJIbCOM Po = 2.
MCnOJIb3YH OTBeTbI n. I, MbI MomeM nocTpoHTb rpa$HKH 38.BHCH-
MocTeii x}(t) H Pl(X 1 ) ,lJ,JISI .lI;BHJKeHHH KBapK8 1 (pHC. 102, CTIJIOmHaSl
JIHHHSI; pHC. 103 a).
AnSI KBapKa 2 cHTyaU;HSI CXO,AH8H, n03TOMY HeTpY,lJ,HO nOJIytIHTb
COOTBeTCTByroIIJ;He rpafPHKH. OHR nOK838HbI Uft pRC. 102 (mTpHxoBaSI
JlHHHSI) II pRC. 103.
MaKCHMaJIbHOe paCCTO.HHHe d Mem.n;y ,lJ,BYMH KBapKaMH RaK HCHO
R3 pRC. I 02  p8BHO
d = 2L = 2 PoC = M c 2 .
f f
3. npeo6paaoB8HHH JIopeH:Q;8 ,lJ,JUI ,lJ,syx CRCTeM OTCqeTa S H S' , ,lJ,aH-
HhIX B YCJIOBHH 3a.n;a'lll aanHmeM B Bu.u;e
x' = y{x + pet),
t' = r(t+ ).
r.n;ep=E.y= 1 .
c .JI - p2
146

3 5
IIpn v = 0,6 c J3 = 5' y == 4'
J1.nH l\BSpK8 1:
CHCTeM80TCQeT8 S CHcTeM8 OTClleT8 S'
xr = y(x 1 + pet l ) = ' ( Xl )
t l =y t 1 +7 =
Xl t 1 5 3
== 4X1 +4 Ct1 5 3 xJ
="4 tl +"4c
0 0 0 0
L t 2L 2't
0 2'[ 2.L 5
-1"
2 2
-L 3T L 3T
0 4'[ 3L 5T
.l(nH KBapK8 2:
X 2 t 2 x; == y(x 2 + Jkt 2 ) t 2 == Y (t 2 + P:2 )
0 0 0 0
-L -lL 1
't -'t
2 2
0 2t 2.L 5
-'t
2 2
L 3'£ 1L 9
-1"
2 2
0 4't 3L 5't
3.u;ech L = P,C = 2 ; t = o = C .
lIoCKOJIbKY npe06p830BaHHH JIopeHa JIHHeiiHhI npSlM8.R JIHHIUi
x(t) npeo6pa3yeTC.R B np.RM}'IO JIHHHIO x'{t'). HymHo TOJILKO paCCTIHT8Th
TOQKH nOBopoT8 B CHCTeMe S' .
147

3,5L
3OL
25L
2,OL
1,5L
1,OL
0,5L
-0,5 L
x;, X.2
.
I I
I I , I I I I I I I
-----'---r-----1---i-----'---i---r---------
I I I I I I I I
Me 2 I I , , I I I I
L - I I I I I I I I
- 2 f I I I I I I I I
I I I I I I I I
I I I I I I I I EI I
-----i-----i------r--------A------------
: : : : : : : : I: \ :
I I I I I I I I' I \ I
I L I I I I I I I I
I I I I I I I I' t \1
-----f-----------t-----l---+---
: : : : : : : ,: : :\
: : : : : : : I: : : \
I I I I I I II I I I'
__J___l___L_____J___l__Jt__J___ ____________
: : : : : : ,: : : :
: : : : : :': : : :
I I I , I " I I I I
I I I A' I I I I I I
--------:-- - I---- --t---:--- - __---:----------
I I I I I 'I I I I
: : : : : I: :::
: : : : :': :::
__J___l___ 1 __...:___ J2___ ---.:___l____________
'I I I I I I
I I I I I I I
I' I I I I I
:: : , : :::
: I I :' I I :::
--1--- ---r--J---'---I---I --'---T---.---------
I I I I I ,D I I I I
I I I 'I I I I I I I
: : : I: : : : : : :
: : :, : : : : : : :
--i---(-----------r--i---T---r---------
1'1 I I I I I I I
I I I I I , I I I
I 'I I I I I I I I
:,: : :  : : : 
I I I I
I I I I I I I I
\ :0  5-r.1 , 0-r: 1  5'[:2  0'[:2,5'[:3,0'[:3, 5't:4, 0-r:4  5-C:5  O't t'
I 'I I I I I I I I I Me
\ 1 I I I I I I I I I Y=-
I' I I I I I I I I I 2(
-- .v-- -- - ----:- - - ---- -- -- -:---- -- --- - -- ---
BI I I I I I I I I I
I I I I I I I I I I
, I I I I I I I I I
I I I I I I I I I I
I , I I I I I I I I
I I I I I I I I I I
I I I I I , I I I I
o
PHC. 104

C yqeToM nOJIyqeHHLIx pe3YJIhT8TOB HCKOMhIH I'pac1>HK X'(t') "D;JI.f1
,D;BYX KBapKOB RMeeT B, nOKa38.HHhIA aa pRC. 104 (cnJIOIDH8.SI JIH-
BUS! - ,lJ;JIB KB8pK8 1, IDTpUXOB8.H - ,lJ;JIS! KBSpK8 2).
MaKCHM8JILHOe paCCTOS!HHe d' Mem,ny KBapKllMH 6yeT B MOMeHT
BpeMeHH t' = .! t :
2
2
d' = L = Me .
2(
4. ECJIU Me30H C 8HeprHeii nOKO.K M c 2 = 140 MaB ,lJ;BHmeTC.R co CKO-
pOCTLIO V = 0,60 C B na60p8TopHOH CHCTeMe OTC'tJeT8 S', TO ero 3HeprHH
E' B CHCTeMe S' paBHa
, Mc 2 Mc 2
E- - .
- 1- p2 - H 2 '
1--
c 2
E' =  MsB == 175 M8B.
3a.a;aQa 2
1. B npoMemyrKe BpeMeHH (t. - t 3 ) conpOTHBJIeHHe r = O. CJIe,lJ,OBa-
TeJILHO, na.n;eHHe HanpSlmeHH.H HB MBrHIITe paBHO HYJIIO:
U M = L dI. = 0,
dt
1
II ;;:: const = -1 0 ,
2
1
1 2 :; 1 - II == 1--1 0 .
2
B npOMe>KyrKe BpeMeHH (t 3 - t..), TaR K8K 1 2 = 0 B MOMeHT t = t g , 8.
1
I = -1 0 nOCJIe t = t g , HanpHmeaHe aB .M8rHHTe paBHO
2
U..=I 2 r n =O,
149

a
0,5 :  - - - - - - - - -, - 
o
tl
t2.
t 3
t4 t
6
 t- - -  - -- - - - J
[
o
t 1
t 2
t 3
t4 t
6
0,5 :  - - - - - - - - - - - - - - - - - - 
t I .
o t 1 t2 t 3 t 4 t
z
1 2
 -------------------
O5 10
t
o
-0,51 0
PRC. 105
a CJIeAOBaTeJIhHO CHJIhI TOK8 II H 1 2 He H3MeH.HIOTC.H:
1
II = 2 10 ,
1 2 = O.
9TH pe3YJIbT8ThI nOK838HhI H8 pHC. 105 6, 6 z.
2. )J;JI.H npOMem}'TR8 BpeMeHH 0-1 MHH, nOCKOJlbKY r= O
dI}
U M =L-=O
dt
H
150
II = const = II (O) = 0,
12=1-Il=05A.

a
:: ---- r
o 1
3
t, MHB
r
2
6
o t----I
f ---------
. .
o
3
t. MEB
1
2
8
I t
O,5A ---- (

o
t. MHB
1
2
3
z
12 
05A f- -- - - - -- - -----
0,3 A ---- 
.
o 1 2 3 t. MHH
PRC. 106
B MOMeHT BpeMeHH t == 1 MHB conpoTHBJIeHHe r CKS'lKOM MeH.HeTC.H
6
OT 0 p,o rn' CHJI8 TOKa 1 MrHOBeHHO H3MeH.HeTC.H OT 3Ha'leHH.H R /1,0
 
R + Tn . Y'lTSI, '1TO R = 0,5 A, R = 7,5 OM, Tn = 5 OM, nOJIycIHM, 'lTO
CHJI8 TOKa 1 YMeBLmaeTC.H p,O 0,3 A.
B npoMemyrKe BpeMeHH 1-2 MBH CHJlLI TOKOB I, 1 1 ,1 2 npn6JIHma-
IOTC.H K 3B8TJeBHBM
B
I ;; - = 0,5 A,
R
II=I=0,5A,
1 2 = o.
151

BpeM.R YCT8HOBJIeHH.R npO:q,eCC8 paBHO
-c =  = L(R+r n ) .
RdJus. R r ll
llPHL= 10ruR= 7,5 OM, r n = 50M t = 3 c.
B npOMe)KyrKe BpeMeHH 2 - 3 MHU, nOCKOJILKY r = 0, CHJILI TOKOB II
HI 2 He 6YAYT H3MeH.RTLC.R. TaKHM 06pa30M, 1 1 = 0,5 A, 1 2 = O.
Bee OllHcaHHhle npo:q,ecchI npeACTaBJIeHLI Ha pRC. 106. a. tJ 8, e.
3. OnepaJJ;HoHHhIe m8.rH CJIeIIPIe:
n e p B hi H mar. 3aMKueM KJIIO'll K H yBeJIHtIHM o61J:ii TOR I 11.0
20 A, T.e. C11.eJIaeM ero paBHLlN I l' TaR KaK conpOTHBJIeHlle cBepxnpo-
BOA.R.IIJ;ero KJIJOqa B npOMemYTOK BpeMeHIl 3-6 MIlB paBHO UYJIJO (r = 0),
dI
TO UN = L--1... = 0 1'1, CJIeAOB8TeJILHO, TOK 1 1 = const = 20 A, a 1 2 H3Me-
dt
HS:leTCSI 0'1' -20 A ,lJ,O UYJI.Jl.
152
a
20 t---------
o
3
6
12
t MHH
9
6
 t------r---r-----
o
3
6
9
12
t. MHH
(3
II I
20A.
o
9
12
t. MHH
 ----- -. -
3
6
e
1
20A 2 - -- - -- --- - -- -- -----
o
t. MHH
12
9
-20 A
Puc. 107

B T 0 pO it mar. ConpOTHBJleHHe KJIIO"l8 r YBeJIHQHBaJOT OT 0 1I.0 r n.
T peT B it mar. IIocTeneHHo YMeHhm8.H CHJIY TOKa 1 11.0 ° B Te'tle-
BHe npOMem}'TKB BpeNeHH 6-9 lIIIHB, noro:r;epmHBaeM 1 2 < 0,5 A. TaR
U dI
KaK 1 2 = -!!.. HUM = L ---.1.., TO B CJIyqae, Ror.u;a L = 10 rH, r n = 0,5 OM,
TII d t
Tpe60BaHHe 1 2 < 0,5 A COOTBeTcTByeT BhIUOJlHeHHIO YCJ10BHH
dl
-2.. < 0,25 A/c, T.e. CROpOCTL H3MeHeHH.H CHJIhI TOR a AOJI>KH8 6hITb
dt
MeHhme 15 A/MUB.
dI dI
HapHc.l07 -  0,1 A/c = 6 A/MIDI H --.!..  0,1 A/c, T.e. Tpe60-
dt dt
BaBBe 1 2 < 0,5 A BLInOJ1BeHO.
a
301  -- ::=::------ --- -----
20 A ---7. ---.- ,,-----
....
o 3 6 9 12 15 t MHB
6
  ------ c==J ---------
. ... ....
o
6
9
12 15 t. MUH
3
(3
301 1 r --------- -
20A  ---------
- .... .
o 3 6 9 12 15 t. MUH
Z 1 2
30 A - - - - - - - - - - - - - - - - - - -
20 A --- ----------- -----
o
-20 A
-30 A
3
15 t. MliH
PRe. 108
153

cI> II B a:n & H &I fi mar. COnpOTHBJIeHHe r CTRHeT paBHIaIM BYJnO,
KorA8 UN = 0 H ICJIIOTI K OTKJIIOQaeTC.B.
Pe3ynLTaThI nOIC83aHLI Ha pHC. 107, a, 6,8, t.
4. llepBhIA H BTOpoA marH cOBnaIOTcnpeAhI)J;HM:nYHK-
TON.
T peT H it mar. YBenH'IHBaeM CHJIY TOKa I OT 20 .0;0 30 A co
CKOPOCT&IO, He06xo.o;HMoit .o;nSI BLInOJIHeHHSI ycnoBHH 1 2 < 0,5 A
(  <6A/MHH).
B npONe)K}'TKe BpeMeHH 6-9 MRH yseJIHTIeHUe 1 ua 10 A STONY yc-
JIOBHIO YP;OBJIeTBOpSleT...
q e T B e p T LI it mar. ConpoTIIBJIeHlle rCTBHOBHTCSI paBHbIM ay:TIlO,
Korp;a UN = O.
IT $I T LI A mar. .l(OBOP;HM CHJIY TOKB 1 .0;0 BY JISI, cn:JIa TOKa I. = 30 A
Be H3MeHSleTCS, T8.K !CaR U.. = 0, 1 2 = 1 - 1 1 = -30 A.
 H H a JI L H LI it mar. BKJIIO"lHM KJIlOqK. Marmrr pa60TBeT B 3B.H-
HOMpemHMe. no.nyqeHHhIepe3YJlLTaTLIoT})8jKeHhIHapac.l0B, a, (5, 8, t.
3a.n;aQa 3
a) B MOMeHT CTOJIKHOBeHHH AHCKa A C .o;HCKOM B H306pa3HM HOP-
M8JILHlaIH Ii II TBHreHQ;HaJI&HhIA i BeKTophI (pHC. 109).
11:3 pn:cyaKa .HCHO, qTO b = (R A + R B ) sin qJ.
y
x
he. 109
154

CnpoepyeM HMnyJIbChI.D;HCKOB 11.0 CTOJIKHOBeHHH HS HanpSBJle-
HH.R BeKTOpOB ii Hi:
mv An = mvcos Q>j mV Sn = 0,
mv At = mvsin Q>; mVSt = O.
0603HS1:JHM npoeKHH HMnyJlhCOB ,[J;HCKOB A H B H8 3TH HanpaBJle-
BH.R nOCJIe CTOJIKHOSeHH.R 1:Jepea mv, mv Sn ' mvA,' mvs t .
0603H8QUM qepea CD A H CD B yrJIOBhIe CKOpOCTH ,D;HCKOB nOCJle CTOJIK-
BOBeHH.R, 1 A Ii 1 B - COOTBeTCTBYIOHe MOMeHThI nHepItHH:
J A = imR, I B =  mRj.
3anHmeM aRKOH COXp8.HeHRH HMIIYJIhCa B npoeKIUIX Ha HRnpaB-
JIeHDS Ii H t :
mvcoscp = mv + mVBFJ'
mvsincp = mVt + mVt .
(1)
(2)
SannmeN 38.KOH COXp8.HeHRH MOMeHTa HMnYJIbCa OTHaCHTeJIbHO
TO'lKH 0:
mvb = mv(RA + R B ) + JAW A + JBW S '
(3)
MOMeHT HMnyJIhCS, coa.n;SBseMhIH CH.JIOH TpeHHJI, ,lJ,eACTBYIOII:{eA HS
11.RCR B, paseH
mVStRB ;:;: I B CD B .
(4)
no YCJIOBHIO 3SqH COCT8BJIHIOIItHe CKopOCTeH ,lI;nCKaB, nepneH)J;H-
KYJI.HpHhle JIHHHH, COe.D;HBHIOII:{en ux eHTphI, B TO''lKe conpHKOCHOBe-
HRa nO' MO.n;YJIJO pB.BHhI ,lq)yr APyry:
v:V - CDAR A = Vt + (OBRs,
(5)
a OTHOCHTeJILHhIe CKO'pocTH ,D;HCKOB B,lI;aJIh 3TOH JIHHHH O,D;HHSKOBhI ,D;O
H naCJIe COY,ll;apeHHH T.e.
VCOscp = VBn - Vn'
(6)
113 BhlpaiKeHH:ft (1) R (6) nOJI)"IseM:
v = O
V Bn = v coscp.
155

Ha Bldp8meHHH (2) H (5) nOJl}"lseM:
, 5.
vAt;::: -vslnq>,
6
, 1.
v Bt ;::: -v81nq>,
6
V sin q>
CDA;:::
:A
V sinq>
CD B =-.
3R B
lIpoeRItHH cKOpocTeA Ha KOOp.n;HH8THlde OCH OX H OY HecJlO)KBO
paCC'QUTaTL:
" , . 5vb 2
VAx = vAn COSq> + vAt smcp;::: ;
6(RA + R B )2
, ,. , 5Vb ( R A + R B t - b 2
VAJI ;::: -vAn smq> + VAt COScp ;:::
6(RA + R B )2
v% ;::: Vn COSq> + Vt sincp ;::: ( 1 _ vb 2 ) v ;
6 (RA + RS )2
,. , 5Vb ( R A + R B )2 - b 2
V1I ;::: -V Sn Slncp + VSt COscp ;::: .
6(RA + Rs)2
6) IIocJIe CTOJIKHOBeHHH KHHeTlf1IecKaH SHeprH.R .n;HCKa A paDBa
, 1 ( ,2 '2 ) 1 2 3mv 2 b 2
E A = -m VAx + VAll + -I A CD A ;::: 2 .
2 2 8( R A + R B )
KHHeTH'QeCKaH SHeprH.fI AHCK8 B paDua
E B ;::: .!.m( VB + VB) + .!:.lsCD ;::: .!:.mv 2 ( 1- Ilb2 2 ) .
2 2 2 12( R A + Rs)
156

1N\eYHapOAHag
.M3M'IeCKGR OnMMnMaAa
1995 r _, ABCTpanlt1l
3a.n;a'la 1
1. €POTOH C'tIaCTOTOH f HMeeT 84>eKTHBHyro HHepTUYIO Maccy m,
onpe.n;eJI.HeMyro ero 8BeprHeii. ITpe,o;nOJI03RHM, '1TO rp8.BHTaHOHHa.H
MaCca 4>oToHa paBHa HHepTuoA. B COOTBeTCTBHH C 3THM 4>OTOH, H3JIY-
qeHHLIii C nOBepXHOCTH 3Be3.n;LI, 6y.n;eT TepSITL SHeprHIO npH y,o;aneHHH
H3 rpaBHTHoHHOro nOJI.fI 3Be3.n;LI. IIoKIDKHTe, qTO eCJIH c.n;BHr qaCTO-
T:LI t!f c}>OTOHa npH yxo.n;e ero C nOBepXHOCTH 3Be3,lJ,:LI us 6eCKOHe'tlHO 60JIL-
moe paCCTOHHHe OT Hee MHoro MeHLme f, TO
M_ GM
,-- Rt? '
r,n;e G - rp8BHTOUHlUI nOCTOSIHHaSl; R - pa.D;HYC 3Be3AhI; C - CRO-
poCTb CBeTa; M - Macea 3Be3,lJ;h1.
CJIe,n;OBaTeJIbHO, 4KpaCHoe. CMen::teHHe K8.KOit-JIH60 H3BeCTHOH JIH-
BHH cneRTpa, H3MepeHHoe Ha 60JIbmOM paCCTOSlHHH OT 3Be3,lJ;hI, M03ReT
M
6h1Tb HCnOJIL30B8.HO Jl,J1..H H3MepeHHS! OTHOmeUH.H - . Sua.R R, MomHO
R
onpe,n;eJIHTb Maccy 3Be3,n;hI.
2. ABTOMaTH'IeCRHit KOCMlI'IecRHH Kopa6JIb 3anyeB C eJIhlO H3-
MepeHHSI MaCCH M H P8,Q;HYC8 R O.n;HOH H3 3Be3,n; B HameH ranaKTHKe.
IIpH npn6J1HJKeHHH Kopa6JI.H K 3Be3.n;e B,n;OJIb p8.}J;nyca 4>OTOBhI, H3JIY-
qaeMhIe C nOBepXHOCTH 3Be3,lJ;LI HOH8.MH He+ , Ha6JIIO,o;B.IOTC.R C nOMOm;hlO
pe30H8.HCHOrO B036Y>K.n;eHHSI nyq:Ka HOHOB He+ B HcnLITaTeJIhHOH Ka-
Mepe BHyrpH Kops6JI.H. PeaOHaHCHoe nOrJIOII:{eHHe npOHCXOAHT TOJIbKO
B TON. CJIyqae, eCJIH HOHhI He+ nOJIytIaroT CROpOCTb, HSnpaBJIeHHyJO K
eBTpy 3Be3ALI, 'ITO n03BOJI.ReT CKOMneaCHpOBaTb cRpaCBoe. CMeII:{e-
HHe (C,lJ;BHr 'IaCTOThI B CTOPOHY MeHbmHX qaCTOT).
CKOpoCTb (V ::;:; f3c) HOBOB He+ BuyrpH Ropa6J1JI OTHOCHTeJIhHO 3Be3-
,lJ;LI npH pe30H8HCHOM nOrJI0n::teHHH H3MepHeTCS! K8.K 4>YHKH.R paCCTO-
.HHHS d AO nOBepXBOCTH 3Be3,n;hI.
157

SRcnepHNeHTaJI:bHLIe aHHLIe npHBe.n;eHLI B Ta6nH:Q,e. HCnOJI:b3YR
IIX onpe.D;eJIIITe rpa4»JltleCRHN cnoc060M Macey M II p8,ZJ;IIYC R 3Be3JJ;hI.
OneHK8 norpemHOCTII He Tpe6yeTC.R.
Ta6AUija
ITapaM:eTp CKOpOCTH
p=, 10- 5 3,352 3,279 3,195 3,077 2,955
c
PaCCTO.RHHe OT nOBepx-
HOCTH 3Be3)];hI d, 10 8 M 38,90 19,98 13,32 8,99 667
):(.n.R onpeen:eHHJI R H M B TaRON 3RcnepUMeHTe Of)LfC{HO paCCM8T-
pHBalOT nOnpaBRY R '!IaCTOTe, BhI3B8.HHYIO oT.n;aqeii H3JIyq8.IOID;erO aTO.
Ma. (TenJIOBOe BH)KeHHe paCmHpJIeT nOJIOChI H3JI}"IeHUR, HO He C.n;BU-
raeT nOJIOmeHHe MaRCHMyMa, U CJIe.n;OB8TeJlhHO, MLI MomeM CqUTaTh,
qTO TeIIJIOBLIe a4xl>eRThI yq:TeHhI.)
3. llYCTh AE - pa3HOCTL 9HeprHH .ZJ;BYX aTOM:HldX ypoBReH Ror.n;a
aTOM: HaXOJJ;HTCH B nOKoe. llpe,ltIIOJIOmHM:, 'iTO nOKOaID;HH:CJI aTOM H3-
JIyq:aeT 4»OTOH H nOJIyqaeT onpe.n;eJIeHBy1O CROpoCTh. BLIBe.n;HTe PeJIR-
THBHCTCKoe BLIpIDKeHue /I,JIB. 3HeprHH hI H3JIyqeHHoro 4»oToHa B 38.BH-
CHMOCTH OT AE H MaCCLI m IICXOJJ;Horo aTOMa.
4. C.n;eJIaitTe 1iHCJIOByJO OneHJ(Y peJIHTHBHCTcRoro C,ABHra 'tI8CTOThI
( i) ...... AJIJI HOHOH He+.
Barn pe3YJIhTaT .u;OJDKeH 6hITh ropa3AO MeHhme fRpaCHoro. rpaBH-
TanHOHHoro cMeIIl;eHHa nOJIyq:eHHoro B II. 2.
,D;aHHhIe:
CROpOCTh CBeT8 C = 3,0 -10 8 M- c- 1 ; 9Hepl'H.R nOROH He+ m o c 2 = 4 x
13,6Z 2
x 938 MaB; anepreTJfllecKue YPOBHH no Bopy En = 2 sB; rpaBU-
n
TOHH8JI nOCTOHBHa.fI G= 6,7 -10- 11 H. ",2. Kr- 2 .
38Jtaqa 2
CKOPOCTh pacnpoc.rp8HeHIISI 3BYK8 B OKe8.He 38.BHCUT OT rJIy6HHhI,
TeMnepaTyphI II KOHneHTpaH COJIH B BO)];e. Hapuc. 110 nOKa38.Ha 38,-
BHCHMOCTb CKOpoCTH 3BYKa C OT rJIy6HHhI Z 11.J1H CJIyq8J1, KorJJ;a MHHH-
158

N8JIbH8JI CKOpocTb Co COOTBeTCTBye-r cepe,AH-
He paCCTO.JIBH.JI Mem.z:r;y nOBepXHOCTblO H
.z:r;HON OKeaHa. .AmI IIpOCTOTbI no.nO)KHM Z = 0
,lJ,JIHr.ny6HBhI. COOTBeTCTBY1OeA STONY MH-
HIJMYMY, Z = z. aa nOBepXHOCTH H Z = -Z" Ha
.z:r;He oKeaua. lIpH Z ;> 0 ClCOpocTh C onpe.z:r;e-
JIaeTCSI H3 BLlp8.)KeHHH
C == Co + bz.
lIpH Z iii:; 0 OHa onpe.z:r;eJIaeTC.JI BhIpame-
BHeN
B o6oHX CJIY'UiBX b = I :: I. T .e. 3Ha'l&-
HRe b p8BHO rpAAHeHTY CKOpOCTH 3BYKa no
rJIy6Hue. CqHTaeTC.R. "ITO b - nOCTOHHHaJI
BeJIH'lHHa.
Ha pHC. 111 nOKa3aHO Ce'leHHe OKeaHa -z"
B nJIOCKOCTH ZX, r.z:r;e OCIt X ropH30HTaJIlt-
Ha. Bo Bcex TO'llK8X IIJIOCKOCTH ZX 38.BHCH-
NOCTL CKOpoCTH 3BYX8 c(z) COOTBeTcTByeT
rpa4»HKY, fipHBeJJ;eHHONY Ha pHC. 110. HCTO"lHHK 3BYK8 S pacnOJIo.meH
B TOllKe z == 0, x = O. Hanp8.BJIeBHLIit 3BYXOBOH nyqoK, C03,A8.BaeMhlH
3THN: HCTO'lHHROM, OfiHChIBaeTca 3BYXOBLIM JIyqOM, BhIxo.n;aID;HM H3
TO'llKH S nop; Haq8JIbHhlN yrnoM
eo. K8.R nOK838HO H8 pRC. 111.
lIoCKOJILKY CROpOCTIt 3BYKa
NeHSleTC.JI B 38BHCHNOCTH OT Z,
3BYKOBOit IIy'IOK npen:ONJIHeT-
ca, 'lTO npHBo.n;HT K H3MeHe.
BRIO yrJIa e no TpaeKTopHH
JIyq:a.
1. lIoKWKHTe, 'IlTO HaTl8JIL.
HaH TpaeKTopRH 3BYKOBoro
.nyqa, HCXO,lJ,ero H3 TO'llRH S
H pacnpOCTp8HHIOerOC.H B
nJIOCKOCTH ZX, eCTIt .n;yra OK-
PY)KHOCTH P8,Zl;HYCOM R, r.z:r;e
C = Co - bz.
R _ - Co rnTd' 0 ,. 9 ,. 7t
""""<1110 .... 0.... - .
bsin90 2
z
Z.
c
c=co+bz
o
Co
c =co-bz
PRC. 110
o s
-Z"
PRe. 111
X
HX
159

2. lIoJIyq:HTe f1>OPNYJIY cop;epm8YIO z. Co If b )J;JIH MHHHM8.JIhHO-
ro 3HaqeHHJI yrJl8 8 0 ,lJ;J1.H JIyq:eH BanpaBJIeHHbIX BBepx KOTOp:&Ie He HC-
nLIT:&IB8IOT OTpameHH.R OT nOBepXHOCTH OKeaHa.
3. Ha pUC. 111 H306pameB npHeNHHR H, pacnOJIOmeHH:&IA B TO'tlKe
Z = O x = X. BLIBe)J;HTe BLIpWfCeHHe, cop;eJ»lC8D];ee b, X H Co onpe,AeJI.HIO-
ee H860p 3Ha'tleuHit yrJI8 8 0 , He06xo)J;HMLIX 1J)ISI Toro, 'tlT06hI 3BYXO-
BOit JIyq OT HCTO'tlBHK8 S p;OCTHr npHeMHHK8 H.
BhI MomeTe npep;nOJIOmHTb 'tlTO Z. H Zb tl;OCT8TOQUO BeJIHKH )J;JISI
Toro 'tlTo6hI HCKJIIO'IflITb B03MO)KHOCTh OTpameHHB OT nOBepXHOCTH If
)J;Ha OReaHR.
4. BLPlHCJIHTe 'tleTbIpe MHHHMaJIbHhIX 3UaQeHIUI eo npH KOTOphIX
npeJIOMJIeHHLIe JIyqH OT HCTOQHHK8 S .n;OCTHr8lOT npHeMHHKa H  eCJIH
X = 10000 M; C o = 1500 M. c- 1 ; b = 0,020 c- I .
5. BhIBHTe BhIpa)f(.eHHe .n;JISI BpeMeHH, Heo6xo,AHMorO AJISI pacnpo-
CTpB.HeHHSl3BYKa OT HCTOQHHKR S /1,0 npHeHHKa H B.n;OJIh TpaeKTopHH
JIyqa, COOTBeTCTBYIOeH MHHHMaJI:bHOMY 3HaQeHHIO yrJI8 eo nOJIyqeH-
HOMY B ll. 3. B&ItlHCJIHTe 3H8'1eHHe SToro BpeNeEH 1J)ISI yCJIOBHH, yxa-
38HHLIX B ll. 4.
C.rre/1,yIO:r:qee COOTHomeHHe MomeT BaM nOMOqb:
IsiX =m(tg ; }
HaHTe BpeNH pacnpOCTpB.HeHHSI 3B}'KOBoro JIyqa OT S P;O H no
JIHHHH Z = O. KaKoit H3 ,ABYX JIyqeit npH,lJ;eT nepBLIM: JI}"I, COOTBeTCTBY-
1O:r:qIIH eo = rr./2, HJIH .rryq C HaHMeHhmHM 3HSQeHHeM eo BhI'tIHCJIeUHhIM
B n. 4?
3a,n;aq:a 3
2a
1. ByA. COCTOHT H3 TBep.n;oro
IUiJIHH.AP8 P8.,ZJ;HYCOM a H .l+1IHHOH I,
cp;e.rr8.HHOrO H3 JIerKoro 0AHOPOP;-
BOro M8TepHaJl8 UJIOTHOCThlO d H
OW{OpOP;HOro mecTJ(.oro CTep)KH.R
npHKpenJIeHHoro CHH3Y K :qHJIHH-
npy B cepeHe era o6pa3YIOeii
(puc. 112). Macca crepmHSI paDua
M8Cce :qHJIHIIJU)8, ero ,AJIHH8 paDua
JJ;H8MeTpy JIHH,.IU>a, a ero nJIOT-
HocTh60JIhme UJIOTHOCTH MOpcIWit
BO.n;hI. Byit nJI8BaeT B MOpCKOH
BO,lJ,e nJIOTHOCTblO p.
PRC. 112
160

.
.
-----
.
.
---------
.
..
PIIC. 113
PHC. 114
lIoJI}"lHTe COOTHomeHHe, CBH3hIBalOII1;ee yrOJI a B nOJIOmeHHH paB-
HOBecHSI (CM. pnc.112) C OTHOmeHHeM: UJIOTHOCTeii . 06'beMOM CTepm-
p
HS{ npeHe6pe1lh.
2. ECJIH 6yA BCJIe.u;CTBHe B03M}TID;eHHit norpymeH BepTHK8.JIhHO aa
M8.JIOe paccToSlHHe Z, TO OB ua"UIHaeT COBepmaTL BepTHK8.JIhHhIe Kone-
6SHHSI OTHOCHTeJILHO nOJl<»KeHIUI p8BHOBeCHH (puc. 113).
Onpe;D;eJIHTe qaCTOTY 3THX BepTHK8.JIhHLIX Kone6aHHH 1lepe3 a, g H
a, r.u;e g - YCKopeHHe cBo6o;D;soro ll8,Zl;eHH.R, npep;nOJIar8..H, 1lTO B]1II.R-
HHe p;BHmeHHH BO.n;LI sa ;D;HH8MHKY KOJIe6aHHH 6ys: TaKOBO, 1lTO ero
a4>fPeKTHBH8..H Macea yneJlU'IHBaeTC.H Ha 1/3 MaCChI 6y.H.
BLI MomeTe npe.n;nOJIOJKHTIIo, 1:ITO a He MMO.
3. B npe.u;nOJIOmeHHH, 1lTO :Q;HJIHH):UJ 6ya MomeT COBepmaTb KOJIe-
6aHHSI (K8118.ThC.R) OTHOCHTeJILHO cBOeit :Q;eHTp8.JIhHOii ropH30HTftJIhHOH
OCH, onpep;eJIHTe qaCTOTY STHX Ka1:l8HD Qepe3 g n a, npeHe6peraSl B
STOM c.nyqae p;BHmeHHeM BOP;bI H ee B.H3KOCThlO. Y rOJI OTKJIOHeHH.H e
npn KS'laHH.HX Cq}ITaTL ,lJ;OCTftTO"lUO M8.JIbIM (pHC. 114).
4. Hs 6ye YCTaHOBJIeHbI 8.KCenepOMeTpLI KOTophIe MOryT 1I3MepHTh
nepnop;bI BepTHK8.JIhHbIX Kone6aHHH II Ka'laHIIH 6YH II nepep;8B8Th lIB-
4>opM8:Q;HIO aft 6eper. H3MepeHH.R nOKft38.JIH, "lTO B OTHOCIITeJIhHO cno-
KOA:HOii BOp;e nepHo;D; sepTHKaJILHLIX KOJIe6aHD T 1 == 1 c, a nepnop; Ka-
TIaHHit Tl s::=: 1,5 c.
Ha OCBOBe 3THX p;ftHHLIX nOKIDKHTe, 1lTO yrOJI a s::=: 90 0 . Onpep;eJIHTe
pa,zJ;nyc I\HJIHH.n;pa 6YH, ero nOJIHYJO MaCCY, ecJIH .n;JIHU8 :Q;HJIHHAPa l P8.B-
Ha ero pa.n;nycy a (BbI MomeTe HCllOJIL30BaTh ,lJ,JISI O:Q;eHOK TIHCJIOBLIe .u;8H-
HbIe: p = 1000 Kr. M- 3 , g = 9,8 M. c- 2 ).
6-379
161

PeweHMII
3a'l8 1
1. T8.K K8K 3HepI'HJI<}>oTOU8 c qaCToToi: I paDua E :;;; hI, TO, npHMe-
HHB aSKOR COXpaHeHHII DueprHH P;JIH B3aBMo.n;eitCTBHII (}>oTOHa c nOJIeM
THroTeHHII 3Beap;h1, nOJIyqHM
hf _G Mm = h(f+Af),
R
rp;e m - Macea <}>OToHa. Ee JIeneo HaHTH H3 BLIP8'KeHUa
mc 2 = hI,
(1)
(2)
TaK KaK At « f.
H3 BLIpameHH.H (1) CJIep;YeT
hA t =- G Mm
R'
a C ywIeToN BLIpWKeHHJI (2) nOJl}"lHM:
h!!J. f = _ OM hf
R c 2 
OTKyp;a
Af_ GM
,-- &2 .
2. H3MeueBHe DUepl'HB 4>OTOH8 llpH nOp;'beNe OT r,.n;o T, (rp;e r,-
H8Q8JIbHOe p8CCTOHHHe OT c)K>ToBa /1,0 [(.euTpa 3Be3,lJ,LI, 8 r,- KOUeQUOe
paCCTOBHRe OT 4>oToHa 1J.0 D;eBTpa 3Be3,D;LI) paDHO
hf, -hI, = _ GMm, + GM .
'i 'i
¥TITH, 'lITO In, ::t: m, == h ' nOJIytlHM
c
hI, - hi, = - Ghfa ( ! _1. ) ,
c r, r,
OTKYp;&
If = 1- eM (.!. _.!. ) .
It c 2 Ii rf
162

B OfiHC8HHOM B YCJIOBHH 3aqH 8KCnepHMeHTe r l :; R; r, ::;;: R + d,
Cne.D;OB8TeJIbHO,
If :;::;1- GM ( _ ) .
1, c 2 R R+d
(3)
Bcne,ACTBHe peJUITHBHCTCKoro p;OIIJIepOBCKoro s4»4>eKT8, CBII3aH-
HOro c .n;BlDKeHHeN HOHOB He+, npOHCXO,AHT H3MeaeHHe Tl8CT01'M Ha-
6JIJO,u;aeMLIx 4>oTOHOS:
f, ; p-IJ .
 l+P
llpH P « 1
1,  1 - p.
1.
(4)
(BhIpaaceHHe (4) CJIep;yeT H Ha KJIaCCeCKoro ,AOnJIepOBCKoro 84>-
«}>eKT8.) Ha BhIp8JKeHUii (3) H (4) KMeeN
(3:;::; GM ( .!._ ) .
c 2 R R + d
(5)
TIpeo6paaYH BhIpS)ReHHe (5), nOJI)"IHN
1 Rc 2 ( R )
j3;;;: GM (1+1 .
(6)
lIoCTpoHB Ha OCHOB8HHH 3KcnepHMeUT8JILHhIX A8HBhIX rp8.pHK
 ) (PRc. 115) R cpaBHlIBWI C Bblp8>I<eBReM (6), nOJIy'IRM:
111
npH i3 = 0 ,AJIHH8 0Tpe3K8. paBHa OB = d = - R ;
1 1 Rc 2
npH - :; 0 wum8 0T}>e3K8 paBH8 AD = - :;::; - ;
d J3 GM
OA R.c.
tga:;::;-=-.
OB GM
163

1
13
B_
1
-IE
.;" ....A
-c;
o
1
d
PRC. 115
o LE)
hf

mo
mo
PRC. 116
H3Mep.HHllOrpa4>HKytga(tga=3,2 -10 12 M)HOA(OA=2,9 -10 4 ),
JIerKO onpe.n;eJIHT:L R:
R = tga R = 3.2 . l012N: = 1104.10 8 M.
OA ' 29 .10" '
)J;8.JIee HftXO.n;HM Maccy 3Be3,D;hI:
M -. M - 1.104.10 8 . (3-10 8 )2 Kr - 5 11 . 10 30
- ,- - , Hr.
G. OA 67 .10- 11 . 29 .10 4
3aMeTHM, 'lITO C)J;BHr qaCTOThI 4>OTOH& npH ero yxoAe C nOBepxHoc-
TH ,lJ,aHHOH 3Be3,lJ,hI aa 6eCKOHeQHO 60JIbmOe paccToSlHHe OT Bee p8Ben
I1f _ _ GM _ _ 67 .10- 11 . 511.1030 _ -10 5
f - Re 2 - 11. 10 8 . (3 .10 8 )2
3. H306pa3HN pacn nOKOHerOCH ftTOMft (pHC. 116). llpHMeHHM
3aKOH coxpaHeHHH 3HeprHH )J;JIH aTOMa 11.0 H nOCJIe paCll8,lta:
11lu C2 =  p2C2 + C4 + hf,
(7)
r)J;e HMIIYJIbC ftTOMR nOCJIe paCIIB,Aa p8.BeH no MOJIIO HMIIYJIhCY <}>oToHa:
hI
p==-.
c
(8)
164

M3BHpameBuA(7)u(8)uonyqHM:
(m o c 2 - hl)2;: (h/)2 + n\fc 4 ,
2 '2
1110 c 4 - 2hfm o c 2 = 17\j c 4 ,
h/(2m o c 2 ) = (- ni:)c 4 ,
hf(2m o c 2 )=(m o - )c2(mo + )c2.
3aMeTHM, '1TO l1E = (m o - )C2. CJIe,n:oB8TeJILHO,
hf(2m o c 2 ) = AE [2m o - (m o - )] c 2 = f1E (2m o c 2 - LV:)
H
hf==AE ( l- J .
2mo C2
(9)
4. C yqeToM COOTHOmeHH.H hfo = /)£ II BhIpameHH.H (9) nOJI}"IHM
Af =
fo 2moC2.
.D;JI.H HOHa He+ npH uepexo.n;e co BToporo ypOBHH fia nepBLIA
M = 13,6Z t -  JaB = 13,6. 4(1- }B = 4O,8aB.
SHeprHH nOKO.H HOHa He+ m o c 2 = 3,752 .lO!l3B, CJIeAOB8TeJILHO,
( : )_ = 5,44.10- 9 .
3aMeTHM, 'IlTo 3Ta nonpaBKB, BhI3B1U1Ha.a OTAs-qeit H3JIyq8IOero
aTOMa, MEoro MeH&me t:KpacHoro. rpaBHTaHoHBoro CMem;eHH.R 4>0-
TORa ( 11' ) = 10-5 (MbI ee OIteHHJIH B u. 2), n03TOMY B 3KcnepHMeH-
, Itp. 01
Tax C rpaBHTan;nOHHhIN t:KpaCHhIM. CMeeHHeM M03CHO 3TO:A nonpaB-
Koii npeBe6peQL.
165

38.Q;8-qa 2
1. COrJI8CBO 38KOHY npeJIOMJIeUH.R,
sinG C
-=-
sin 8 0 Co
P8CCNOTpHN 3JIeNeUT J1yt18 dB, KOTOpLIH NomBO C'IlHT8TL ,lJ,yroii OK-
PY>KHOCTH P8AHYCON R (pHC. 117):
(1)
dB
ds==Rcte, -==R.
de
I1po,lJ;H4>tPepeuIUIPyeM BLIpaaceHHe (1):
G o. sinB(I de
COB u.u= - .
Co
Y'IlTB, qTO C = Co + bz H de = bdz DOJIytlHM
9 "0. - sinG o bd
COB I.W - - z.
Co
CJIe.u;OB8TeJlhHO,
Co 1
dz :::::; --C06e de.
sin 9(1 b
(2)
MLI NomeN T8K>Ke 3BnHC8Th
dz = ds cosS
(3,ZJ;ecL p;yry dB C"fIUTaeM npRMoii JlHHHeit).
dz
Puc.tt7
166
(3)
z
z, - - - - - - - - = ::;:.
-,,,,-
,
,
o
R sin 8 0
x
Pue.tt8

...---....,............
-....
.......
.....
...
...
"
"
',X
H X
z
he. 119
H3 BMpameHUfi (2)H(3)nonyqaeM
Co 1
d8;;;: e - b d9,
SIn 0
oTKy.o;a
ds Co
de ;;;: R - b sineo const.
(4)
2. H3 pRC. 118 BH,D;HO, '1TO
z. = R - R sin8 0 ;:::; R (1 - sine o ).
(5)
lIo.n;cT8.BHB B BMpameUHe (5) 3HaqeHHe R R3 BhlpaaceHR.R (4),noJIY-
qHM
z. =  (1- sineo),
b sm9 0
oTKy.o;a
e . Co
o = &resm
bz. + Co
3. Ha pRc.1I9 BHBO, !lTO
2c o 2c o
X == 2Rcos9 o - b . 9 cose o = - b ctg90 J
SIn 0
OTKYAa
bX
ctg9 0 :::;-.
2co
167

z
s
x
Puc. 120
Bo3M02ICHLI H p;pyrHe TpaeKTopHH nyqa JJ.ng AOCTH2ICeHHSI npHeM-
HHKa H (pRC. 120).
X 2c
Ha pHCyHKe 2":; 2Rcose o :; b 0 ctg9 0 '
OTRYAB
bX
ctge o = 4c o .
7t
Bcero AJUI 9 0 < 2: MomeT 6LITb H860p 3HaqeHH:ft 9 0 , Heo6xoAHMbIX
1I,J1SJ. Toro, QTo6LI 3ByKOBO:ft JI}"I OT HCTOQUHKa S AOCTHr npHeMHHKa H:
( bx ) 2nco
8 0 = arcctg - ;;;; arctg-,
2neo bx
rAen == 1, 2,8,4,... .
1t
8aMeTHM, 'lJTO, KorA8 n = 00, 9 0 == - . B STOM CJIyqae JI}"I AocTliraeT
2
npHeMHHKa no np.JIMoii.
4. )];n$l3HaqeHHft: n = 1, n = 2, n = 3, n = 4
9 01 = 86 19°; 9 02 = 88,09°; 8 03 = 88,73°; 9 04 = 89,04 0 .
5. 113 pHC. 121 BHAHO (TpaeKTopHs 1-2-3):
3 ads 2 ds
t== fdt= f-C=2 I-c'
1 1 1
I[
2cJs 2Rd8
Ie= I c .
I e
C
lIop;cT8.BHB R = - b . e ' nOJIyqHM
Sln

1 2 dO
t l2 == - J.
b eo sme
168

2
Z V..
S 1 H X
PRC. 121
¥=SI. "TO I :::x = m( tg ; ). 3lUIHmeM BLiplUi<eHHe 11JU< t 12 B BH)le
1 ( e )I 1t/2 1 8 0
t 12 :;;; - lntg- = - -lntg-,
b 2 90 b 2
a TaK KftK f l3 = 2t 12 , TO
2 eo
3 = --lntg-.
b 2
ITOACTaBHB 3HaQeBUe 9 0 = 9 01 = 86, 19°, nOM t l3 = 6,6617 c. Bpe-
M.JI npOXO)KJJ;eBIUI 3ByKa no oceBOMY' .nyqy paDRO
X 10000 M
t == -; t:;;; = 6,6666 c.
Co 1500 M/c
TaKHM o6paaoM, BpeMSI npoxoacAeHHSI 3ByK8 no oceB0MY' JIyqy 60JIh-
me, qeN no ,lJ,yre oKpymnocTH AJIa n =:: 1.
3a,n;aQa 3
1. Macca ItHJIHHp;pa mil. no YCJIOBHJO paSHa Macce me CTep)KHSI:
mu = me = 1ta 2 Id. 06m;aa Macca m
6YJl (IJ;HJIHH,D;pa CO CTep>KHeM)
p8BH8
m = 21ta 2 ld. (1)
ll3 yc.nOBHSI paBHoBecHH 6ySl
cnep;yeT
mg=F A , (2)
I'p;e FA - apxHMe,n;oB8 c:HJIa ,neH-
CTBYX>Iq8g aB 6yi!;
FA = pSFAg, (3)
I',n;e S - nJIoIq8,Ab cerMeHTa H306-
paJKeHHoro Da pnc. 122.
p
PRe. 122
169

H3 reoMeTpUU .RCBO, 'QTO
S :; a.a 2 - .!.a 2 sin (20.)
2
(4)
(TaR K8K Z Mano, TO H3MeBeHueM yrJIB a MLI npeHe6peraeM)a
lIo.n;CT8.BHB Bhtpa)l(eBH.R (1), (3), (4) B YCJIOBUe pBBHOBecHSI (2). no-
JIytlHM
2rr.a 2 ldg = Plda 2 [ a -sin(2a)l
OTKYAB
d :; -1-sin(2a).
p 21t 47t
HJIH
d 2a - sin (2«)
p 4n
(5)
2. lIorpY3HM 6yii ua He60JILmOe paccToSlHHe z. npH STOM Ha Hero
H8'tJ:HeT p;eHCTBOBaTb .l{OnOJIHHTe.nLH8SI apXHMep;OB8 CHJIa F.ii. HBllpaB-
JIeHHaJi BeprHRaJIbHO BBepx (pUCa 123), npoeK{tlul KOTopon HB OCh Z
peHa
F. :; -pg . 2a (sin a) lz.
HCnOJIL3Y.R BTOpoit saROH HblOToHa, aBllumeM:
m*z" :; - pg. 2alz sin a,
rp;e m * - s4>4>eKTHBH8H MaCC8 6YBj
m- :; m =. 21ta 2 UJ :;  1ta 2 Id..
333
lIo.n;cTBBHB BhIpa)l(eHHe (7) B yp8.BHeHHe (6), nOnytlHN:
!7ta 2 Idz" :; -p g . 2alz sin a .
3
(6)
(7)
1tadz" = -p gz sin a..
3
(8)
Ha nOJIytleHHoro ypaBHeuHSI THIIR
z" = - OJ2Z
CJIeP;YeT, q:TO 6yji COBepmaeT BepTHK8.JIbBhIe rapMOHH'tIecKHe KOJlOOa-
HH.R C q:aCToToit OJz:
2 3pg sino.
OJ -
Z 41tda.
170

I
I
Z I
2a sina
Z mcg
Puc.123 PHc.124
d
nOp;CT8BHB B aTO BhIpameHHe 3H8't1eHHe - (tl>oPMYJIB (5)] nO.rryqHM
p
co 2
z
3g8m a
47ta
4n
2a - sin(2a)
3gsina
[2a -sin(2a)]a t
oTlCy.u;a
m =
z
3gsina
a[2a -sin(2a)] .
(9)
3. lIpH OTKJIOUeBHH 6y.H UB M8JILIit yrOJl e H8 Hero .n;eiiCTByeT NO.
MeRT M CHJIhI T.HmecTH CTepmHft (pHC. 124):
M = -mcg. 2a sin e.
Y-qT.R. TITO yrOJI e OTIeHh M8JI no YCJIOBHIO. nOJIyqHM
M = -2mc: ga e.
(10)
B COOTBeTCTBHH C OCHOBHhlM aaKOHOM .J];HH8MHKH BpB.I:qaTeJIbHOrO
ABHmeHHH TBepp;oro TeJIa aanHmeM:
M= le"
(11)
r1I.e I - MOMeHT HHepItHH IUIpa H CTep)KHj{ OTHOCHTeJIbHO TO'lKli O.
0" - yrJIOBOe YCKopeHHe.
171

MOllJleHT HHepH CTepmH.R OTHOCHTeJIhHO OCH nepneHJ1;HKYJIHpHO:H
CTepmHIO II npoxo,Am:IJ;eii 'Ilepe3 ero cepe1J;HHY  pueR
2
I _mea
1--.
3
STOT peaYJIbT8T MOO .n;OKaaaTh:
a M 8 D 2
I. = I (me dx)x 2 ;:;: I = mea .
-0 2a 6a -0 3
B COOTBeTCTBHH C TeopeMofi illTefiuepa MOMeHT HHepD;HH CTepmH.R
OTHOCHTeJIhHO TO'llKII 0 puen
2
2 mea 2 13 2
Ie = II + me (2a )::: - + 4mca = -mea . (12)
3 3
MOMeHT HHepD;HH D;HJIHpa OTHOCHTeJIbHO D;eHTpa paBeH
1 2
In. = -mila.
2
H3 BhIpameHHii (11) II (12) 1I0JIyqaeM
(13)
13 2 1 2 29 2
I = Ie + Iu, = gme a +2' m e a ;:: 6 mea .
II01J;CTaBHB BLipsmeHHH (10) H (14) B ypUHeHHe (11), 1I0JIytlHM
(14)
2 e 29 2 9 "
- mega = -mea .
6
STO JJ;HcpcpepeHD;H8JIbHOe yp8BHeHHe THlla e" = - ro e CBH,D;eTeJIbCTBY-
eT 0 TOM, 'tITO 6yii 6y.n;eT COBepmaTh rapMOHHQeCKHe Ka'I8HHSI C D;HKJIH-
qeCKO:H 'tIaCTOTOH, paBHOH
mo=  .
4. IIo YCJIOBIIIO aa,n;a'IH, T 1 = 1 c; T 2 = 1,5 c, a cpOPMYJIhI JJ;JUI paCqeTa
nepHo,u;OB T. H T 2 6LIJIH BLIBe.n;eHLI B npe,AbI,lJ;ym;Hx TIyJlKT8X:
T. - 2 [2a -ain(2a)]a
1- 1t ,
3gain a
 9a
T 2 = 2n -,
12q
172

( 7J. ) 2 4 2a - sln(2a)
oTKy.u;a - =-.
T 2 29 sin a '
a
2a - in(2a) _ 29 ( 7i ) 2 = 29 ( ..!- ) 2  3,22.
sma 4 T 2 4 1,5
7t 2a - sin(2a)
ECJIH a =- 2 ' TO . - 7t.
Sina
7t
Tax K8.K 3,22 ::t: 7t, TO yrOJI a  -.
2
P&JI;HYC a rtHJIHH.n;pa MOJKBO HaitTH us BhIpameHHJI
a = ( 72 ) 2 128 = ..!.g ( 72 ) 2;
21t 29 29 1t
a  0,23 N.
T 1t d 2a-sin(2a) 1
a1\!tax a2' TO p= 47t 4.
Macey 6y.R onpe.u;enHM H3 BhIpameHHH
m = 21ta 2 ld.
Y-qT.H, '1TO, no YCJIOBHIO, l ;::: a, nOJI}"IHM
m;: 21ta 3 d == 7t a3 p .
2 '
m  19,11 Kr.

IIMeYHGpOAHaA
cl»M3MlfeCKGA OnMMnMGAa
1996 r _, HopBerHI1
3ap;aQa 1
1. lIaTb pe3HCTOpOB conpOTHBJ'leHHeN 1 OM Kam)J;LI:i: coe)J;HUeHLI
T8.K, K8K nOKa38HO U8 pRC. 125. ConpoTHBlIeHBe Coe,a:mmTeJIhHhIX npo-
BO.n;OB npeHe6pemHMo MaJIO. Onpe.u;eJIHTe pe3YJIbTHpYIOm;ee conpo-
THBJIeHHe R Me)K.n;y TO'IK8MH A II B.
2. JILImHHK cTspTyeT H3 COCTOaHH.R nOKOSI B TO'lKe A H CKOJIL3HT
no CKJIOHY 6e3 nOBopoToB R npHTOpMWKHB8.HHii. Koa4>4>HItHeHT Tpe.
HilS p8.BeB IJ.. Kor.n;s OB OCTaB8BJIHBaeTCH B TOQKe B, ero ropH30HT8JIb-
Boe CMeeHHe p8BRO 8. KSKOBS p83HOCTL BhICOT h Mey TO"lK8MHA II
B (pRC. 126)1 CKOPOCTh JIhI)KHHK8 M8JI8 HaCTOJILKO, '1TO MomBO npe-
He6peQL .n;onOJIHHTeJILBhIM .n;8BJIeHHeM Hft CHer, B03BHK81Om;HM BCJIe,Q;-
CTBHe KPHBH3BhI TpaeKTopHH. ConpOTHBJIeHHeM B03.u;yxa H 38BHCHMO-
CTLIO J.1 OT CKOpOCTU T8.Kme MOmHO npeHe6pe"lL.
3. TenJIOH30JIHpOBaHHhrlt KYCOK NeT8.JIJI8 H8rpeB8eTCB npH aTMO-
cfl>epBoM ,Q;8BJIeHHR 3JIeKTpR'tleCKHM TOKON T8KUM 06P830M, TlTO M:Om;-
HOCTb P HCTO'IHHK8 TOKa OCT8eTCSI nocTOSIHHOfi. 3TO Be,tteT K POCTY TeM-
nep8TypLI T MeTanJIa B 38.BHCHMOCTH OT speMeHH t no 3aKOHY
T(t) = To[l + aCt - t o ]l/4.
8,Q;eCba, toH To - nOCTOSHHhIe BeJIHHLI.
OnpeAeJIHTe 38.BHCHMOCTh TenJIOeMKOCTH Cp(T) NeT8JIJI8 OT TeMue-
paTyphI B 06JIaCTH TeMnepBTyp COOTBeTCTBYIOm;eH .n;8.HHOMY ODldTY.
4. '1:lepHBSI nJIOCKBSI nOBepXHOCTb, uaxo.n;sm;aSlcSI npH nocToHHHOH
TeMnep8TYpe T l' napBJIJIeJIbHB ApyrOH qepHoA nJIOCKOH nOBepXHOCTH,
A B
PHC. 125
174

A
I J
8
PRe. 126
1
2
-...Tl
[ijl'!
b I
PRe. 127
HaxO,lte:A:c.a: npH nOCTOmlHOii 60Jlee RH3RoA TeMDepaType T 2. Mem-
1I.Y ,lJ;BYM.fI nOBepXHOCTSfMB H8XOP;HTCII B8.KYYM.
QTo6hI }'MeHhmHTIt TenJlOBOA nOTOR H3JI}"IeHHB, Nem,tty nJlaCTU-
BaNK nONecTBJIH Ten.noBoA &ICPaH, COCTOJIIIpdt H3 ,ltBYX TOHKHX ep-
HLlX wrACTHH, napaJIJ'IeJIltHhIX HCXO.n;HhIM nJIaCTHBaN (pRC. 127). lIo
HCTeqeHRR HeKOToporo BpeNeHH YCTaHaDJIHB8eTC.H CTIlItHORapRoe co-
CTO.RRHe. Bo CKOJIltRO paa YMeRbmaeTC.R TenJIOBOA nOTOK 3a CqeT Ten-
JlOBOro 8KpaHa? KpaeBhINH &4HPeKTaMH, 06ycJIoBJIeHHhINH KOUeQHhI-
NH p83Nep8MH nJl8CTHB, MO)l(.RO npeae6pe"llt.
5. .ItBa nPJIMbIX H OqeBb JIHHHLlX HeN&rHHTHhIX npoBOAHHKa C+
R C, H30JIHpoBBHHldX ,lU)yr OT .n;pyra, aecyr O,lJ;HB8.KOBlaIe TOKR 1 COOT-
BeTCTBeSHO B nOJIO)l(.HTeJIhHON H OTpSrtBTeJlhHOM Hanp8.BJIeBHSIX OCR
Z. CeQeHH.H npoBO,D;BHKOB (aamTpHXOBaHhI sa pRC. 128) orp8HHlleHLI
OKpyawOCT.RMH ,lJ;H8MeTpoN D, JIemaJ:ItHMH B nJIOCKOCTH XY, npH1leM
paCCTO.HHHe Me ax IteSTpaMH paDRO D/2. TaKBM 06paaON, DJIOm;a-
y
PRe. 128
Puc. 129
175

,lJ,H nOnepe'IBOrO CeqeHHS npOBO,D;OB p8.BHld ( 1 7t +  J3) D2. TOK B
K8..)It,D;0M npOBO,D;HHKe p8.BHOMepHO pacnpe.D;eJIeH no CeqeHHIO. Onpe,D;e-
nHTe HBJJ;YKQ,HIO B(x, y) MarHHTHoro nOJIS B npOCTp8HCTBe Mem,ltY npo-
BO,D;HHKaMH.
3ap;aQa 2
B npOCTp8HCTBe Mem.D;Y napoA K08RCH&JIbHldX ItHJIHH,o;PHqecKHX
npOBO,ltHHKOB C03J];8H BSKYYM. Pa.n;HYc BHyrpeHHero ltHJIHH,ltpa paBeH
a, 8 pa.zJ;HYC BHemHero ltHJIHH,ltpa p8.BeH b (pHC. 129). Ha BHemHeN :Q,H-
JIHHJJ;pe, H83hIBae:MOM 8HO,ll;OM, MO.H<HO C03P;8TL nOJIOmHTeJILHhIit nOTeB-
:Q,HaJI V no OTHomeHHIO K BuyTpeHHeMY ItHJIHHJWY. CHcTeMa nONem;e-
H8 B CTSTHqeCROe O,ll;Bopo,ll;Uoe NarHHTHOe nOJIe C HYK:Q,Heii B, ns-
paJIJIeJILHoit OCH :Q,HJIHHJq>OB H Hanp8BJIeHHoit K '1HTBTeJIlO.
HCCJIe,ltyeTCSI ,lI;HH8MHKft 3JIeKTpOHoB C M8CCOit nOKO.H m H 38p.H-
)];OM -e. 8TH SJIeKTpoBLI HcnycK81OTC.H nosepxHoCTLIO BHyrpeHHero :Q,H-
JIHHp;pa.
1. IIYCTL nOTeH:Q,H&JI BHemBero :Q,HJIHHJU>8. p8BeH V, HO B = O. 8JIeK-
TpOH HcnycK8eTcSi nOBepXHOCTLIO BHyTpeHHero :Q,HJIHHApa C npeue6pe-
.>KHMO N8.JIOit CKOpOCTLIO. Onpe,lteJIHTe ero CROpoCTL v B MONeHT, Korp;8
3JIeKTpOB ,lI;OCTHr8eT 8HO.n;8. J];aitTe OTBeT ,ltJISI ,lI;BYX CJIytlaeB: B BepeJIX-
THBHCTCKON H peJUITHBHCTCRON npH6JIHmeHHsx.
B n. 2-6 STOit 3a;:J;8'IH HCnOJlL3yeTCS: HepeJI.HTHBHCTCKoe npH6J1H-
meHHe.
2. nYCTL Tenepb V = O. HO npHcyrcTByeT O,ll;HOpO,ll;Hoe N8rHHTUOe
none C HHP;YKItHeit B . SneKTpoH cTapTYeT C H8TI8JIbHOH CKOpOCTLIO V o
B P8,lJ,H8JIhHOM H8np8.BJIeHHH. npH HHJI;}'K:Q,HH N8rHHTBoro nons, npe-
BLImarom;eit KpHTH'IecKoe 3Ha'leHHe Be. 3neKTpOH HHKOr,lta He ,ltOCTHr-
HeT BHO)];8. H306pS3HTe cxeM8T1NeCKH TpaeRTopHIO 3JIeKTpOH8, Korp;a
B HeMHoro 60JILme 'IeM Be' Onpe.u;eJIHTe Be.
B ll. 3-6 ytlHTldBseTCSI K8K nOTeHItH8JI V, T8R II O,ltHopOAHOe M8r-
HHTHoe nOJIe C HIIJ.{yK:Q,Heii B .
3. MamHTHoe nOJIe coa,o;aeT HeHYJIeBOH yrnoBoH MOMeHT L 3JIeKT-
poBa OTHOCHTe.nLHO OCR :Q,HJIHH,u;P8. HanHmHTe ypaBHeBRe 1JJI.SI CKOPO-
dl
CTH H3MeHeHHH yr JIOBorO MOMeHTa dt . fiOK8..>KHTe, 'ITO H3 HerO CJIe,n;y-
eT  'ITO BeJII{"qHB8
C=L- keBr
176

(k - 6e3pa3MepuLIA Koa4>4>B:qHeHT, r - paCCTOSlHBe OT OCB ItHJIBB,lUJa)
OCT8eTCSI nOCToSlHBOit B npo:Q,ecce .o;BH2ReBHSI 8JIeKTpOH8. OnpeAeJIHTe
3aaQeUHe k.
4. PaccMoTpUM 8JIeKTpoR, HcneHHLlit C npeRe6pemHMo MBJIOit
CKOpoCThlO BHyrpeHHHM :qHJIHBAPOM. STOT SJIeKTpoH YA8JISleTC.H OT OCH
:Q,HJIHHApa aa MaKCHMWIhHoe paCCTOSlHBe r m' He AOCTHrM 8HoAa. On-
PeAeJIHTe CKOpOCTh 3JIeKTpoHa VB TO'lKe M8.KCBM8JlbHOrO YAaJIeHH.H B
3aBHCHMOCTH OT r m.
5. MLI XOTHY HCnOJIL30BaTh MarHHTHOe nOJIe p,JISI pery.nBpoBKH
BBoAHoro TOKS. lIpH UH)J;YK:Q,HH B, npeBLImaIOII{.ei: Be' 3JIeKTpOH, HC-
eHHLIit C npeae6pemHMO MftJIOH CKOpOCThlO, He AOCTHrHeT aHo.o;a.
OnpeAeJIHTe Be.
6. ECJIH 8JIeKTpoHIaI BhICB060m.n;aIOTC.H C nOBepXHOCTH BByTpeHHe-
ro :Q,HJlHHApa 3a C'IeT HarpeBa. TO B 06II{.eM CJIyqae 3JIeKTpoH aa STOH
nOBepXHOCTH HMeeT BeHYJIeByx> HaqWIbHyIO CKOpoCTL. nYCTb COCTaB-
JI.HIOII{.8.R H8TJftJIhBOH CKOPOCTH. napBJIJIeJlLHWI if  eCTL VB. a COCT8B-
JI.HIOII{.He. nepneHJJ;HI<YJISIpHlaIe K B . ecTb Dr (B p8,ZJ;H8.JILHOM HanpaBJIe-
HUH) H V" (D 83HM}'TftJILHOM Hanp8BJIeHHH. T.e. nepneHAHKYJI.HpHO pa-
AHEUILHOM}THanpaBJIeHHIO). Onpep,eJIHTe B 8TOH CHTYawm RpuTlfClecKoe
3HaqeHHe Be HH:Q,HH M8I'HHTHOro nOJI.H. npH KOTOpOM 8J1eKTpoa He
AOCTHraeT RHop,a.
3a,D;aQa 3
B 38,ZJ;a'tJe paCCMaTpHBaIOTCJI HeKOTopLle 06II{.He CBOHCTBa OKeaH-
CKHX npHJIHBOB H OTJIHBOB aa BeMJIe. J];J1JI ynpOII{.eHH.R np06JIeMIaI CAe-
JIaHhI CJIe,AyIOII{.He npeAIIOJlOmeHH.H:
1) BeNJUI H JIyua C'lHTBIOTCSI H30JIHpOB8.HHO:H cHcTeMoii;
2) pacCTO.HHHe Mem,zJ;Y 3eMJIeA H JIyuoit C'IUTaeTC.H nOCTOJIHHLIM;
3) C"lHTaeTCSI. 'fIT0 BeMJI.H nOJIHOCTLlO nORpLITa OKeaHOM;
4) npeae6peraeTC.H AHHaMHQeCKHMB a«l><l>eKTaMH. 06YCJIOBJIeHHhI-
MH BpaII{.eHHeM 3eMJIH BOKpyr cBoeH OCB;
5) rp8BHTau;HOHHoe npHTHmeHBe 3eNJIH CQHTaeTC.JI TaR, K8K eCJIH
6LI BCH ee Macca 6hIJl8 COcpe,lJ,OTO'leHa B :Q,eHTpe 3eMJlH.
Ba.n;aHhI BeJIH'flHHhI: Macca 3eMJIu M = 5.98.10 24 Kr. M8cca JIYHhI
M m = 7.3.10 22 Kr. P8AUYc 3eNJIH R = 6.37.10 6 N. paCCTO.HHHe Nem.n;y
:Q,eHTpaMH 3eMJIH H JIyHLI L = 3,84. 10 8 N. rpaBJlTaQ,HoHH8.R nOCTO.JIH-
HaSI G= 6.67.10- 11 1'43. Kr- 1 .c- 2 .
1. JIyua H 3eMJI.JI Bpan:(8IOTC.H BOKpyr HX 06II{.ero .ueHTpa Macc C C
yrJIOBOH CKOpOCTLIO (0. Onpe.o;eJIHTe STY yrJIOByIO CKOpocTL. Hailp,HTe
p8CCTO.HHHe l Mem,Ay C H :Q,eHTpOM 3eMJIH.
2. BYAeM TenepL HCnOJIL30B8TL Bp8.IQ,aJOII{.yIOcSI BOKpyr TO'lRH C
CRCTeMY' KOOp.AHHaT, OARa H3 oceit KOTOpoA COBnB,D;aeT C JIHHHeH. npo-
177

(!):
r
PRC. 180
XO,lJ,.sm;eit 'tJepoo n;eHTpLl SeMJIH H JIYULI. B 8TOit CBCTeNe ICOOp,u;RH8T
(}>OpM8 3KKOit nOBepXHOCTH SeMJIH JlBJI.s:eTC.R CT8TecKOit.
B nJlOCROCTH P, npoxo,u;.s:m;eit 'lJepe3 C B nepneJl,lUlKYJIIipHoit It OCB
Bpam;eHB.H, DOJlO3KeHHe M8TepHaJIbHOA T01JRR Ha mJl,ZUCoit DOBepXHO-
CTH 3eNJIH N03KeT 6L1TL ODBC8HO nOJI.RpHLlNH ICOOp,u;HBBTBNH r, q>
(pHC. 130, r,u;e r - paCCTomme OT n;eHTpa 3eMJIH).
MLI 6y,u;eM H3YQ8TL 4>opMY r (q» 3KIIJJ;ICoii nOBepXHOCTB SeMJIH B
nJIOCKOCTB P:
r(q» = R + h(q».
PaCCMOTpHM N8TepRlUlbHyJO TO'tJKY (N8CCO:H m) HB3OI.AK0H nOBepx-
HOCTH 3eMJIH (B DJI0CKOCTH P). B aame:H CHCTeNe OTCQeT8 H8 Bee .u;eii-
cTByeT Q,eHTp06emH8.a CHJIB, 8 T8K3Ke CHJlLI npHTSl3teHH.R co CTOpoHLI
JIYULI H 3eMJIH. H8nHmHTe BLlpameHHe P;JISI nOTeHQ,HBJIhBOH 8HeprHH
3TOH M8TepHaJIhHOH TOq}(H.
Y K 838 H He. JI106a.H CHJIa F(r), HanpaBJIeHHBSI pa..u;U&.nhBO no OT-
HomeHHIO K HeKOTopoMY H8'IJaJIY KOOPAHHBT, ecTL npoH3BO,ltB8JI OT He-
KOTOpOH c(}>epH"lecKH-CHMNeTpH"IaOH nOTeHD;HaJILHOii 8HeprHH W(r),
B3.HTOH co 3H8KOM MHHYC: F(r) = - W'(r).
3. BLlpa3HTe npU6nH>KeHHO 4>oPMY npHJIHBHoA BOJIBLI h( q» (CN. n. 2)
qepea 38.)J;8HHLle B ycnoBH.f.IX 38.)J;8"lU BeJIHtlHHLI. KaKOB8 p83HHIJ;8 (B
MeTPax) Me>Kp;Y ypoBH.f.IMH npHJIHBB H OTJIHB8 B ,u;8.IlH0it MO,u;eJIU?
BLI MOmeTe HCnOJIh30B8Th npH6JlH2ReRHOe BhIp8JKeHHe
.J 1  1 + acose + _ 2 1 a2(3cos2 e -1).
l+a 2 -2acos8
npuMeHHMoe npu a « 1.
PeweHMSI
38.)J;aQa 1
1. ,IJ;8BH8JI CHCTeMa pe3HcTopOB MomeT 6LlTb npe06pa30BBBB B 8K-
BHB8.JIeHTHYIO (puc. 131). B 3KBHB8JleHTROii cxeNe conpoTHBneHHe
Mem.n;y TOqK8MH C H A a T8Kme Mem.ny TO'tJKBMH D H B PaBHO 0,5 ON.
06II1;ee conpoTHBneRHe R Mem.ny TO'lK8MHA H B paDRO 0.5 OM.
178

A e  B
c::::J
Pac. 131
2. ,D;JIB NaJILIX As R AL (pIlC. 132) CHJIa TpeHH.JI NO.meT 6hITh BLIpa-
meHa TaK:
As
F'IP = IJ.mg AL '
a pa6oT8 CRJIhI TpeHRSI 38 ,u;aHHladt BpeMeHHoA HHTepB8J1 ecTL
As
M. = Jlmg AL AL = J.l.mgAs.
nOJIHa.&: pa60Ta CHJ1LI TpeHRB paBHaA = J.l.mgs.
113 3aKOH8 COXPaHeBRSI SHeprRH CJIeAYeT, 'lITO pa60T8 CHJIhI TpeHHSJ:
paBHB R3MeHeHHIO nOTeHIUI8JILBOA 3aeprHH mgh JlhlHtHHKa. CJIe,u;OB8-
TeJlLHO,
h = J.ls.
3. 0603H8QHM R3MeHeHHe TeNlIepaTypLl 3a MBJIwA npONe.mYTOK
BpeMeBH dt q:epe3 dT. 3a STOT npONe3:KyroK BpeNeHR KOJlHqeCTBO Ten-
J10ThI, nOJIyqeHHoe NeTa.nJION PUHO Pdt.
Tor ,u;8
Pdt P
C p = dT = dT .
dt
HCnOJIL3Y$l yc.nOBRe 3aq:H, Haii;:!;eM
dT = To a[l+a(t-Io)] -3/4=
dt 4
= Tu( ; r
CJle,u;OB8TeJlLHO,
c == 4P T a .
P dT a7: 4
_ 0
dt
TaKRM 06pa30M, B HeKOTopoN,u;H-
anBaoae Te:r.rnepaTyp TenJIOeMKOCTb
MeT8JIJIOB nponOpItHOHBJILH8 T3.
/).s
PHC. 132
179

J

J J
.... 
T 2 m
4. B CT8.ItHOHapHON COCTOHHHH TenJIO-
BoA nOTOK Be3p;e OHH H TOT me (pRC. 133):
J::; a ( - Tt ),
J::; 0' (T - Tt),
J::; 0' (T: - Tl).
OTCIO,D;ft CJIe.r:tyeT, 'tJTO
3J; a (Tt- T:> ::;Jo'
li'
r,n;e J 0 - TeIIJIOBOA nOTOK npR OTCYTCTBHH
TenJIOBoro 3Kp8.Ha. TaKRM o6paao:M, Ten-
JIOBoA nOTOK YMeBbmseorC.R B 3 paaa.
5.araHTHOenOJIeTaKOronpoBop;HH-
Ka MomeT 6L1Th npeCTSBJIeHO K8K cynepn03UD.H$I MarHHTHhIX nOJIeH
P;BYX D.UJlHlI)J;pWiecKRX npoBop;HHKOB, no KOTOphIM TeKYT B npoTHBO-
nOJIoaatLIx HanpaBJIeUURX paBHLIe TORH. lIo ItHJIHH;D;pHqeCRHM npo-
BOP;8N p;OJImeH npOTeKftTh TOK l' T8.KO:A, 'tJTo6LI B TOH 06JIftCTH, re npo-
Boa He tnepeKphIB8.1OTC.tU, TOK no-npemBer.ty OCT8BaJIC.R p8BHhIM I.
TaK K8K TORR paBHOMepHO pacnpep;eJIeHLI no CeqeHHIO, TO
Ts T..
PHc.133
it D2
4
BuyrpH IUl1IH1I.ZU>HtlecKOro npoBOAHHKa, no KOTopoMYnpoTeK8eT TOK
I' , HHJ];YKItH.tI NarHHTHOrO nona NO)KeT 6LITb Hmp;eHa no 4»oPMYJIe
I
I'
( 7t + J3 ) D2
12 8 _ 2n + 3../3
- 61t
.Eo - )..1.0 l'nr! _ 2)..1.oI'r
- 2w D2 - 1tD 2 '
4
rp;e r - paCCTOSlHRe OT OCR npOBO,D;HHKS.
B npOeKD.JUIX HS OCR X H Y:
Bz ;;:: Eo 1!.. ;;:: 2J.1 0 I'Y ; B ::;;;.Eo  ::;;; 2J.1 0 I'x .
r nD 2 , r nD 2
Cynepn03HItBH noneR, cOap;8BSeMhIX TOKRMH %]' , onpe.o;eJIeH811.11H
D -
X;;:: =1="4' COCTaBJI.HIOII\Re Bx B38HMHO YHH'IlTOmaJOTcH, T8.K K8.K nOJIH
cHMMeTpu-qHLI, a KOMfiOHeUTS
Bil ;;:: 2J.1o [ I' ( X + D ) -I' ( X _ D )] ; J.1 o I' ;:: 6J.101 
rr.n 2 4 4 nD (2ft + 3.J3)D
180

T.e.IIHJJYK£(R$I NarHHTHOrO nOJIB UOCTOBBH8 no NOP;YJIIO R HanpaSJle-
H8 SP;OJlh OCR Y.
3a,a;8W18 2
1. lIOTeH8JlbHa.R 3HeprHB eV nepeXOART B KHHeTH'lecKYIO:
! mv 2 = eV (HepeJUlTUBHCTcKH:A CJI}"Iaii),
2
me 2
H 2
1--
e 2
me 2 = eV (peJI.1ITHBHCTCKd CJIyqaA).
CJle,lJ;OB8TeJlbBO,
! J2;; (uepeJISlTHBHCTCKHii CJIy«aA),
v = C 1- ( 2 ) 2 (peJI.RTRBHCTCKu:A cJIyrlau). (1)
me + eV
2. Kor.n;a V = 0, 3JIeKTpoH ,lJ;BHmeTCH B nOCTO.RHHOM MftrHHTHOM none.
CHJI8. JIopeH£(8 nepneH,lJ;HKYJI.fIpH8 CKOpocTH ABRmeHliB SJIeKTpOHa. B
STOM c.nyqae OH8 COo68eT 3JIeKTpoHY £(eHTpocTpeMHTeJIhHOe YCKope-
HHe, T .e.
mv. 2
eBG> ::: ---!l
R.
OrCIOJJ;8
B = mvo .
eR
113 pRC. 134 BHP;HO, 'IlTO AJISI
KpHTHqeCKOrOCJIyrlaH
.J a 2 + R2 ::: b - R,
a 2 +R2=b 2 - 2bR +R2
H
b 2 _ a 2
R= .
2b
(2)
PHc.134
181

nOCTasHB aTO BhIpWKeHHe 1I.JJ.B. pa,D;Hyca B pMYJIY (2), nonyqaeN
3H8'11eHHe HH,lI,yKH MarBHTHoro nOJIH A1111 I<pHTlAecKOro CJI}T'ias::
B := mv o = 2bmv o .
e eR (b 2 _ a 2 ) e
3. A3HMyr8JIbH8a KOMnOHeHT8 FfP CHJJ.hI JIopeH8 F = (-e)[B, vI
06YCJIOBJIHB8eT MOMeHT Bpam;eHHIl FfPr. OR 3aBHCHT TOJIbKO OT pa.p;:u-
u M
BJIbHOIi KOMIIOHeHThI CKOpoCTH V, := dt .
CneJJ;OB8TeJIbHO,
dL dr
- = eBr-.
dt dt
ITpeo6psayeM 8TO BhIpWKeHHe:
:t (L- eB;2) = O.
CJIe,lJ,OBaTeJIbHO,
1 2
C = L - -eBr = const.
2
(3)
OrCIOA8 6eapaaMepH8.f1 BeJIHqHH8 k, 3B.,D;8.HHall B YCJIOBHH aa,n;a"tl:u,
paBHa 1/2.
4. BOCnOJIhayeMcSI nOCTOIlHCTBOM C (CM. n. 3) ,lI,JIII onpep;eJIeHHSI
M8KCHMaJIhHOro paCCTOSlHHII r mOT OCR IJ.HJIHHApa:
0- leBa! = mvr. - .!eBr. 2
2 m 2 m'
OrCJO.o;8
_ eB (r  -a 2 )
v- .
2mr m
5. llpH KpHTlAeCKOM 3HSQeHHH Be HHAY.KHH M8rHHTHoro nOJIH
M8KCHMaJIbHOe 3H8qeHHe r m paSHO b, n03TOMY
_ eBe(b2 - a 2 )
V - 2mb .
KHHeTH"IeCK8H 8HeprHSl nepexo.n;HT B nOTeHHaJlbayro (HcnOJILay-
eM peaYJIbT8ThI, nOJIyqeHHbIe B n. 1):
v;: 2e .
182

HCnOnb3YBo68BwpameHHBnBv,nonyqaeM
eB(' (b 2 -.' ) g
::;:: 2e-.
2mb m
KpHTKtleCKOe 3H8'qeHHe Be HH,z:QTKIUIH M8I'HHTHorO nOJIa paBHo
B::;:: 2b .J2mv
C bl - a 2 e.
6. CHnaJIopeHI\B He HMeeT COCTaBJISIIO:m;eA, nap8.JIJIeJIbHOH HanpaB-
JIeHHIO HYKBH NarHHTHOrO nOJlB, H, CJIep;OS8TeJlhBO, COCTaBJlHIO-
:m;8SI CKOpocTH, DapaJ1JIeJIbBBJI HH.AYKIJ.HH OCTaeTC.H nOCTOJIHHOH B npo-
ecce ABHmeUHSI:
VB ::;:: const.
0603Ua'lHM 'lepe3 v 83HMYT8JIbHYIO CKOpOCTb 3JIeKTpoH8 B6JlH3H
aHOAa. li3 38KOB8 COXpB.HeHHH SHeprHH nOJIY'laeM
1 ( 2 2 2 ) 1 ( 2 2 )
2 m VB + Vtp + v,. + eV :;;; 2 m VB + V .
OTCIOAa
2 2 2eV
v::;:: V r + Vcp + -.
m
licnoJlL3ya nOCTO.RHCTBO C [cpOPMYJI8 (3)], sannmeN:
1 1
mvcpa - 2 eBe al :;;; mvb - "2 eB c b2 .
Pemaa COBMecTHO c (4), nonyqaeM
B _ 2m(vb- v"a) _ 2mb [ 2 2 2eV a J
v,. + V" + - - v" - .
c - e(b2 _a2) - e(b2 _a 2 ) m b
(4)
(5)
38.)J;a qa 3
1. TaR RBI< TOqKB C .HBJI.ReTCJJ eHTpOM MaCC CHCTeMhI 3eM.Hn:-
JIyua (pHC. 135), TO M())KHO 3aUHcaTh:
Ml:;;;Mm{L -l).
OrclOAa
M m 6
l :;;; L:;;;. 4,63.10 M.
M+M m
(1)
3TO 3H8qeHHe MeHhme 3H8'leHH.R p8.AHyca BeNJIE, CJIe,n;OBaTeJIhHO,
o6it: rr;eHTp Macc Haxo,n;HTC.H BuyrpH 3eMJIH.
183

r m
JIyna
PHc.135
Q,eHTpo6emHhIe CHJlhI ypB.BHOBemHB8IOTC$I rp8BHTaItHOHHbIMH:
Jl.K 21= o MMm
.lV..I(O L2 
n03TOMY
= JGM.. = JG(M + M..) = 2 67 . 10 6 -1
CO L 2 1 L3 ' C .
(2)
(3'1'0 COOTBe'l'CTBYeT nepHOAY 21t = 27,2 cyr. )
(0
2. TIoTeHnn8JILHa.R 3HeprH.R MBTepHBJIbHOH '1'O'llKH onpep;eJI.HeTCJI
CYMMOH Tpex BeJIHqHU:
- nOTeHaJIbHOH 3HeprHeii, 06YCJIOBJIeHHOH BpeHHeM H paB-
HOR
1 22
-2 mro 'i ,
rJ];e r. - paCCTOHHHe 0'1' C. 3'1'0 COOTBeTcTByeT neHTp06emHoii CRJIe
mw 2 r., HanpaBJIeHHOH 0'1' C;
- nOTeHItHaJIbHOH 3HeprHeii, 06YCJIOBJIeHHOH rpaBHTHOHHhIM
BJIH.HHHeM 3eMJIH Ii P8BHOH
_G mM ;
r
- nOTeHnn8JIbHOH 3HeprHeH, 06YCJIOBJIeHHOii rpaBHTanHOHHbIM
BJIH.HHHeM JIYHbI H p8.BHOH
_o mMm ,
r m
r.n;e r m - paCCTOHHHe ,lJ,0 JIYUhI.
OnHmeM nOJIOmeHHe MaTepHWIbHoH TOqKII B nOJI.HpHbIX KOOPAH-
HBTax r Ii <p. 113 pRC. 135 BIi.n;HO, QTO
ri11 2 = Ii - 2 = r 2 - 2rlcosa + 1 2 .
184

CKJI8,.Q;LlB8JI BCe TPH KOMnOHeHTLI nOTeHItHaJILHoii: 8HeprHH M8Te-
pHanLHOATOKH,nonyq8eM
( ) 1 2 ( 2 2 ) mM mM m
V r =--11UJ.) r -2rlcostp+l -G--G-.
2 r 
(3)
Ilo.n;cT8.BJISleN l us BLIpameHH.H (1), H
Tm = L_i'12 = .Jc -2Li'+r2 =L 1+( :)" -2( : )008'1"
r
3. TaK KaK OTHomeHHe - ::; a MaJIO, MomHO BOCnOJIbSOB8ThC.H BLI-
L
paaceHHeM
1 = 1 + acost:p + .!:..a 2 (3cos 2 t:p -1).
.Jl + a 2 - 2acoscp 2
IloJJ;cT8BJI.H.R. B BLlpameHHe (3) )l;JIH nOTeHaJILHOH 8HeprHH, no-
nyqaeM C TO'lHOCTLIO )1;0 nocToHHHOH
V(r, <p) ::; _ .!:..olr2 _ GM _ GM m r 2 ( 3008 21 1\ -1 ) .
m 2 r 2L3 "t" (4)
c.t>OpM8 nOBepXHOCTH )KH)J;KOCTH .HBJI.HeTC.H 8KBHnoTeHH8JIDHOii
nOBepXHOCTLIO p8,AHYCOM
r::; R + h.
T8K KaK BeJIlAHH8 h MHOro MeHLme R, MOmHO SRnHC8TL
1 _ 1 _ 1 1  1 ( h ) _ 1 h
;- R+h - R - 1 h;::: R I- R - R - R 2 '
+-
R
8 T8K>Re
r;::R2+ 2Rh + h2R2+ 2Rh.
HCnOJIL3YH 8TH COOTHomeUHH U BLlpWKeHue I1.JI.H 0) (2), t}>OPMYJIY
(4) npHBep;eM K BHJJ;Y
VCr, cp) == G (M+M".)R h + GM h- GMmr2 ( 3COS 2 tp_l ) (5)
m C  2C
onllTb C TO'lUOCTLJO 1J.0 nOCTO.RHHOH.
185

nepBoe CJIaraeNOe B 4JopMYJIe (5) HMeeT nOp.H,ZJ;OR BeJIWlBHW
M + M m ( R ) 8 r:::: 10- 5
M L '
II ero BRJIS)]; no CpRBHeHHIO C pyrHMII Be.nH'lHHRMH Man. OCT8.BmUeCH
J];Ba CJIaraeNLIX B c}>oPMYJIe (5) KONneHcllp}'1OT pyr pyra.
TI08TOMY
M r 2 R2
h;;;: 2ML3 (3cos 2 cp -1).
3aMeH.H.f.I r ua R 2 , nOJIyqaeM, 'lITO BLICOTa npHJIHBHo:A BOJIBLI C TOq-
HOCTLIO o nOCTO.RHHOA PUSR
h - M m R4 (3 2 _ 1)
- 2ML8 COS <P .
M R"
H8H60Jlbmee 3SaQeUUe h max ;;;: ;L 3 COOTBeTCTByeT yrJIY cP = 0
M"Jt*
HJlH cp = 1t, H8.HMeHhlllee 3Ua-qeUUe h m1n = - 2ML 3 cOOTBeTCTByeT yrJIy
cP = 1t/2 HJIH <P = 31t/2. P83HHa Ne*.n;y ypOBH.HNH npUJIHBa H OTJIHB8
COCT8BJIHeor
h - = 3M m R4 = 0,54 N.
mu 2ML 3

XXVIII Me)I(AYHGpOAHGR
cl»M3H'IeCKGR OnHMnMGAG
1997 r _, KGHGAG
3aAa8 1
1. He6oJILmOe TeJIO no.n;BemeHO sa KOHIJ;e seBecoMoii HJJ;eBJILHOA
nPYJKHHLI II KOJIe6JIeTCSI BBepx II RHII3 C -qacToToH cof)cTBeHHhIx KOJIe-
6SHUit I. KaKo:A CT8HeT HOBas TIaCTOT8 KOJIe6aHHH I', ecJIU npy:mHHY
YKOpOTHTL B ,lI;Da pa38 II no.n;BeCHTh K HeA TO me TeJIo?
2. B COOTBeTCTBHH C MO,ll;eJIhlO aTOMa Bopa p8JJ;uyc aTOMa BO,lJ,opo.n;a
B OCHOSHON COCTOmIHH au = 0,0529 HN. qeMy paRes PAAHYC a' aTOMa
.NIOOHsoro BO,ll;opo,ll;a., B KOTOpoM 3JIeKTpoH 3aNeUes HB :MIOOS? 3a-
pH,lJ, M:100aa peeH 3ap.H,ZJ;Y SJIeKTpOHa, a ero :Macca B 207 paa 60JILme
MaCCLI SJIeKTposa. C1'{HTaATe, 'liTO Macca npoTOHa 3B81:JUTeJlLUO 60JIL-
me MaCChI NIOOHa.
3. CpeJJ;H.R.H TeMnepaTypa 8eMJIH T = 287 K. K8.Ko:A 6hI CTaJIa HOBBSI
TeNlIepaTypa 3eMJIH T' , ecn:H 6hI cpe.n;Hee paCCTO.HHHe MemAY 3eMJIe:A II
COJIHIJ;eM Y:MeHLmBJIOC:b sa 1 % ?
4. B O)J;HH H3 ,lI;Re:A B03)];YX 6hIJI a6COJIJOTHO CYXHN H IINeJI DJIOTBOCTL
p = 1,2500 Kr /r. Ha cJIe,ll;yIOJdt )];eBL BJI8JKHOCTh BO;3)];yXa YBeJIII1iH-
naCL H B03.n;yx CT8JI cO)];epJKaTL 20/0 (no Macce) BO;l];.HHOrO napa. ]J;aBJIe-
HHe II TeMnepaTypa B STaT .n;eHL OCT8JIHCL T81CHMH me, "ITO II B npe,u;hI-
,D;yIIJ;uit .n;es:b. 't:leNY paDHa DJIOTBOCTb B03,D;yxa p' B STOT JJ;eHL? Cpe.n;USlH
MOB.HpH8H Macca cyxoro B03JJ;yxa 2,88.10- 2 Kr/MoBb. CqHTaATe Boa-
JJ;YX H)];e8JJLHLIM ra30M.
5. BeproJIeT MOJKeT 38BHCuyTL B Boa,u;yxe, ecJIU MeX8.HH'lecK8JI MOIIJ;-
HOCTh ,lI;DHraTeJIH peua N. KaKo:A .n;OJUKH8 6LITL MeX8.HH'IIeCR8S1 MOIIJ;-
BOCTb J];BHraTeJISI N' Y YMemmeHHoA Mo.n;eJIH STore BePTOJIeT8 (Bee JIH-
Be:ftHhIe pa3MepLI KOTopoA B )];Da paa8 MeB:Lme pa3NepOB OpHI'HB8JIa)
QTo6:L1 OB Mor 38BHCHYTh B Boayxe?
38a'l8 2
Bee sueprHH B STO:ft 3a'lle BhIpaJKeHH B MeraSJIeKTpoHBOJIbTaX
(MsB): Macca M 8TONBOro .H,lU)8., COCTO.Hm;,ero U3 Z npoTOHOB H N BeRT-
pOHOB (T.e. .RJU>8 C NaCCOBhIM 'IIHCJlONA = N + Z) p8BBa CYMMe Macc
187

B M3B
A  HYKJIOH
.. r ur. -r  r.. . ur .r. -r A:_ A A - ":" -"r -.r."r  ...r. r.:..r. . nr. .r -r..r 
9
. II ,.. ,.... .II ..... .. III ,.
--:--:--:-..,.. - --"1-" --1--1-:--1-- :-..:--:--:- --1--1---r.-:--
--:--1"-" 1-- II ,. III,. : -.---:-.. --:."1--:--:-- --:--:--:-..:--
--1---:- : --:-- ..-:....1--1..-1-- -:..-:--: . -- -_Or" -:--:--:-- --:--:--1--:-"
..:-.. -:--:-.. --1"-1---r 4 -1-- --1"-:---:-"-:-- -1--1-- --:-":--:-1-
8
.. . ". III 110 .. ..,.. .......
-- -:--:---:-- ---:---:---:---- ---:---:-.."!---:-... ---:-"1---:--;- -- . ---:--1--
MM: -i--i--i-- --iui--i--i-- --i--i-i--i-- --1--1--i--1-- --i--1--: ;--
--- -7....1....1-- --1"-1--1--1-- --1---:--1--1-" --:--:-:--:- ---:..-1--1-..1....
-- -1--1 0 '-1-" -:..-:-..1..-:-- -".r--:--:--:-" --1--1-:--1-- --:--:---:....1-..
7
o
50
100
150
200
250 A
PRC. 136
OT.n;eJlhHLIX HyKJIOHOB, COCT8.BJIHIOHX H,D;PO (npoToHoB II HeHTpoHoB)
B
MHBYC SHeprHH CBH311 B, .n;eJIeHHaSl Hft c 2 , T .e. '2"' CJIep;OBaTeJIhHO,
c
Mc 2 = ZmJlc 2 + Nm n c 2 - B.
B
SHepI'H$I CBH311 sa O,ltHH HYKJIOH A H83hIB8eTC$I YJJ;eJIhHOH SHepl'Hel:
CBH3H. 'tIeN 60JI:bme y.n;enhHasJ 3HepI'HH CBa3R, TeN cTa611JIhBee pO.
Ha pRC. 136 H306pameH8 3aBRCHNOCTL y,JJ;eJIhHoii: BHeprHH CB.R3H OT
M8CCOBoro "lHCJl8A.
1. ECJIH MaCCOBoe 'IHcJIO s.n;pa npeBLImaeT onpe,JJ;eJleHHOe 3ua-qeBBe
Au, YJJ;eJlLH8.J13HeprHa CBa3R O:K83h1BaeTCa ,lJ,OCT8TOQBO N8J10:A, TalC "ITO
CT8.HOBBTCS! B03MomHo:A 3MHCCHJI 8.JIh4>8-qaCTH (A = 4), T .e. aJILc}»a-
pacn8Jl;. HcnoJlL3Y.8 JIHBe:AHYI<> annpOKCHMaIJ;HIO H306pameHBoii: sa
pRC. 136 KpHBOii: 11.11.8 3B8'1eHHii:A > 100, on,eHHTeA u '
,n;JISI pemeHH.8 38JJ;S'lJH HCnOJIh3yHTe CJle,lJ,YIOee:
3SSQeHH$I y.n;eJILHOH SHeprHH CB.8311 ,JJ;JI.R aTOMBhIX s.n;ep ,lJ,O II nocJle
pacua;J;s NOryT 6hITh nOJlyqeHhI 113 rpac}»RKa;
nOJIHaS! SBeprHSI CB.H3H aJIhcpa-qaCTHnLI B pe COCTaBJI.HeT B4 =
= 25,0 MsB (STO 3Ha-qeSHe He MO)fCeT 6hITL nOJIyqeHo 113 rpac}»HKal).
2. SHeprHH CBSI311 aTONHoro .RApa, COCTO.RerO H3 Z npOTOHOB II N
neHTpoHoB (A = N + Z), OUHChIBaeTCH cJIe.n;YIOIIJ;eH nonYSMnHpHQeCEOA
clJOPMY JlOH:
(N Z Y
B = a A - a A 2/3 - a Z2 A -1/3 - a - - B
v . C cA.
IIapsMeTp B paBeH:
+ a Jl A-3/4.n;l1SI a.n;ep C ne'IeTHLIM N II neqeTHLIN Z;
(1)
188

HynlO AnSI .HJJ;ep c qeTHJ,lM N H HeqeTHLIM Z HnH C HeqeTHhIM N H
qeTHhIM Z;
- a-3/4 AJIH SlJ];ep C 'tIeTHJ,lM N H lleTHJ,lM Z.
KOs«}>«}>H-qHeHThI B «}>opMyne ,lJ,JIH sHeprHH CHSIau COOTBeTCTBeHHO
paBHL!:
a v = 15,8 M3B; as = 16,8 MsB; a c = 0,72 MsB; aa = 23,5 MaB; a p =
== 33,5 MsB.
2.1. ITOJIytlHTe BhIpameHHe JVlSI onpe)1;eJIeHH.H qHCna npOTOHOB Zmax B
.HJJ;pe C HaH60JILmeH sHeprHeii CBHaH npH 3B.,D;aHHOM M8CCOBOM qHcJIeA.
llPH oTBeTe Hft STOT Bonpoc npeHe6perHTe BJIH.HHHeM (J-napaMeTpa.
2.2. Onpe.n;eJIHTe tlnCJIO npOTOHOB Z B HAPe C H8U6oJILlIIHM 3H8qe-
EHeM y.n;eJILHOH 9HeprHH CBH3H  npHA = 200. Y'tITHTe B STOM CJIyqae
BJIHHHHe S-napaMeTpa.
2.3. PaccMoTpHTe TPH SIJU>8 C M8CCOBhIM 'IHCJIOMA = 128 (CM. Ta6JI.).
Onpe,lJ;eJIHTe, KaRHe H3 3THX HAep 9HepreTHTIeCKH cTa6HJILHJ,I, a KaKHe
MoryT pacna,u,aTbcH no OJ];HOMY H3 rrepeTDICJIeHHhIX HHme npo-qeCCOB.
H8ii,D;HTeZ max (cM. fl. 2.1) H 3anOJIHHTeTa6JIHY.
Ta6.nuqa
llpo:Q;ecc
RAPO 13- -pacn8):\ 13+ -p8CD 13-13- -p8CDBJ( BaXB8T
aneKTpoaa
lI
lXe
ICS
ECJIH npo-qecc 3HepreTH':IeCKH B03MomeB, nOCTaBLTe B Ta6JIH-qe 3H8-
1.IOK V.
ECJIH npo-qecc 3HepreTH'IeCKIf HeB03MomeH, nOCTaBLTe O.
npo-qeccJ,l pacnB.,D;a:
13- -pacnB.,D; - 9MHCCHH 9JIeKTpOna Ha SlJij)a;
13+ -pacnB.,D; - SMHCCHSI n03HTpOHa H3 .HJij)a;
13-13- -pacnB.,D; - o,ZJ;HoBpeNeHHaSl3MHccHSI J];BYX MeKTpOHOB H3 .H)J;pa;
a8XB8T 9neKTpOHa - aaXB8T 8TOMHoro SJIeKTpOHa H)(POM.
8HeprHSI nOKOSl3JIeKTpOH8 (H n03HTpoHa) mec2 = 0, 51 MsB; npoTo-
HS - m p c 2 = 938,27 MsB; HeHTpoH8 - m n c 2 = 939,57 MaB.
3ap;8118 3
HymHo paapa6oT8TL C8MOJIeT, KOTOpJ,lH CMor 6J,1 y.n;epmSTbCH B B03-
.n;yxe HCnOJIb3YSl HCKJIIOTIHTeJIbHO SHeprHIO COJIH:Q;8. HaH60nee 9«l>«}>eK-
THBHSH KOHCTPYK:Q;HH TaKoro CBMOJIeTa .n;OJI»tH8 HMeT:b OqeHb J];JIHHHbIe
189

BCmpeltnbtU nomOlc B03iJyxa
I I I I I I I I I
.  rr _   _ _ . . . .
r_. ... .r.r..r .__r_ n_ _ __ _._ .. __.""_... "_" .....
..r.. or. ..... _..r __ _ __._. ..__.... .. __" .....
r..... .. or.... .r .r.r_. _ _ __"__" __ ""_"_"  .. .....
.. ... ........ .r....... _ _ __"_._ _ ____ _ . _. .. .
. _..........r...........r _r_ or. 'r_ ___ _. ..__. ..
_" "_" ...... .........r_ _ __ _r_ _ .___ _ .  .
__. __ __ 4._. ..._ _ ........... ...... ...r. _. r.
- - - - - --  . ..
......... r  r... r .. _ _.. r ... r r__ .._. _ r  _ r..... __  r
. r.... r..... r _ r._ _.... r _ _. r. r.. _ __ ... r r_ r._......_
............. ... ... r r r .. r_  r_.. .. r_. _  r. ... r.... r
. ... . .  _ _ _ . _ r _ _ . _ _ _ _ _ _ _ _ _ _ _ r _ _ r _ _ _ r  _  _ r r _  _ _ r   _ r _ _ _ _ _ _....... r 
_ _ _. _ __.. __ _. _ __  __. _ __ _ _ _ __ _. ___ _ r_ _. _  r _ _ _.... _ _ _ ___ 
 _.__ _ ___. _ _. _ ____ ____ _ __ _. _ _ r_ __ _.__ _ rr __. r_._ ___ ___.......
_ __ __.. __ _ _ __ _ __  r_ _  .._ _. ___  _ _& _. __  __ _ __ 4.. r4____
- - -  --
I
I
I
I
I
I
I
I ·
I
I
I
I
I
I
I
"I
l
he. 137
TOHKHe KpbIJILH" BepXH.HSI nODepxHocTL ICOTOphlX nOJIHOCTLJO nOKpbIT8
COJIUeQH:hIMH SJleMeHT8MH. 8TH SJIeMeHTbI BhIpa6aThlBaIOT SJIe:KTpo-
3HeprHIO" KOTOp8..fI llHTseT SJIeKTpOMOTOp" npHBo.u;.H:m.;HH BO Bpa.m;eHHe
nponeJIJIep.
PaCCMOTpHM ,lI;JIHHHOe TOHRoe KpLIJIO np.RMoyrOJILHOH 4>oPMLI C paa-
MaXON l H mHpHHoit c" ,lI;Bu:mym;eecs B B03,u;yxe co CKOpOCTLIO 0 OTKO"
CHTeJILHO OKpy:maIOero B03.u;yxa (puc. 137). MhI MOmeN: nOJIyqHTJ.
npH6JIHmeHHOe npe.u;cT8BJIeHHe 06 s4>cJJeKTHBHoCTH .u;eHcTBHs &Taro
KphIJI8 ecnH paCCMOTpHN nOTOK B03,l{yX8 mHpHHo:ft. x" HaJIeTaJOd
ua aTO KpLIJIO H OTKJIOH.fIIOII(UitC.R no,u; He60JILIIIHM yrJIOM E (pHC. 138).
H3MeHeHHeM MO.u;YJI.H CROpoCTH B03JJ;yxa npH STON: MomBO npeBe-
6peQL. YnpaBJI.RlOII(Ue nOBepxHocTH KphIJILeB MOryT 6LITL HcnOJILaO-
B8HLI ,lI;JIsr. Taro" TlTo6L1 no,u;06paTL OTITHM8JILHOe 3H8'qeHHe E, Heo6xo-
,lJ;HMOe ,lJ,JI.K nOJIeTB. fua npOCT8..fI Mo,u;eJIL ,lJ,OCT8TO'qHO 6JIH3KO COOTBeT
it1
cTByeT ,u;e:ft.cTBHTeJILHOCTH" eCJIH x = 4" II DaM CJIe.n;yeT pemaTL ,n;aHHytO
- .
L 
1 . I
. I
... I
..  ,
II" I
- -----=::: I
v ,
BCmpeltH.btU Cellenue nomolC BepmU1CaJLb
nomOK KpbtJla n03aiJu "pbtJla
he. 138
190

38)(aqy B p8.MK8X &ToA NOJJ;eJIH. 06Iq:8JI M8.cca CaMOJIeTB M. IIJIOm;BJJ;h
nOSepx.SOCTH Kp:w.m.eB S = cl, nOCTO.RBHaB. KpLIJIaA = 1/ c. B CBOHX BLI-
'IlHCJIeHH.fIX yrIHTWBaitTe roJn.KO B03p;yx. 06TeK81Om;Hii KphlJIO.
Ha pRC. 137 npe.a;CTaBJIeH B H8 C8.MOJIeT cBepxy (C8MOJIeT H306-
pBeH B co6cTBeHBOH CRCTeNe OTC'qeTa).
Ha pRC. 138 H30t5plUReu BH.lt HB KphlJlO COOKY (B CHCTeMe OTC'IleTB,
,lJ,BH)Kym;eAcs BMecTe C CBMOJleTON).
ITpR pemeHHH aa.u;aqu He yq:HThtB8.A:Te H3MeueUHe B03,l1;ymBoro 110-
TORa, BROCHMoe nponeJIJlepoN.
1. PaCCNaTpHBaJI H3MeueUHe HMIIYJlLC8 nOTOKa BOa,ll;yxa, o6TeKaIO-
m;ero KpWJIO, nOJIyqHTe BldplUReHHs JI.R nO,J:rLeMHo:A: CHJIhI L H roPH-
aOHT8.JIhHoA CHJIhl COupoTHBJIeUHS Dl , ,AeAcTBYIOIq:HX Ha KphlJIO, B 3a-
BHCHNOCTH 0'1' pB3MepoB Kpwn:8, MOP;YJIH CROpoCTH V H nJIOTHOCTH B03-
JJ;yxa. C''lHTaATe, "ITO uanpaSJIeUHe B03.u;ymHoro nOTORB Be3e
nap8JIJleJIhHO DJIOCKOCTB qepTe2K8..
2. CecTBYeT em.;e .u;onOJIHHTeJIhH8a. ropH30HT8JlhHM CRna D2,
06YCJIOBJIeBB8.R TpeHHeM B03.u;yxa, 06TeR8IOIq:erO 1I0sepXHOCTh KpLIJlB.
Bo3,o;yx BeMBoro S8.Me)];JI.ReTCS, T8.K 'liTO H3NeBeuue ropH30HTWIhHOA
KOMlIOBeBThI CKOpoc'l'B nOTOKa B03,IIyXR COC'1'B.BJUIeT I'1v = (  « 1 )
OTUOCHTeJIhHoe H3NeBeUUe CKOpOCTH onpep;eJISeTC.R KaK
Av _ f
v- A .
BeJIH1IHHa f He SaBHCHT 0'1' E.
HaA:.zr;BTe BwpaeHBe I<8R 4»YBK:q;HIO M, f, A. S P;JIH CROpOCTH no-
neTB V o ' COOTBeTCTBYIOIq:eH NHHHMBJIhHOH MOIq:aOCTH, ROTOp8JI Heo6-
XO,ll;HNa)];JI.R TOro, QT06H y.u;eP'f(HBBTh 3'1'0'1' C8.MOJIeT ua ueH3MeUHoA BLI-
COTe H C HeH3MeHHoA CKOpocThlO. ('1JIeUBMH nop.BJ];K8 Elf HJIH 60JIee BLI-
COROro nopKa CJlep;yeT npeue6pe'flh.)
Bid MO)KeTe HCnOJIh30BaTh IIPH BLftlHCJIeHHSlX npH6nHeHHoe BId-
pameHHe, Cnp8Be,l(JIHBOe AJISI M8JIhlX yrJIOS:
. 2
1- COSE R:j Sl E .
3. :Hso6pa3HTe CXeMaTH'IeCKH rpa4»HK 38.BHCHMOCTH MOIIJ;HOCTH P
0'1' CKOpOCTH v. Y"lTHTe T8R:JKe BKna.n,hl B MOm;HOCTh 0'1' ,ABYX CHJl conpo-
THBJleHH.R (D J H D 2 ). HaAAHTe BhIpaJKeuHe KaK 4»YHKItHIO M, f, A, S.
P H q p;31$1 NHHHMaJIhROI: NOID;ROCTH P min.
4. CoJIUe'lHLIe 9JIeMeRThI MOryT Bhlpa6BThIBaTb CTOJIhKO 3HeprnH, 'lITO
MOTOphl H npOlIeJIJIepw pa3BHB8IOT MexauH'IecKytO MO:I:qHOCTh N = 10 BT
191

C Kamoro KBa,zq>aTHOrO lI4eTpa KphIJl8. BLI1IHCJlHTe lII8.KCHMa..m.Bym
nO.o;"LeMH}'IO CHJIY KphIJ1a, .o;eACTBYIOJ1WIO ua KWl(,ALlii KB8JU>8THhIA NeTp
m(Mg/S), H CKOpOCTL nOJIe-ra V o npH ,D;aHBoA Mex8HHqecKoA MOOC-
TH. OIHTaHTe, q'fO P= 1,25 Kr/M 3 , f=O,004,A= 10,g= 9,81 M/C 2 .
PeweH".
38j1;a-qa 1
1. ITYCTL npY>KHHa B HCXOHON COCTO.RHHH HMeeT .o;JIHHY I H )l(ecT-
KOCTL h. QacToT8 f KOJIe68.HHit TeJla N8CCOit m, nOBemeHHoro a8 K08-
enpymHBhI,onpeAeJlHCHBhIpeHHeM
f ==.1... Ik .
27t ;n
mecTKocT!. k nOK83LIBaeT, KaRaH CHJl8 F Tpe6YeTcs, 1lTo6L1 paCT$I-
HyTL npymHHY ua BeJImIHHY 8x:
F
k=-.
lu
ITo,u; .n;eitcTBHeM Toii )l(e CHJlLI F cpe)J;HSlSI TO'IJK8 npymHHhI C,ABHBeT-
CH Bcero JIHm!. us BeJIH'tJ:HBY 6x/2, T.e. meCTKOCTL nOJIOBHHLI npY'f(H-
HLI pUBa
h' :;:::  = 2k.
(llX/2)
qacToTa KOJIOOaHHH TeJIa TOH me MaCCLI Ha npy)KHHe nOJlOBHBBOA
.n;JIHHLI 6y.n;eT p8.BBa
r :;:::  [2k = J2f.
2nV-;;;
2. MOl\lleHT HMnYJIbC8 KB8HTOB8.H (YCJIOBUe Bopa HJIH,lJ,e BpoiiJIH):
mvr =nli.
8JIeKTpOCT8TIJtleCK8.fI CHJI8 BhI3LIBSeT :c;eHTpOCTpeMHTeJILHOe YCKO-
peRHe:
mv2 = he"  v = J he"
r r 2 mr '
.Jke 2 mr :;::: h, T.e.
1
r--.
m
192

PHYC M100HHorO Bo.o;opO,lJ,ono.o;o6Horo aTOMS onpe,lJ,eJUleTCJi BhIpa-
)KeHHeM
a o
a p ::::; 207 = 0,256 fiM.
3. lIycTh MOm;aOCTL H3J1yqeHH.fI COJIHI(,a paDHa P, a pa,lJ;HYC op6HThI
3eMJlH paBea R. TeMnepaTypa T onpe.o;eJuleTc.fI 113 YCJIOBHSI paBeHCTBa
MOm;HOCTH IICJIyIqeHaoro II norJIOm;eaHOro H3JIyqeHH.fI:
(1- r)'  :::; 41tEcrT4,
47tR 2
r,lJ,e r - OTpamaTeJILHa.fl cnoco6aOCTh 3eNJIH no OTHomeHHJO K COJIHeq-
BONY R3JIytleHHIO (8JIL6e.A0),  - P8.AHYc 3eNJlli, E - H3JIytlaTeJIbH8.H
cnoco6HOCTL 3eMJIH, 0" -nOCTOHHHaH CT8.Ha- BOJTh8.Ha. H3JI}"I8-
TeJILHaH cnoco6HoCTL HBJI.HeTCSI (aapBHee HeH3BeCTHOH) c})YHReH TeN-
nepaTypLI. ITpe.o;nOJIarSeTCH, T,lTO E cJIa60 3aBHCHT OT TeNuepaTypLI.
O'leBHJ\Ho, 'ITO T - {f; . n03TOM}' yMeHhmeHHe R HB 1 % npHIIO-
ART K POCTY T aa 0,50/0, T .e. aa 1,4 K. B pe3YJILTBTe
T'= 288,4 K.
4. Yp8.BHeHHe COCTOSlHH.H H,lJ;e8JILHOrO ra38 113 N MOJIeKYJI HMeerr BH,lJ,
pV = NkT. ,l];B8 O,Q;HH8KOBLIX 06'beNa raa8 npH O,Q;HHX H Tex me p;aBJIe-
ERH II TeMnepBType co.o;epJKBT 0,lJ,HH8KOBoe 'IRCJIO NOJIeKYJI, nOSToMy'
nJIOTHOCT:& raa8 B K8.mJJ;oM 06'beMe nponopI(,HOHaJILH8 cpe.o;aeA Macce
MOJIeKYJI ra38.
nYCTh Hap;eKchI CC. II 48. 03H8Q8lOT .cyxoih II .BJIIDKHblih. ,D;JI.H
cyxoro B03.o;yxa Cpe,D;H.RH MaceR NOJleKYJI KOToporo pSBHB m c '
N c mcp
Pc :::; me V:::; kT .
AJISI BJI8)KBOro Boa.o;yxa, Cpe.o;HHH Maccs MOJIeKYJI KOToporo pBBH8 mB'
N. mBP
P. :::; m B IT:::; kT .
ECJIH M - Macea cyxoro Boa.o;yxa, TO
M
N e - 288 .
ECJIH M' - Macca BJISmHOrO B03,nyxa, TO
( M' M' )
N. -- 0,02- + 0,98- .
18 288
7 -79
193

TaR KaR N c = N., TO
M' 1
P. = - = = 0 9881;
Pc: M 28 8. ( 0,02 + 0,98 ) ,
, 18 28,8
p' = P. = O,9881p c = 1,2352 Kr/r.
5. MeX8.HKqecKASI NOm;HOCT HraTeJI.H BepTOJIeTa N, HeOOXOAH.
MaR 1J)ISI Toro, "'iT06LI OB Nor 3URCHyrh B B03e, paBHa npoH3Be.D;e-
BRIO NO)l;YJlSI BanpaBJIeBBoA: BBH3 CIIJ!.hI ,u;UJleHHSI T nonacTeA poTOpa
(paRHOA: no NOA}'JIIO Becy BePTOJIeTa P) H cpe):tHeA: CKOpocTH V)I;BHXCY-
eroCJI BHU3 cToJI6a Boa,u;yxa 1I0,u; J10IIaCTSlNH poTopa:
N = Tv.
JIonaCTR COa.D;8IOT nOTOR B03J:WX8, ,u;BH)I(HitCSl co CKOpoCT:hIO v,
dm
TaK 'lTO nepeHoCIIM8Jl38 e)tHYBpeNeHH N8.cca pUBa dt . lIJIo8Jth
a8XB8T8 J10nacTeA pasHa S. TalCuM 06paaON,
T=v dm ; dm =pSv=p=pSv 2 .
dt dt
Ec.n:H pa3NephI BepTOJIeT8onpe,zteJUIIOTCSI JIHBeA:BLIM pa3MepoN L, TO
P - L3; S - L"  v - g -Ii  N = Pv - Ill",
T8KHM 06pa30M, MOID;HocThABHI'aTeJUI YMeBLIDeHHoA No,u;e;m )l;OJPK-
H8 6hlT& pUBa N' = 0,0884 N.
3a'la 2
1. npoecc a-paclIa.zta aanDmeN B BH.D;e A -. (A - 4) + a.
9HepreTDllecKHH KpHTepHA: B03NO)KHOCTR a-pacna.u;a:
m A - m A -4 - m 4 > O.
't{UCJIO H Tun II}'KJIOaOB B pacne COXpaBJlIOTCJI, T8.K "lTO CJIe,nYeT
paCCNOTpeTL TOJIhKO saeprRH CBfl3U:
-BA+B A - f +Bf> O.
B
EcJIH A = a + M, r11. e a H b - KOHCT8.HThl, KOTOphte CJIe,u;YeT onpe-
}teJIDTL U3 rpa4»uKa, TO aepaBeHCTBO AJISI 9HepruA CBa3H IIpHNeT BBA
-A(a + bA) + (A - 4) [a + b(A - 4)] + B4> 0,
T.e.
-8 bA - 4a + 16b + B4 > o.
194

RaK cneP;YeT H3 rpa4»mc8, xopomee JIHHe:ft.Hoe npH6JIBmeHHe ,Q;JIa
B B OKpeCTHOCTR A = 100 T8KOBO:
A
B
A :; (9,6 - O,OO80A) MaB,
T.e. a = 9,6 MsB R b= - 0,0080 MaB. IIosToNYycJIoBHe npHHRMaeT BH,A
O,064A o - 38,4 - 0,1 + 25,0 > 0,
OTKy)1;8
 > 13,5/0,064 -= 210,94;  = 211.
2.1. TaR KaK A 4»HKCHpoB8HO, Be06xOP;HMO paCCMOTpe'l'b B BLIpa-
meHHH (1) YCJIOBH.H 38JJ;a"lH TOJIhKO )1;B8 npe)1;nOCJlep;HHX CJIaraeMLIX,
KOTOplaIe 38BHCJlT OT Z:
dB ;;; -2Za A -V3 _ a. ( -4A + 8Z )
dZ  A '
Z - 4a.. - AI . Z 79 25
max - 8a - 2 ( A 2/3 J ' max;;: , .
2a A -1/3 + ..--.!... l ac
C' A +-
4a..
2. 2. npO,u;R4»<J>epeHpyeM ypaBueuHe (1):
dB ;;: -2Za A -1/3 - au ( -4A + 8Z ) :i: 2a A-3/4
dZ cAP.
nOcJle)1;uee CJIaraeM:oe nOJIOmHTeJIhHO, ecJIH R3MeHeBRe Z us + 1
nepeBOT JI,Iij>O H3 "leTBO-"le'l'HOro B Ue"leTBO-Ue'IeTuoe, H oTpHTeJIb-
HO B 06psTHoM CJIyqae. 3aMeTHM, qoro npH STON A nOJlO:H<RTeJIbBO.
'tJTO )1;eJI8Tb c nOCJIep;BHM CJIaraeNhIM?
't{HCJIO Zmax )1;OJ1)KHO 6HTh D,eJIhIM, npH'IeM 'IeTHhIe 'IRCna npe,Q;noq-
THTeJIhHee HeqeTHLIX, n03TOM}' M03tHO npe,u;nonO)KHTh, '1lTO Zmu ;;; 80.
AJIa npoBeplCH B:&IqHCJIHM 3H8q:eUHSI CYMNhI TpeX nOCJIe)1;HHX cnsrae-
MhIX AJI.H p8.3HbIX 3HS'IeUHA Z:
Z
77 979,241
78 975,915
79 976,295
80 975,341
81 978,093
82 979,512
83 984,637
dB
BR,u;HM, 'liTO dZ NHHHM8JIbHO npH Z .... 80.
3TUM nOJJ;TBep*,ZJ;8eTCSI, '1TO Zmax = 80, T.e. aTO q:eTHO.'tJeTUoe .ft.ZU)o.
195

2.3. PaccNOTpHM TOJIhKO TpH nOCJle.n;HHX CJIaraeMIUX B ypaBHeHIUI
(I) II 0603HaQHM HX CYMMY Qepe3 X (CM. Ta6JI. 1). OCTaJIhHhIe "'IJleHhI
.nalOT nOCTO.HHHLIA BKJI8,A npH nOCTO.RHHOMA. QTo6hI onpeeJIHTht HB
R.HIOTC.fI JIB p;8BHhIe pa cTa6HJ1bHhIMH aeo6XOlJ:BMO BaATH pa3BOCTR
3UaQeBHH X JISI COCe,lJ,UHX ep II Cp8BHHTh 3TH pa3HOCTH C 3HepreTH-
qeCKHMH YCJlOBHSlMH, onpeeJUIIOH:MH B03MO>KHOCTb Toro HJlH HHOro
pacn8,Aa:
Tat1.nuqa 1
H,u;po X, MaB
l'I 491,06
lXe 489,16
lCS 492,54
13- .pacna..n;: n -+ p + e- t
Tpe6yeTC.H tJ.X> -1,30 + 0,51 = -0,79 (MaB);
(3+ -pacn8,A: p ---+ n + e+ ,
Tpe6YeTCSI tJ.X > 1,30 + 0,51 = 1,81 (MsB);
13- 13- -pacn8,A: 2n -+ 2p + 2e-,
Tpe6YeTc.KM> 2(-1,30 + 0,51) = -1,58 (MaB);
SJIeKTpoHHhIA. 38XBaT: e- + p -+ n,
Tpe6yeTC.K I1X > 1,30 - 0,51 = 0,79 (MaB).
HCnOJIL3Y.K pe3YJlbT8TIU H ,D;8IlHLIe Ta6JIHI{1U 1, 3BnOJIH.HeM Ta6JIH-
y2.
"Tat1.nulta 2
npoIteCC
.H.o;po P - -pacnB,D; p+-pacn8,lt 13- 13- -pacnB,D; 3axB8.T
3JIeJ<TpoHa
lI V 0 V 0
lXe 0 0 0 0
ICS 0 V 0 V
196

3aAa-q8 3
dm
1. QT06hI H3MeUHT:h B8 Av CKOpoCTb B03.n;ymHoro nOTOK8 --;it. ue-
06XO,AHMa CHJIa
F = Av dm .
dt
dm
B BameN cnyq:ae DOTOK "dt p8BeH
dm 7t 2
dt = xlpv = 4"Pl v.
BepTHKaJILH8.R KONnOHeHT8 Av paDH8 Av. = V sin t:, rOpH30HT8JIh-
BaR KOMUoaeHT8 6.v r = v(1- CDS E).
Teneplt NomBO 38nHC8TL BhIp8meUH.R .n;n.R Mo.n;YJIeA no.n;'beMHOfi
CHJIhI i H ropH30HTaJIhBOH CHJIhI conpoTHBJIeHHB VI :
L = -i pv 2 l 2 sinE,
 = 'ipv212(1- eosE).
MomHO npH6JIHmeHHO 38MeHHTh sin E Ha E.
2. MOIQ;HOCTL, Heo6xo.n;HM8g 1I.1IS1. TOro, 'lJTo6L1 y.n;epmHB8Th C8MO-
neT B ropH30HTaJIbHOM nOJIeTe. pUBa
P=Dv =(D I +D 2 )v.
MO.n;YJIh ropH30HT8JIhHOH CWIhI eonpOTHBJIeHHB  (06YCJIOBJIeH-
HOB TpeHHe:M) onpe.n;eJI.ReTC.R CKOpoCThlO H3NeHeHHSI. 38 C'IleT TpeHH.R HM-
nyJIhC8 B031J.YX8, o6TeK8IOero KpLIJIO:
dffl] drn,.
D 2 = VI lit - V 2 -;jt'
dm. dm
r.n;e lit - nOToK, HanpaBJIeHHbIit K KphIJIY, d/ - nOToK, CTeKalO-
HH C KpbIJIS.
TaK K8K KpLIJIO He .RBJI.HeTCSI HH HCTO'IJHHKON, HH CTOKOM, TO
dffl] dm 2 dm
lit = lit = dt = xlpv.
nO.n;CT8.BJI.H.H VI = V H v2 = V - !Jr.v, 11114001\1
D 2 = vxlpv - (v - !Jr.v)xlpv = xlpv 2 - xlpv 2 + v!Jr.vxlp =
= xlpvAv = 7tl 'pVAV = ttf pv 2 l 2 .
4 4A
197

p
.
.
..... ... DI
'-:':-.............
......
Pmm
o
Vo
v
PRc. 139
(ST8 CHJI8 conpOTHBJIeHH.R Bcer,lJ,a Hanp8s.neH8 B;D;OJIh nosepXHOCTH
KphlJI8. Kor,lta KpLIJIO H8.KJIOBeBO no,lt yrJION E, ropH30HTaJIbH8JI KON-
nOBeHT8 CHJlhI pUBa
!!!....pv 2 l 2 roSE RI .!!:!.... PV 2 l 2 ( 1_ £ ) RI .!!:!....pv 2 l 2 + 0 ( £2/ ) ,
4A 4A 2 4A
TaK 'liTO C 3a.n;aHBoA TOquOCThJO Ham npOCTOH OTBeT BepeH.)
nOJIHaS CHJI8 conpOTHBJIeHR.H D = DI + D 2 38BHCHT OT yrJI8 H8.KJIO.
Ba KpLIJIa £ H KOatPcPH:qHeHT8 conpoTHBJleBH.H f, ee NO,lJ;YJIL paBeH
D ::::;; 'It P V 2 l 2 [ (1- COSE)+ L ] RI pv2l2 ( !Sin 2 E + L )
4 A 4 2 A .
Tenepb BhIp83HN D Qepe3 M8.ccY, CKOpoCTb B pa3MRX KphIJIheS ca-
MoneTa. 3aMeTHM, 'liTO B c.nyqae rOpR30HTaJIbHoro nOJIeTa nO)];'heMH8JI
CHJIa )];OJUKH8 6LITh paBBa no MO,lJ;Ymo CBJIe T.HmeCTH, ,n;efiCTByK)II1;eA B8
C8NOJIeT:
L =  pv 2 l 2 sin£ = Mg  sinE = 4Mg .
4 npvq2
Teneplt MO)l(HO MHHRNH3Bp0B8TL NOHOCTL no v B.nR no E. Mid
BLl6RpaeN NHBBNH38IU1JO no v:
P = Dv = pv3l2 [ 1... + 1 (4Mg)2 ] = !!..pv3l2 L + 2(Mg)2
4 A 2 (npv 2 , 2 )2 4 A 1tpvl 2
dP = 3n pv 2 l 2 L _ 2(Mg)2 .
dv 4 A npv 2 , 2
198

dP
llpH V = Vo npOH3BO,ltB8JI dv paDBa HYJIIO, T8K 'lJTO CKOpoCTL noneT8
C nOCTO.HHHOH CKOpoCTLIO npH MHBHN:8JILHOH MOHOCTH P min onpe.n;e-
JIJleTCfi BLIp8JKeHBeN
4 8(Mg)2 A 8 ( M6 ) 2
V O = Sn 2 p 2 l4f = 3At npS .
3. IPa41HK MOBOCTH KaK WYHKD;,HH CKOpOCTH HMeeT BH,D;, nOK8-
38HHLIft HB pHC. 139, npH aTOM
P.. =:: !: po 3 1 2 [ L+ 1 (Mgfl. ] = 7t p o312L = 7t p v.8Sf.
mm 4 0 A 2(npv12)2 0 A 0
llO)];CT8BJIIUI va' no.nyqaeM
8 3 /4 (Mg)3 / 2 ( 8 ) 3/. (Mg)3/2
P.. == 1tpSf - - f l /4
mm {3At)3/4 (npS)1/ 2 3A {npS)1/2 .
4. llpHp8BHJleM BLIpBHteHHe )];JIJI. MHHHM8JILHO:A MOIItHOCTH (CM.
II. 3) pa3BHBseMoft MeXftHH'IeCKo:A MOII(HOCTH: P ..ex = NS = P min' OTKY-
.n;8
( Mg ) 3 / 2 = N ( 3A ) 3/4 (7tp)1/2
S 8 1/4'
Mg = N 2 /3 ( 3A ) 1 /2 (7tp)1/3 .
S 8 ,1/6
'tJHCJIOBLIe OTBeThI:
Mg/S = 35,6 H/M 2 ; V o = 8,60 M/c.

XXIX MeJKAYHapOAHGR
c!»M3MlleCKag OnMMnMGAG
1998 r., MCnaHAMIi
38}];a.,18 1
P8CCMOTpHM AJIHHHYIO TBepAYIO meCTKYIO npaBHJIhHYIO meCTH-
yronhHYIO npH3MY, HanOMHHBIOII(YIO 4»opMOH 06:&:['tJHhIH K8panAam
(pHC. 140). Macca npH3M:bI M pacnpe.n;eJIeHS paBHOMepHO. lIonepeq-
Hoe CeqeHHe npH3MhI HMeeT 4»OPMY mecTHyrOJlhHHK8 C )J;JUIHOH CTOpO-
RbI a. MOMeHT HHepn;HH I mecTHyrO.JIbHOH IIpH3MhI OTHOCHTeJIhHO ee
n;eHTp8JIhHOH OCH pBBeR
5 2
I=-Ma.
12
MOMeHT HHepn;:nH I' npH3MbI OTHOCHTeJIbHO ee pe6pa p8BeH
l' = 17 M a 2 .
12
1. IIpH3Ma OCh KOTOpOH ropH30HT8JIbHS, BU81l8JIe nOKOHTC.R aB HS-
KJIOHHOH nJIOCKOCTH, pacnonomeHHoH nOA ne60JIbmHM yrJIOM e K 1'0-
pH30HT8JIH (pHC. 141). (.n;onYCTHTe '[{TO nOBepXHOCTH npH3MbI HeMHO-
PRC. 140
200

hc. 141
ro BOI'ayTLI, TBK qTO OHa conpHKacaeTCH C nJIOCKOCThIO TOJIhKO pe6pa-
MH. BJIHaHHe 3TOH BorHyrocTH Ha MOMeHT HHep:qHH He npHHIIM8eTCH
BO BBHM8Bne.) 3aTeM npH3MY BhlBO,ltHT H3 COCTO.HHH.H nOROH, H OHa H8-
tlHBaeT HepaBHOMepHoe KBqeHHe BHH3 no nJIOCKOCTH. )J;onYCTHTe, qTO
TpeBHe nOJIHOCTblO HCKJ110'lae-r CKOJIbmeHHe npH3MLI II'ITO npH3Ma He
TepSleT KOHT8KTS C BaKJIOHHOH nJIOCKOCThlO. IIYCTh yrJIOB8..H CKOpOCTh
HenocpeJJ;CTBeHHO nepe.n; TeM KaR JJ;8HBoe pe6po YJJ;SPHTC.H 0 nJIOCKOCTh
6y,neT ro, a 0), - yrJIOB8Sl CKOpOCTh cpa3Y nOCJIe y,u:apa.
IIoKaJKHTe, '1TO
OJ, = S (Ot
II Bbf'lIHCJIHTe KosttMPHD;neBT s.
2. KRHeTH'lecKYIO 3HeprHIO npH3MhI HenOCpe)];CTBeHHO nepe.n; y)];s-
pOM II nacJIe Hero 0603HaqHM COOTBeTCTBeHHO K, II K,. IIoK8.mHTe, '1TO
K,= r K,
II BhlqHCJIHTe Koac}»q.H:qHeHT r.
3. ,lJ;JIH Toro qTOOLI npOH30meJI CJIe,lJ,yJOnpi:H Y,lJ,ap, 3H8QeUHe K, )];OJIJK-
HO npeBhICIITh HeKOTopoe MHHHMaJIhHOe 3UaQeHHe K i . min' KOTopoe MO-
meT 6hITb npe)];CTftBJIeHO B BH,lJ,e
Kj,mln = SMga,
re g:::; 981 M/C 2 - YCKopeHHe cBo6oJJ;Horo na.n;eHH.H.
Ha:H.n;HTe KOBc}»q.H:qHeHT 0, Bhlpa3HB era Qepe3 yrOJI H8.KJIOH8 e H KO-
a<pqmnueHT r. (HcnOJIh3yATe 3JJ;eCb 8.JIre6pB.1NeCKH:A CIIMBOJI r, B He ero
'tlHCJIOBOe 3H8'1eHHe.)
4. ECJIH BLIllOJIHeHO YCJIOBHe n. 3, TO KHHeTH'leCK8H 3HeprHH KI
6Y,lJ,eT npn6JIHmaTbCH K nOCTommOH BeJIWlHHe K,.o no Mepe Toro, RaK
npH3M8 KBTHTCH BHH3 no HaKJIOHHOH nJIOCKOCTH.
IIpHH.JIB, "lTO 3TOT npeAeJI cym;ecTByeT, nOKWKHTe, 'lToK,.oMomHo
3anHcaTb B BRJJ;e
K,.o=kMga
II HaTe Koaq.cpHD;neHT k BhIpa3HB ero '1epe3 e II T.
201

5. BhI1:IHCJIHTe C TO'laOCTItIO 11.00,1 0 MHBHNaJIbHLIit yroJI HaRJlOHa
eo' npH KOTOpOM HepaBHONepaoe KaqeHHe, Ha'l8BmHCb, 6YAeT npoAOJI-
:JK8TbCa 6eCKOHe1inO 1J.OJIro.
3aJJ;aa 2
JIeASUIoit HT - aTO TOJICTLI:ft CJIo:A JIbP;8 (TOJIIItHHOit 1J.0 HeCKOJIb-
KHX KBJIONeTpoB), KOTOpLlA nOKOHTC.f.I H8. nOBepXHOCTH 3eNJIB, npo.
CTHp8JICL ropH30HTaJIhHO sa AecSITKH H COTHH KHJIOMeTpoB. B .u;asHoA
38.,AaQe paCCM8TpHBaJOTca T8JlHHe JIb)J;a H nOBe.u;eHHe B01J.Ll no.u; CJlOeM
JIbJ:J;8, Hax01J.HMC.H npH TeNnepaType, 6JIH3Koit K TO'lKe nJIaBJIeSH.H.
By,u:eM CqHTaTL, 'tJTO B TaKHX YCJIOBH.HX JIe.u; C03.u;aeT .n8BJIeHHe RaK B83.
Raa )l(BJ(KOCTL, HO Ae4>oPNHPYeTC.H R8K XpynRHH MBTepHaJI, rJlaBHLIM
06pa30M nYTeM BepTHKaJILHOrO CMeIIJ;eUH.R.l1.JI.H pemeHH.H 38,AaQH H8.N
H3BeCTHhI cJleAYIOe 1J.8HHLle:
nnOTBOCTb BO)J;L1
P. = 1,000. lOa Kr/M 8 ;
PJI = 9,17.10 2 Kr/M 8 ;
ell = 2,1 . l0811.m/{Kr. K);
LJI = 34 . l0511.)K/Kr;
P D = 2,9.10 3 Kr/M 3 ;
nnOTHOCTL JIbp;a
Y)J;enbHBa TelIJIoeMKocTb BO,A1d
Y AenbH8a TelIJlOT8 DJI8.BneHHH Jlb)J;8
nJIOTHOCTb nopo.A H MarMbI
y.o;eJlbH8S TenJIoeMKocT!> DOpO.u; H M8rMbI
C n = 700 l1.m/(Kr. K);
Y,lteJIbH8H TeDJIOT8 nnmmeHHH nopoA
H M8rMbI
L D = 42 . l0511."JK/KI';
TeMuep8Typ8 Dn8BJIeHHH JIha
t = 0 0 C H nOCTOJlHHB;
nOBepXBOCTH8S1 nnOTHOCTb 'l'eUnOBOro
nOTOKBQepe3fiOBepxHocTL3eMJlH
J Q = O06 BT/M 2 .
1. PBCCNoTpHTe TOJICTLI:A CJIOH JIb,ZJ;a, pacnOJIOmeHHLIH HB nOBepx-
HOCTH 3eMJIH, Qepe3 KOTOpYIO nocTynaeT CpeP;HHH TenJIOBoit nOTOR.
nOJIL:3y.RCL .u;RHHbIMH Ta6JIHQ;hI, BHCJIHTe TOJIHy d TaJOII.tero 3& ro,u;
CJlO.H JIhAa.
2. PBCCMOTpHTe BepxHlO1O 1I0BepxHoCTb. TB. IToBepxHocTL 3eM-
JIB H8KJlOHea& nop; yrJlOM a K rOpH30HTY. BePXHH.H nOBepXaOCTL tqHTa
06pa3YeT yrOJI pc ropH30HTOM, KBK nOKa3aHO aa pUC. 142. TOJIII:tHBB
JIL,lJ,B B TOqKe x = 0 paBHR hoe TaKHM 06pa30M RU)KBHS H sePXHSUI no-
BepXHOCTH JI{HTR .MOryT 6LITh OIIHcaHhI ypaBHeHH.HMH
Yl = xtg a,
Y2=h o + xtg p.
(I)
202

y
,
,
JIeOSl.HOU Ulum
--
x
Puc. 142
IIoJI)"IHTe c}>OPNYJlY ,Q;JI.R .n;aBJleHHH p Y HHJKHeii KpoMKH HT8 B
a8BHCHMOCTH OT ropH30HT8.JIhHO:A KOOP)];HU8TLI x. lloJl}"lHTe COOTBO-
meBHe Mem.n;y yrJIRMH a H p, npH KOTOpoM Bo.n;a B CJIoe Me)l(JJ;Y JIL)J;OM H
3eMJIeli He 6Y,D;eT .u;BHraT})CSI BH B O,lJ;ROM HBnpaBJIeHHH. IIoK8)RHTe, 'liTO
:no ycJIOBHe HMeeT BHJJ: tg P = B tga. HaA.zJ;HTe Koacp4JHeHT B.
JIUHH.R, OnHCLIBfteMftJl yp8.BHeHHeM YI = 0,8 x, Uft pRC. 143 nOK83hl-
BseT nOBepXHOCTL 3eMJIH no,D; JIe.n;.RHLIM HTOM. TOJIHHft Jlh,Q;a kg npH
x = 0 PUB8 2 KM. IIoJIRr8JI, T{TO BO.D;a IIO,Q; HTOM H8XO)];HTC.R B p8BHOBe-
CBB, Hft rpac}>HKe H306pa3HTe JIHHHJO Yl H )];068BhTe JIHBHJO Y2' nOK83hI-
BaIOytO BepXHJOIO nOBepXHOCTh Jlhp;ft.
y
",'
,.
,.
y=h o
JI eORHou
Ulum
3eMJUl
y=o
x=o
x
he. 143
203

3. BHYTPH TOJICTOrO CJIOH JIhp;a, H8X0.AamerocH Ha ropH30HT8.JIhHOA
nOBepXHOCTH 3eMJIH nepBOH8'11aJILHO HNeBmero nOCTOHHHyIO TOJIIqH-
BY D = 2, 0 KM, B pe3YJILT8Te Ta.RHH.H JIL)J;a BHeaanHO 4»opMupyeTclI
KOHHQeCKa.H nOJIOCTh, nOJIHOCTLIO a8nOJIHeHHRH BOJJ;Oft C BIdCOTOft
H = 1,0 KM H p8.JJ;HYCOM r = 1,0 KM (puc. 144). BYJJ;eM C'1lHTaTL, '1lTO oc-
TaJILHO:H nep; pear:npyeT aa 06p830B8HHe STO:H nOJIOCTH TOJILKO BepTH-
KaJIhHhIM CMeeHHeM.
3annmHTe 8UaJIHTHQeCKOe BhIpameHHe H Ha MHJIJIHNeTpoBo:A 6y-
Mare H306pa3HTe rpact>:nQeCKH 4»opldY BepxHeit nOBepXHOCTH nep;HHoro
HTa nocJle YCTaHOBJIeUHH rHJU>0CTaTHQeCKOrO p8BHoBecHH.
4. EmeroJJ;HftSi 3Rcne.o;HD;HH Me)K,l(YHftpoJJ;HOH rpyrrnhI yqeuwx HC.
cJIe,lJ;YeT JIeJJ;oBhIH HT B OJJ;BOM H3 paAOHOB AHTapKTHKH. 06hrt1ao STOT
paAOB npep;CT8BJIHeT co60it o6mnpHoe nJIaTO, HO B STOT p83 Y1IeHhIe
o6uapymHJIH rJIy6oKyIO KpaTepoo6pS3HyIO Bn8)l;HHY B 4>opMe nepeBep.
HYTOro KOHyC8 rJIy6HHoH h = 100 M H p8,l);HYCOM r = 500 M (puc. 145).
TOnII1;UHa JIhp;a B aTOM MecTe 2000 M.
IIocne JJ;HCRYCCHH yqeubIe np:nmJIH K BhIBOJJ;Y, TlTO CKopee BCero nop;
HTOMnpOH30mJIO ue60JIhmOe BYJIK8UUQecKoe H3BepmeHHe. He6oJIh-
mOH 06'heM M8rMhI (pacnJI8BJIeHHoit nopoP;LI) npOHHK 'Iepea HH)KHIOIO
KpoMKY ne,a;SlHoro HTa, 3aTBep,n;en H OCTLIJI, paCTonHB HeKoTopoe RO.
y
--------- ---y=D-----
JI eiJllHou
14 um
-- y=H
3e.M.Jl..1l
x
PRC. 144
204

x=r
JI eiJJl1tOU
14 um
x
x=o
he. 145
JIHqecTBO JIb.n;a. YqeHhIe nbIT810TCH OI:l;eHHTb 06'beM SToro . BTOp)KeHHSI.
(npOHHKEoseHHe NarMLI UOA JIeA.REo:A HT) H 06'b.RCHHTb, 'lITO npo-
H30mJIO c paCTOnJIeHHo:ft So.u;oft, CJIe)];YIOID;HM 06paaOM.
lIpeAnOJIO)KHM, qTO JIe,lJ; .D;BHraJIC.H TOJIhKO BepTHK8.JIhHO. By)];eM
CQHTaTh T8.Kme, qTO BHaQ8.JIe MftrMa 6hIJIa B UOJIHOCTblO p8CnJI8BJIeH-
RON COCTO.HHHH npH TeMnep8Type 1200 DC, qTO COOTseTcTByeT TeNne-
paType WIaBJIeHH.H N8rNhI. }J.JI.H ynpoID;eHHH 6y.u;eM CQnT8TI., qTO BTOP-
)KeHne M8rMhI nOA ne.u;$lHo:ft HT nMeno cbopMY BepTHK8.JII.HOrO KOUYC8
C KpyrJIhIM OCHOS8HneM, pacnOJIOmeHHhIM SepTHK8.JILHO UOA KOHHqec-
ROft BOpOHKO:A: aa nOBepXUOCTH. BpeM.H nO,lJ;lUlTHH MarMhI 6hIJIO M8.JILIM.
no CpaBHeHHIO C BpeMeHeM TenJIoo6M.eH8. npe.u;nOJIOmHM, qTO TenJlOT8
pacnpOCTp8H.ReTC.H rJIaBHLIM 06p8aOM no BepTHK8.JIH T8.K qTO 06'beM
BOAhI, 06p83YIOm;e:ftcB npn llJI8BJIeHHH JIb.u;a Bce speNH OrpaHHQeH KO-
HH'QeCKOH nOBepXBOCThlO JIh,lJ;8, p8CnOJIOmeHHOro a8)l; I:I;eHTpOM BTOp-
)KeUHH MarNhI.
B STHX YCJIOBH.RX Ta.RHHe JIL,lJ;8 npOHCXO,lJ,HT s .u;B8 STana. BuaqaJIe
o6paayrom;8S1CSI npH UJIaBJIeHHH BO)];8 He HaXO.u;HJI8Ch B COCTOSlUHH rHp;-
pOCTaTHqeCKoro paBHOSecHSI a8 nOBepXHOCTH M8rMhI H nOSTOMY yre-
Kan8. MO)KHO CQHT8Th, qTO 9T8 Bo.n;a IIMena TeMuep8Typy 0 DC. 3aTeN
6LIJIO ,lJ,OCTnTHyro paBHOSeCHe H BO.n;8 nepecTaJIa Te'lL H CTaJI8 H8R8rr-
JIHSaThC.H a8,ZI; npOHHKmeH MBrMOH. HafiJ:l;HTe CJIe.u;yrom;He BeJIH1:IHUhI
nOCJIe .n;oCTH.>KeUH.H TenJlOBOrO paBHoBecHH:
BhICOTY H BepmHHhI Bo.n;.HHOrO KOHYC8, C«t»OpMHpOB8Bmeroca no.o;
ne,lJ,.HHhIM m;HTOM OTHOCHTeJIbHO nepBOHaQ8.JIhHoit nOBepxHocTH RH)K-
Hero OCHOB8UH.H JIe.o;Baoro HT8;
BLICOTY hI BTopmeHHH M8rMhI (CM. pnc. 145);
205

o6IqyIO Maccy 06pa30B8.BmeAcK BOALI m oOJlll H Macey m' BO,ltLl, KOTO-
p8JI YTeKJla.
Ha'lepTHTe B NacmTa6e ua NHJIJIHMeTPOBOH 6YNare nOBepXHOCTb
BTopmeHHK 06pa30B8BmeAcK BO.u;LI. HCnOJIh3yii:Te T8KytO me CHCTeNy
KoopAuBaT, KaK ua pRC. 145.
3aq:a 3
B 3ToA 38.Q;a'tJe NLI aB8.JIH3HpyeM R RHTepnpeTHpyeN H3MepeBHJI
pa,u;uoH3.nyq:eHHA OT COCTasHOro HCTOq}fHKa B HameA raJI8.KTHKe, BLI-
nOJIUeUHLte B 1994 r. lIpHeMBHK fiLtJI HacTpoeH ua wupoKuA .u;Rana-
30B p8.):(HOBOJIH A1IHHOA B HecKOJILKO C8HTHMeTpOB. Ha pHC. 146 nOKa-
3aRB cepUK H306pWKeBHA, aBnUC8HUWX B pa3Hoe BpeNa. JIUBRR no-
CTO.HHHOA HBTeBCHBHOCTH o603ua'tJeHhI KOHTypaMH - npHMepHo T8K
me, KaK KOBTypLI BLICOT ua reorpac}>lNecKo:A K8pTe. ABa M8KCHM)1M8
HHTeHCHBHOCTH nOUHMaIOTca K8.K a oCYbeKTa, y.u;8JIalOHeCH OT 06-
ero qeHTpa, KOTOpLIA 0603u8QeH ua pHcytrKe KpeCTHKON. (eHTp KO-
TOPHi: ClIHTaeTCB aa<l>HKCHpOB8HHhIM HenO,ltBH)KHO B npoCTpaBCTSe,
TaRme .RBJI.ReTC.R MOHLIM HCTO'tJHHKOM H3JIyq:eUH.R, HO npeHMym;e-
CTseUHO )J;pyroit ,AnHHLI BOJIHhI.) H3NepeHHII B pR3HLIe )];HH 6LIJIH CAe-
JlaHhI B O)];UO BpeN.JI cyrOK.
M8cmTa6 pHCytrKB ,lt8eTC.H B BH.n;e OTpe3K8 B 1" ( I" = 30 ). Pac-
CTOHHHe1l.0 HOOecHOro TeJI8 B qeHi'pe pHCytlK8, nOK838HHOM KpecTHKOM,
On;eHHB8eTCH s R = 12,5 Knc (1 KUC = 3,09.10 19 N). CKOpoCTh CBeTa
c = 3,00.10 8 M/C. OrteHKR norpeIUHocTeA DpH pemeHHH He TpOOyeTCJr.
1. 0603B8'1JH:M yrJIOBLle nOJIOmeHHH 1l.BYX y.P;aJImoHXC.H HCTOTJHH-
KOB p8.,lJ;HOH3JIyqeHHII no OTHomeBHIO K o6m;e:MY n;eHTpy Qepe3 8 t (t) B
8 2 ( t), r.p;e HH/l,eKC 1 OTHOCHTC.H K JIeBO:MY IICTOQHHKY, a HH,lteKC 2 - K
npaso:MY; t - speNs Ha6nlO.u;euHH. YrnoBhIe CKOpoCTH HCTO'tJHHKOB,
Ha6JIIO)];aeNLlX c SeMJIH, 0603Ha'lHN 'lepea co 1 H 0>2. COOTBeTCTSYIOIItHe
KS)KYHeC.H nonepeqHLIe JIHHeAHLle CKOpOCTH .u;syx HCTO'tJBIiKOS 060-
3H8'lHM 'tJepea V'I,.1. H v' 2,.1. .
C nONOLIO pRC. 146 nocTpoiiTe rpa«t»RKH 38BHCHMOCTeii 8 1 (t) H 8 2 {t)
AJISI onpeJteJIeHHH 3H8'tleHHA C0 1 H co 2 B yrJIOBLlX MHJIJIHCeKyu.zI;8X B .u;eHIJ>
(:MC/ AeHb). T8Kme HsA,ZtHTe 3H8.'leHH.H V'I,1.' v' 2.1.. (HeKOTOphle pe3YJIb-
T8ThI MOryT BaC YAHBHTb.)
2. QTo6Ll pa3pemHTh B03HHKwee B IT. 1 HeOYMeHHe, paCCMOTpHTe
HCTO'lHHK CBeTa.1l.BHmym;HAcJr H3 TO'lKHA co CKOpoCThlO V no.u; yrJIOM
(J) (0 " qJ  7t) K nBnp8.BJIeHHIO H8. Y1l.8JIeHBOrO H86JlIOJ(8TeJIK 0 (pHC. 147).
CKOPOCTb Momuo 38IIHC8Tb KaK v = pc  r.u;e C - CKOpOCTL CBeTa. Pac-
CTOSlHHe)];O HCTO'iHHK8, H3NepeHBoe Us6JlI01l.8TeJIeN, PURO R. YrJIO-
206

'.-'!
-:...
27 NapT8
3 anpeJIJI
9 anpeJI.R
I
I I
 16anpeJUI
I I
* 23anpeJU1
, I
, =-- I
 30anpeJUI
1"
PHc. 146
Bas: CKOpoCTL HCTO'qHHKB C D03HItHH Ua6JDO,u;8TtUUI - co, a K8.)Kym;a.aC.R
JIHHeAH8.s: CKOpoCTL, nepneHJJ;HKYJl.s:pH8.a: JlHHHB Ha6JIIO.n;eHH.R, - Ul..
Onpe)];eJIHTe co H II 1..' BhIpsmeHHhIe'tJepea p, R H q>.
3. .I{onycTHN, "iTO )];B8 pa3.neT8lOBXCa 06'heKT8, OnBcaHHLIe BO
BBe,u;eHHH H n. 1, ,ltBHmyrclI B npOTHBOnOJIOmHLIX H8.IIp8BJIeBHBX C OJUl-
HSKOBOA CKOpOCTLIO v = pc. Tor.n;8 peaYJlLTBThI n. 2 nOaBOJI.JDOT nOJlY-
qHTL p H t:p 113 yrJIOBLIX CKopoCTeA co. H CO 2 B paCCTOIIBHII R. S,lteCL qJ -
yrOJI, onpe.n;eJIeBHLIH B n. 2 )];JIll JIeBoro 06'LeKTB, COOTBeTCTBYIOero
HHJ];eKCY 1 (CM. D. 1).
BbIBe,ttBTe 4>OPMYJIhI ]J.1I11 J3 H <p 'Ilepea H3BeCTHhIe BebI H onpe-
,l1.eJIHTe ux 3UaQeSH.H UB )];8HBhlX D. 1.
4. J(JI.R NOJ(enH TOJILKO C O)];HHM 06'LeKTOM (CM. n. 2) HSHAUTe YCJIO-
Bile, npH KOTOpOM KBmy:IIl;as:c.s: nepneU,l(RKynHpHBII CKOpOCTb V'.l. 6y-
,l1.eT 60JILIIIe CKOpOCTH CBeTa c.
o .
A A
R
PRe. 147
207

8annmHTe aTO YCJIOBHe B BH,Ae p > f{qJ) H ,u;aHTe auaJlHTWleCKoe
BbIpameHHe $YHKn;HH f.
Ha MHJIJIHMeTpoBoft 6YMare BhI,AeJIHTe Ii 38mTpHXY:HTe 06naCTh
nJIOCKOCTH p, <p B KOTOpoii BhInOJIH.HeTC.R YCJIOBHe v'1. > e.
5. ):tJI.R 3TOii me MO)];eJIH, TOJIbKO C O,u;HHM 06'LeKTOM (CM. u. 2), HaA-
,lJ,HTe BbIpameHHe ,AJI.H M8KCHMBJIbHOrO 3H8'1eHH.R (u'..l)max K8myn::J;eitC.R
nepUeH.AHKYJIHpROH CKOpOCTH u1. ,AJIH 38,lJ;aHHOro p. 06psTHTe BHUMa-
BIle, tlTO ST8 CKOpOCTh CTpeMHTC.fI K 6eCKOHeQHOCTH npH p - 1.
6. On;eHx8 3H8t1eHH.R R, npHBe.n;eHH8S1 BO BBe.o;eHHH, He OtleHb H8-
.n;eJKH8.. II03TOMY yqeHhIe Ha1laJlH HCR8Tb 60JIee Ha,u;emBhIH II np.R-
MOit MeTO,A onpe)];eJIeRHSI 3U8'1eHHH R. OAHR H3 MeTOAOB Momer 6hITh
TaKHM.
AonYCTHM, 'liTO MbI MomeN H3MepHTh CMeII.1;eRUhIe no )'];onJIepy.o;JIH-
HbI BOJIH Ai H A. 2 ,lJ,JI.R H3JIeHHA ABYX pa3JIeTaIOIItHXC.R 06'LeKTOB, COOT-
BeTcTBYJOHe H3BecTBOA nepBH'tJBoH ,AJIURe BOJIHbI A.o )];JI.R STHX 06'LeK-
TOB B COCTO.HHHH nOKOH.
HCnOJIb3Y.R yp8BHeHHe ,lJ,JUI peJI.RTHBHCTCKoro st}JcpeKT8 AonJIepa
A=A o 1-(3
.Jl - (32
Ii npHHHB, RaK paHee, 'liTO 068 06"beKTft ,lJ,BHJKYTCH C OHB.KOBOH CKO-
pOCTbJO v. nOK8mHTe "lTO HeH3BeCTBoe 3H8'tJ.eHHe J3 = vIe Nomer 6LITL
BhlpameHo "lepea AO AI II A:a CJIep;yroII\HM 06paaoM:
(3= 1- a#. .
(AI +#.2)2
qeMY paBHO "lHCJIOBOe 3HftqeHHe Koacp<l>HD;HeuT8 a?
BLI MomeTe a8MeTHTL, '1TO npe,u;JIOmeHHhIft cnoco6 H3MepeHHSI AJIH-
HLI BOJIHLI Uft np8KTHKe n03BOJIHT nonyqHTb HOBYJO OJJ;eHXY paCCTO$.l.-
HRH.
PeweHMII
3a.n;aq:a 1
1. II epBblu cnocot5 peULenUJl. lIpH y,u;ape npH3M8 H8"lHH8eT BparqaTb-
C.H BOKpyr HOBOH OCR, T .e. BOKpyr pe6ps. KOTopoe B STOT MOMeHT y,u;a-
pHJIOCb 0 UJIOCKOCTb. CHJI8 peaKH nJIOCKOCTH He C03,n;aeT MOMeBT8
OTHOCHTeJIbHO T8KOft OCH TaK QTO 38 RopOTKoe BpeMSI coy,u;apeHH.K 1140-
MeHT HMnYJlhCa OTHOCHTeJIbHO pe6pa COXp8.HSleTC.R. HMUYJIhC npH3MhI
208

PHC. 148
KBR :Q,eJIoro Hanp8.BJIeH TaK me RaI< U CKOpOCTL n;eHTpa Macc (p = Mv c 
l'.ZJ;e HH.ZJ;eKC C COOTBeTCTByeT n;eHTpy Macc) H 3'1'0 HSnpaBJIeHHe JIerKO
YCTaHOBHTb eCJIH MLI aRaeN OCb BpaeHHS B .n;8HHLIfi: MOMeHT BpeMe-
HR. nepe)); caMbIM coy.n;apeHHeM HMUYJIbC Pi nanp8BJIeH nOA yrJIOM 30°
BHH3 no OTHomeHHIO K nJIOCKOCTH HO naCJIe YAapa HMnYJlbC PI Hanp8B-
neH nop; yrJIOM 30° BBepx (puc. 148).
QT06bl HaRTH MOMenT HMUYJIbCa OTHOCHTeJILHO pe6pa, KOTopoe
HcnblTbIBaeT cOY,lJ,apeHHe B MOMeHT BpeMeHH nepe.n; caMblM y.n;SPOM,
HCnOJIb3yeM yp8BHeHHe, CBSl3LIBBJOee MOMeHT HMnYJIbCa L OTHOCH-
TeJIbHO npOH3BOJIbHOH OCR C MOMeHTOM HMnYJIbCa 4 OTHOCHTeJIbHO
napaJIJIeJIhHOH OCR, npoXO,tJ;Hm;eH qepea eHTp M8CC:
i = 4 + M[rc, Vc]'
(1)
B ,A8HHOM CJIyqae 3Ta opMYJIa npHMeHseTCH KOCH B TO'tlKe coyp;a-
peRHS, TaK 1:1'1'0  eCTb BeKTOp, HanpB.1UIeHHblH 0'1' 3'1'OH '1'OQKH K n;eHT-
PY MaCC (CM. pHC. 1(8). 06a BeKTopa B npaBoH qaC'1'H BLlpameHH.R (1)
Hanp8.BJIeHhI 0.ZJ;HH8KOBO. n03TOMY .ZJ;JIS 3THX BeJIHqUH nepe.n; COYAape-
HHeM nOJlyqaeM*
1[ - - ]1 - . 30 ° - 1 2
  Vcr - Vcr sIn - -u rot'
2
* 3,n;eCh MOJRHO BOCnOJIJ.30B8TbCIi H TeopeMOH IIlTe:HHepli npHMeHHB ee
JJ;BamJ{bI npH nepexoJJ;e OT npe.o;hI,n;yl1.teii OCH coy.n;apeHHII K OCH npoxo.o;SilI.{eH
1lepe3 eHTp Macc, 8 a8TeN K nOCJIe.o;YIOl1.teH OCR coy.o;apeHMS.
209

1 2 ( 5 1 ) 2 11 2
1, = Iffi, + -Ma ffi, :;; - + - Ma rot = -Ma ffi j _
2 12 2 12
C )];pyrOH CTOpoHhI MOMenT HMnYJIbCa OTHOCHTeJI&HO pe6pa cpaay
nOCJIe cOY,lJ,apeulUI paBen*
L I I 17 M 2
 = OJ, = 12 a OJ"
r1J,e HHp;eKC f BCer1J,a OTHOCHTCSI K cHTyaItHH nocJle YAapa.
CJIeT OTMeTHTh, q-ro pa3HH B 3USQeHH.HX L, H L, B03HHKaeT
H3-3S pa3Horo Hanp8BJIeHHJI vel H vel . TIOCKOJlhlCyyrBep)K1J,aeTCSI,llTO
MOMeUT HMllYJIbCa COXpB.HSleTCSI, T.e. L, = L" nonyqaeTcSI COOTHome-
BHe Me)K.lO' yrJIOBhIMH CKOpOCTSlMU:
11/12 11
OJ, = 17/12 w, = 17 COt.
II03TOM}T
11
8= 17 .
OrMeTHM, llTO 8 Be 38.BHCHT OT a, CO j H 9.
Bmopou en-oeoD peWeHUJl. npH Y)];ape pe6poM npH3Ma no.nyqaeT OT
UJIOCKOCTH HMnyJIbC p. B TO me BpeMSl pe6po, nOKHJJ;8JOee WIOCKOCTb,
He HcnhITLIBaeT BHRaROH pe8.KI\HH.IIMnyJIbC MomBO pS3JIO.>KHTh Hae
COCTa.BJI.moHe: COCT8BJISIIOm;YJO Pn , napaJIJIeJIhHytO H8.KJlOHHO:A nJloc-
KOCTH (nOJIO>KHTeJILHOe HSnp8BJIeHHe BhI6HpSeTCSI BBepx B1J,OJIL Ha-
KJIOHHO:A nJIOCKOCTH HS pRC. 148), H COCT8BJIIIIOII{yJO Pl.' nepneHKY-
JISpHytO nJIOCKOCTH (nOJIO>KHTeJILHOe HSnp8BJIeHHe - BBepx, nepneu-
J];HKYJIHpHO UJIOCKOCTH U8 TOM)Ke pHCy:HKe).
MO:mHO COCT8BHTL TPH ypsBHeHHSI C TpeMH BeH3BeCTHhlMH: PI ' pJ..
CO,
H 8:;: - . 3UaQeHHe PI onpe.n;eJIHeTc.H H3:MeHeHHeM nap8.JIJIeJlbBOH KOM-
CO.
* AJIhTepH8THBHO:
L, = 1m, + MI[, Vcl]1 = 1ro, + Ma 2 w, sin90° =
( 5 ) 2 17 2
= - + 1 Ma ro, = - Ma co,.
12 12
210

nOHeHTLI HMIlYJlLCa npH3ML1, a 3B8QeBUe P.l paDBO COOTBeTCTByIOII1;e-
MY H3MeHeBHIO nepneH,D;HKYJIsr:pBOa KO:MnOHeBTbI. HNeeld:
PI = M (W t - w j ) a .J3 ,
2
PJ. :::;: M (w. + W j ) a.!..
2
H8.KOBe}\
pa  - J¥I  = I(m, -0>;).
TaK K81< npaB8JI qaCTlt pUBa H3MeHeHHJO MONeHTB HMnyJIhCa OTBOCH-
TeJIhHO }\CHTpa :Macc. Tenepb MOmBO pemHTb nonyqcHBLIe YPaDUCBHH
w,
H BaaTH OTHomeHHe 8:::;: - . PcaynTaT, KOUe1iaO, cOBn8,A8eT c npe,lUd-
(0,
,lJ,YITI;HM.
2. JIHHCaU8JI CKOpOCTh n;eHTpa MBce nepe.n; coy.n;apeHHeM PUB8 aro,
a nOCJIc coy.n;apeUH.H - am,. H3BecTHO, 11'1'0 Bcer.o;a NOJKBO npe.n;CT8BHTb
KHUeTHq:ecKYIO 3HeprH1O BpaII1;8.IOII1;eroc.R TBep.n;oro TeJIa K8.K cYMMY
KHHeTH1IecKoH 3HeprHH .o;BHmeHHH n;eHTpa Mace TeJIa H KHHCTH'tJecKOH
3HeprHH ero BpaTeJIhHOrO .o;BHmeHHH:
1
KUMA:::;: 2' I c m2 + Mv!.
BHJ];HO, 'lITO B RameN c.nyqae KHHeTH'tJeCK8H3HCprU.H K n 01l.A nponop-
HOH8JIbHa m 2 K8.K nepe coy.n;apeUHeM, TaK H noeJIe Hero, n08TONY
( 11 ) 2 121
K, :::;:rK,:::;: - K. =-K"
17 289
121
r == - s;cj 0,419.
289
3. KHUeTH'IeCK8SI 3HeprH.H nOCJIe coy.o;apeUHH K f .o;OJIJKH8 6bITb .n;o-
CTaT01lHO SeJIHKa, 'qTo6L1 }\eHTp Mace Mor no.o;H.HTbCSI Ha caMYIO Bepx-
RIOlO n03HIJ;HIO npSINO H TO'llKOH KOHT8KTa. YrOJI, Ha KOTOpbIft .n;OlI-
meH .o;JISI 3TOro nOBepHYTbC.H pa)J;HYC-BeKTop . paseH
d
X = "2 - 9,
r.n;e a:;:: 60° - yrOJI npH BepmHHe npaBHJIhHbIX TpeyrOJIhHHKOB cxo.n;a-
m;HXCH B n;CHTpe mecTHyroJIbHHKa. :3HeprHH, Heo6xo.n;HMaH .o;JI.H nO]t"L-
eMa }\eHTpa :Macc, pasHa
Eo == Mga (1- cosx)=Mga [1- cos (30 0 - 6)].
211

OrclO,lJ;8 nonyqaeM YCJIOBHe
K,= r K, > Eo =M ga [1- cos (30 0 - e)].
CJIe,lJ;OB8TeJlbHO,
S = ![l- cos(30° - 8)].
r
(3aMeTHM, qTO cos (30 0 - S) = ..J3 cosB + .!.sinS. )
2 2
4. IIycTb Kt. n H KI' n - 3H8QeHHSI KHHeTHtleCKOH SHeprHH cp8ay
nepe,lJ; n-N coy.u;8peBHeM H cpsay nocne Hero. MLI nOK838JIH, QTO
Kt.n = r K i . n-
121
r.u;e r = 289 JJ;JIB reKCarOHaJILHOH np:H3MbI.
Mem,D;y,l1BYM.H nOCJIeJJ;yIDIII;HMH coyp;speHH.HMH BHCOT8 n;eHTpa Mace
DpH3MLI YMeHLm8eTC.H H8 BeJlHQHHY a sin e B KHHeTH'IeCKBH 3BeprH.R
B03p8CTaeT 38 C'IeT 3Toro Ha BeJIH'llHHY
 = MgasinS.
IIo9TOMY
Kt. n+1 = r K,. n + 1::1.
'tJTo6LI HdTH npe,lJ;eJI, He Tpe6yeTC.H BLIDHCLIBaTb nOJIHOe BbIpWKe-
HRe ,D;JIH Kt. n KBK YHKn;HH Kt. 1 H QHCJI8 n. YTBepm.u;eHHe 0 cym;ecTBo-
BBHHH STOro npe,AeJIa cO,AepmHTCB B YCJIOBHH 38)I;8'1H. Tor.u;a JJ;JISI AOC.
TaTOQHO 60JIbmUX n MomHO C JIJ060H 38.,lJ;8.B.HOft TOQHOCTbJO nOJIO)fCHTL
Ki.. n+ 1 :::::c K i . O. IIo3TOMY npe.o;eJI K , . O JJ;OJImeH YJJ;OBJIeTBOp.HTL TaK H83LI.
BSeMOH HTepBn,HOHHOH OpMYJIe
Kt. 0 = r K i . 0 + /1,
oTKy.u;a MOJKHO nOn}"lRTL BbIpWKeHHe
6.
K,.n = K,.o = _ 1 npH n ---+ 00.
-r
TaRRM 06paaOM, MomBO 3SDHC8Th
6.
- = kMga,
l-r
OTCIO,D;8
k= /1
(1- r)Mga
Mgasine
=
(l-r)Mga
sine
l-r
212

5. )J;JIH Toro 'tlTo5:hI KS'tleHHe npH3MhI npO,lJ,OJI>KaJIOCb 6eCKOUe'tlHO
p;oJlro npe,ZJ;eJlhHOe 3HS'tleUUe K'D' HeHHoe B n. 4, ,ZJ;OJImHO 6hITh 60JIL-
roe MHHHM8.JIbHO neo6xo)J;HMoro 3Ha'tleUHH, HaeHHoro B n. 3:
h. 1 M . a Mga[l- cos(30° - a))
l-r ;::: l-r gasln > r .
r 121
0603Ha'tlUM A = - ;::: -, Tor,lJ;a
1 - r 168
A sin a > 1 - cos 30 0 cos a - sin 30° sin 8,
(A+Sin6+ Jf cos6 > 1.
QTo6hI pemHTL 3TO HCp8BeHCTBO, BBeeM BcnOMOraTeJIhHLI:H yrOJI
u, paBHhIit
A+!
2
(A+l)2+
 35,36°.
TOr,lJ;8
cosusinG + sinucos8 >
1
( A + !) 2 + :
sin(u + e) >
1
(A+)2 + :
a > 8 0 = arcsin
1
-Ul=i::i
I=i::i 41,94° - 35,36° = 6,58°.
ECJIH e > 8 0 II KHHeTH'1IecK8JI 3HeprH.H nepe,lJ; nepBhIM cOY,lJ;apeHHeM
.D;OCTaTOquo SeJIHKa (CN. n. 3), TO npH c.n;eJIBHHLIX npe.n;nOJIo.meHH.HX
CKaThIBBHHe 6Y,lJ;eT npO,ZJ;OJImaThC.H 6ecKoHe'tlHO ,lJ;OJII'O.
213

3a.n;aQa 2
1. Ha OCHOBaHHH 38.KOH8 cOXpftSeHH.R 3HeprHH HMeeM
J Q t = LnPlI. d
(3p;ecL t = 1 ro)];), OTKY,A8
d = JQt = 0,06 ].(,g/(c. M2). 365,25.24.60. 60c = 61.10-3M.
LnPII 3,4.10 5 J];m/Kr- 917 Kr/
2. nYCTL P. - ftTMoc4>epaoe J];B.BJIeHHe, C'lHT8IOm;eecJl nOCTOHHHLlM.
Ha r.rry6HHe Z BHyrpH J1e.o;HHoH waURH .o;aBJIeBHe paDHO
P= Pngz + Pa.
lI03TONY Hft ,AHe ne,ZJ;.HHOH manKH, r.n;e Z = Y2 - Yt' .o;8BJIeHHe
P=Pn g (Y2 - Y.) + Pa =
= P.ng X (tg P - tg a) + Pngho + P a "
QTo6hI BO)];8 B OCHOD8HHH JIe,ll;HHo:A manKH He .o;BHr8JIaCb, 8BJIeaHe
,n;OJIHO 6hITh I'H,lJ;pOCTaTH'lecKHM (aTO yrBepm.u;eUHe TpHBH8.JIhBO, HO
npH )KeJI8.BHH MomeT 6hITh nOJIyqeHo H3 yp8.BHeHH.R BepuYJIJIu), T.e.
P = const - P. KY. = const - P. gx tg a.
ll03TOMY
P.n g x(tg P - tg a) = -Psgx tg a,
OTRYP;8
tgp = P. - P 1i tga = - P tga  - 0091 tga.
pJl P.a
OrCIO,A8 CJIe.o;yeT, qTO 8 = - 6.p = - 0,091, r,D;e BameH 3H8.K MHUYC.
P..
STO e YCJIOBHe MomBO nOJI}"lHTh pyrHM cnoco6oM, paCCMaTpH-
BaSI 3JIeMeHT MaCChI BOJ];hI H8 }:tHe JIe.o;SlHoil m8nKH H Tpe6YH co6JIIOp;e-
HU.R p8BHOBeCHH.
lloJIara.H tg a = O8 UaxO.n;HM tg p = - 0,073 II
Y2 = ho - 0,073x (pRC. 149).
3. nOCKOJIhKY, no npe)];nOJIOmeHHIO. JIe)]; .o;BH>KeTCH TOJIbKO Bep'rH-
KaJIhHO, TO KOHHtlecKHA npoB8JI aa nOBepXHOCTH 6Y,AeT HMeTb TOT me
214

1
.
/"
,,/
/'  Y 1
l/
Rei) /
Y 2  /
7
BcXJa V
V 3eMJUl
 V
/'
/"
Y.KM
4
3
2
o
o
1
2
3
4
5 x. KM
PRC. 149
P8.,lI;HYC 1,0 KM, 'tlTO H p8.,lI;HYC BTOpmeHH.R MarMbI. COrJIaCHO u. 2, rJIY.
6nnR npOBaJIa p8.BHa
h = r tg J3 = dp r tg a = .6p H =
Pn Pn
;:: 0,091.1 KM = 91 M.
1
4. 06"heM KpyroBoro KOHYca paDeH V = g1tr 2 h. lIpeAnOJIOmHM, qTO
BhICOT8 MarMOBoro BTOpmeBH.R pUBS hI. MO)RHO C'tlHT8TL, 'tlTO BHa'tla-
JIe MarMa paCIIJI8BHT KOHYC JIL,lI;a 06"heMOM. paBHhIM co6CTBeHHOMY o6"b-
1
eMY BTOpmeHH.R V; = i1tr2"-t. P8.BHOBeCHe ,lI;8BJIeHHft npH aTOM ee He
.D;OCTHraeorca. CJIeAOBaTeJILHO, BOAa 6YAeT yreKaTL II JIen; BCTynHT B KOH-
TaKT C fiOBepXHOCTLIO BTopmeHHOH MarMhI, qTO B pe3YJlLT8Te BOCCTa-
HOBRT ropH30HTaJILHOCTL BepXHeii nOBepXHOCTR JIb,ll;ft. 3aTeM M8rMa
6p
paCTOfiHT 06".beM, 3KBHB8.JIeHTHLIA KOUYCY BhlCOTOH h 2 = hI -, B pe-
Pn
3YJILTaTe 'tlero 6YAeT )J;OCTHrHyTO p8.BHOBeCHe (CM. fi. 3). Bo BpeMSI BTO-
poii 4>a3h1 paCTaJIBIDIUI BO.J];a TftKJKe YTeqeT. ECJIH npe)J;nOJIOmHTL, 'lITO
MarM8 e:rq,e He OXJI8.AHJI8CL AO 0 °C, TO OHa paCTonHT em;e O,lJ,IIE 06"beM
BO,lJ,LI, 3KBHB8.JIeUTHLIA KOHYCY BhICOTO:H h3 HO 3T8 BOAa Y>Ke He yreqeT
215

H o6pa3yeT KOHYC BhICOTOH h3 = h:3 Pn , eClIH OTC't:IHTh[BSTL OT Bepxym-
P.,
KH BTOpmeHHOH M8.rMhI. lIoJIHaH BbICOTa pSCTOnJIeHHOrO JIep;JIHOrO KO-
HYCR paBHa
h nonH = hI + h 2 + hg.
rJIy6HHa npoBaJIa a8 nOBepXHOCTH
(1)
h = 6p (h:t + 113).
Pn
(2)
JIerqe BCero 3T8 q.OpMyJI8 nOJIaeTCJI npH paCCMOTpeHHH paBHO-
BeCHJI JJ;aBJIeHHA: B KOHeqHOH CTa.u;HH (CHOBa CJIeAYJI paccymJJ;eHHJIM
n. 3). TaKHM 06paaoM, HCKOM8.H BIdCOTa BepxymKH BO,lJ,JIHOro KOHyca
paBH8
H =  + h3 =.&. h = 1,1.10 8 M.
8p
(3)
Y paBHeHHe TenJIOBOrO 68.JI8.HC8 HMeeT BHJJ;
 1tr 2 nh:t (f.u + C n 8t) - P nLnOJiH] = 0,
(4)
rp;e 8t = 1200 °C - 113MeHeHHe TeMnepaTyph[ CKaJILHOro BTOpmeHHJI.
TIop;cTaBJIJIJI 3H8TIeHHe (1) H npHBe,lJ,eHHbIe BbIme BbIpameHHJI)];J1JI
 = P hz II ha = & h 3 , H8XO.n;HM
Pn Pn
huOJlH = hz + t!.p hz + &hj = (hz + hS).
Pn Pn Pn
(5)
TI03TOMY c eTOM (3)
OJIH = &(hz + h3) = PB H ='&h = 1,20.10 8 M. (6)
Pn Pn 8p
OTCIOp;8 BhITeK8eT 't:ITO KOHYC He p;OCTHraeT nOBepXHOCTH JIe.n;.HHOH
mRnKH (pIlC. 150).
TIoJJ;CTaBJIJIJI pe3yJIbT8T B (4), HMeeM
P J. ( L + C 8t ) = PBPnLn h
nUt D D 8p'
oTKy.n;a P;JIJI hi nOJIyqaeM BhIpameHHe
hz = PBPnLn h = 103 M.
Pn8P(f.u + C 11 8t)
216

- -- r- .-- ..- -
r--.. _ 
- -
JIeiJ
J\
/ \
,
/ \
I ,
/ !\
If ,
J \
1/ \
J BoiJa \
I \
J \
/ \
,
J - -
I   i"""" Ma2M.a - -- -- 
...-- ""'-
3eMJl.R.
PRc. 150
06"heM KOHHQeCKOI'O BTOpmeHH.H nopoJJ; H M8I'MhI paBeH
1 2 7 3
 = -11:7  = 2,7.10 M
3 '
a ero Macca
m 1 == PnVl ::: 7,8.10 10 KI'.
217

nOJIH8.SI Macca 06p830B8BmeAcSI BO,ALI p8BHa, KOHe'tJHO Maece pac-
TRHBmero JIJ.,Aa:
1
m = - p 1tr21. = 2 9.10 11 Kr
DWUI 3 B '''nwm  .
Macea yreKmeA BO)];LI
m' =  + Ita m.o.u. = o.u: = 2,7.10 10
huODI P.. huODI
Kr.
aqa 2
1. Ha pHc.146 OTNeTHN K8I( MomBo TO'tJHee D;eHTpld" HCTO'lHHKOB.
ITYCTL 9.(t) - yrnoBoe paCCTOBUHe OT neBoro n;eHTpa)J;O KpecTHKa KRK
41YHKqHH BpeMeHH, a 8 2 (t) - yrJIOBOe paccToBHHe OT npasoro qeHTpa.
HaMepa:eM 3TH BeJIHqHHLI Hft pBcyaKe B )];8.HBld"e MOMeHTLI BpeMeHH H
nepeBO)J;HM B yrJIOBhIe ceKYH)];1d" no 38.A8HBo:A: mKaJIe. PeaYJIhTaThI npH-
Be,D;eHhI B Ta6JIHIte.
Ta6J1,Ulfa
BpeMSI, cyr 0- e
1
0 0,139 -0,076
7 0,253 -0,139
13 0,354 -0,190
20 0,468 -0,253
27 0,601 -0,316
34 0,709 -0,367
OmH6Ka npH C'IHTLIBftHHH nOKa38HHA JlHHeHKH On;eHHBaeTCSI K8.K
:to,S 114M, QTO ,AaeT omH6KY B 3UaQeHWIX a, paBH}T1O :f:O,OIS" . .D;8BHLIe
HaHeceuLI ua rpac2>HK 38BHCHMOCTH yrnoBLIX paccToHHHH 9. If 9 2 OT Bpe-
MeHH (pRC. 151).
lIpoBo)J;SI npSIMble JIHUHH, npoxop;HIqHe Qepe3 HftHeCeHHhle TO'tlKH,
no.nyqaeM:
WI = : = (17,0 :1:1,0) MC/Cyr r:= 9.54.10- 13 P8.A/c, (1)
CO 2 = d;; = (8,7 :1:1,0) MC/Cyr  488 .10- 13 P8.A/ c , (2)
218

B 1 :'
;.
0,8
0,6
o
0,4
0,2
-0,2
-0,4
-O6
o
5
10
15
20
PRe. 151
25
30
35 BpeM.H, CYT
vl..l = <OI R = 9,54.10- 13 .12,5.3,09.10 19 M/C =
= 3,68.10 8 M/C  (123 :t O,07) c, (3)
V2..L = 1,89.10 8 M/C  (0,63 :t 0,07) c. (4)
OrBeT: vI..l nOJIyqHJI8C:L B 1,23 pB.38 60JIhme CKOpoCTH CBeTa c, 'tlTO
Ha nepBLIA B3rJI npoTHBOpe'tlHT TeopHH OTHOCHTeJlLBOCTR.
2. PaccMoTpRM BHa<eHHe HCT01'ilIHKa 3a BHTepBaJI BpeMeHH 6.t OT
TOqKHA K TOqKeA' (pRC. 152). Tora
r AA' == TN - r A ;; v6.t.
(5)
nYCTh 6.t ' - paaBocTL BpeMeH npH6LITHSI CHr:H8JIOB B TOqKY 0 (rAe
HaxoAHTcSI Ha6JIIO,AaTeJIL) 83 A H A'. 1f3-38 p83J1I1'1H.H B paCCTO.HHHSIX
/1,0 A 1'1 A' H KOUeqBOCTH CKOpoCTH CBeT8 c HMeeM
6.l' = 6.t + (r A . - rA)/ c.
(6)
,lJ;JI.H M8JILIX 6.t, T8KHX, 'liTO V 6.t « r A :: R, nOJIyqaeM
r A , - r A ;; -v 6.t cos cp.
(7)
219

A'
o
1\
I \
I \
I \
I \
x I \
I
I
I
I
A
rA
R
PHc.152
H3 ypaBHeHHH (6) II (7) CJIe)U'eT
t' = t(l- (3 cos <p); f3 = .£..
c
(8)
STO 03HSqaeT, 'liTO Ha6JIIO,lJ;aTeJI:b, H8XO,lJ;.RI::qU:AC.H. II TO'IKe 0, 06B8.
pymHT KWKym;yJOC.H nOnepeqHyJO CROpOCTb HCTOqHHKa paBHOii
, _ tJ.x _ tJ.x _ J3csincp
V --- - ,
.L /:"t' lit(l- PCOSIp) 1- J3coscp
(9)
r.u;e MLI yqJIU  '({TO pe8.JIhU8.H nOnepeqH8.H CKOpoCTh B CHCTeNe OTC't.IeT8
S86JIIOJJ;8TeJIH paBua Vi =  = cpsinq>.
t
Ha6JIIO,lJ,aeM8H B 0 yrJIOBaH CKOpoCTb p8BHa
_ Vi _ cf3sinq>
ro - If - R(1- (3 cos q» · (10)
3. CHTY8D;H.R B aTOM. CJIyqae nOKa38.Ha Ha pRC. 153. TRK KaK CRapo.
CTH 06'beRToB p8BHhI no MO)U'JIIO H npOTHBOnOJlmKHLl no HanpaBJleHHIO,
R
-
V2
o
PRe. 153
220

TO V J = V 2 = V II f31 = 132 = 13. 113 pHcyaKa BII,lJ;HO QTO q>2 = 1t - CPl. lIoJlara.H
 = q>1  IIMeeM
sin CP2 = sin q>, COB q>2 = -COS q>.
Tor.n;a 113 (10) nonHM
COI = vi == J3csinq> .
R R(I- pcosq»'
(11)
co = Pc sin q>
2 R(I+J3cosq>f (12)
BeJIH'QHHhI C0 1 ' co 2 II R 3aHhI, 8 Ii II q> CJIe.n;yeT onpe.n;eJIHTh. lIpo-
CTLIe BLI'QnCJIeHHH IIPHBO)];.HT K l}>oPMYJIaM
(1 - J3 cosq» (0. (02 = J3csinq> . (J)2 ,
R
(1 + J3coscp) C0 1 co 2 = J3csincP. ro l .
R
BLIquTaH(13)H3(14),IIonyqaeM
(13)
(14)
ro -(I)
2J3cosq> . CO l co 2 = J3csinq>. 1 2 ,
R
(15)
OTKY,lJ,a
t 2 Rro l 00 2
gcp = ,
c(ro 1 - ffi 2 )
2Rro ro
tV = arctg 1 2 .
c(ro l - ro 2 )
(16)
(17)
Paa,lJ,enllM (13) Ha (14):
co 1 (1- j3cosq» = co 2 (1 + coscp), (18)
oTKy.n;a nOJlyqaeM J3 BLIpameHHoe Qepe3 cos cP II 113BeCTHhIe BeJIIPlHHhI
WI H co 2 :
Ii = (co. :]ro;:OBq> . (19)
lIo.n;cTaBJIHH 3Ha'tleHHH co l II co 2 H3 n. 1, a T8.Kme H3BecTHLIe 3HaTle-
HUH Rile, H8XOJJ;HM
q> = arcctg 2,57 = 1,20 p = (688 :t 2)°  (20)
J3 = 0,892 :t 0,08. (21)
4.113 I}>OPNYJILI (9) BH;J;HO, q'f0 Ha6JIJO.n;aTeJIh o6HapymuT RWKYIIJ;Y-
lOCH nOnepeT.JHyJO CKOpOCTh 60JlLmeH HJIH p8BHOH CKOpOCTH CBeTa TOJIh-
KO npn BLInOJIHeHHH YCJIOBHH
f3 sin q> ;;. 1
1- j3cosq> .
(22)
221

ECJIH P < 1, TO YCJIOBHe (22) 8KBHB8JleHTHO cJleJJ;Y](>I:qBN YCJ10BBJlN:
P sinq> ;;> 1 - (3cosq>.
(3 (sinq> + cosq» ;;> 1,
P.J2 ( sinlpCOS + COSlpsin) .. I,
sin ( '1'+ :) ..  ,
TaR "ITO (22) y.u;oBJleTBOpXeTCJI, eCJIH
(23)
13 > 1('1') = [ J2 am( 'I' + j r
(24)
OqeBHJJ;HO, "ITO YCJIOBHlO (22) MomBO y.u;OBJIeTBOpHTh TOJlhKO IIPH
'I' E [0, ; ], a YCJIOBHe (23) M<»KeT BMeTL pemeHHe AJUI '1', ecJIB 13" l2 ·
PaccMoTpUN n08TONj' nO,lU>o6nee 05JIaCTL
(13, '1') E [} 1]-[0, il
(25)
Oro6pWKeHue
(J3, cp)  PSin( q> + )
(26)
B 3TOA 06JIBCTH HenpepLlBHo. lloaToNY JJ;OCTaTOllHO nOCMOTpeTh UB rpa-
UHY 06JIBCTH, onpeJJ;eJI.JIeMyx> 3HB.KON p8BeHCTBa B (23):
13 sin ( Ip+) =  . (27)
STO OIIpe,lt;eJI.HeT p KaK clJYHKItHIO q> (RapHc. 154 RpUB8JI-P(cp) orpa-
HH"lHBaeT 3amTpHXOB8.BHyJO 06J1aCTh, r.u;e v > c).
5. tITo6L1 HeJiTH 3KCTpeNYM V  RBR I}>YHK CP, .AB4KtJepePyeM
(9) H nOJIytlaeM
 ( vl ) = (3(coscp -(3) .
dcp C (1-(3cosq»2
ITPOH3BO.u;Has: paDHa HYJlIO npH cp = CPm. r.u;e
(28)
COB 'I'm = 13; 'I'm = arccos 13 E [0, ;].
(29)
222

13
1
0,8
0,6
0,4
0,2
o
o 0,5 1 1,5 2 2,5 3 q>
Puc. 154
QTo6w YBeTL, "lTO aTO YCJIOBHe eACTBRTeJlLHO COOTBeTcTByeT
MaKCBNyNy, BHOBL .AHpeHItHPyeN (28) H IIOJIY1laeN
 ( vr ) = _ ( sin<p +2(3Sin<P(COS<P-J3» )
d<p2 c 13 (1- pCOS<p)2 (1- PCOS<p)3 .
(30)
B TO'IIKe 8KCTpeNYK8
d 2 ( VI )I =
dq>2 C CI>.
(3 sin <Pm
(1-(32)2 < o.
(31)
::no .u;OKa3LIBaer, TlTO 'Pm COOTBeTcTByer NaKCHM}TMY.
H3 l}>oPNYJI (9) R (29) HaxOAHN, qoro N8.KCRN8JILHaB KamaBCJI
IIonepeQHaS CKOpoCTL pUBa
lv' ) _ Pe
 .1 max - .Jl- 13 2 .
cmAacyqeTON(29)nonyqaeN,qTO
llm(V) = 00; limqJ,. = O.
11...1 -- 11-+1
(32)
(33)
Ha pRC. 155 rpa«!>l£CIecKH IIpe.u;CTaBJIeHa aaBRCHMOCTL OTHOmeHHJI
v'
 0'1' J3 H <P B 06J1acTB, onpe.u;eJlJleNoA: (25).
c
v'
HCKpHBJleHH8Jt UOBepXHOCTh H306pamaer  K8.K l}>yHKIO PH <po
C
IlnOCKOCTh H30CSpamaeT nOCTOHHHYJO I}>YHKRIO 13 = 1.
223

v'
i
c
4
3
1
2
o
PRC. 155
6. YpaBHeHH.R pen..RTHBHCTCKoro e.KTa )J;onJIepa HMelOT BM
Al,2 1  P cosq>
"I; = Jl- (32 .
(34)
CJIOmHM 3TH ypwmeHH.R, onpe,lJ;eJIHM BcnOMOraTeJIbHOe OTHomeHHe p
H pa3pemHM nonyqeHHoe BLIpameHHe OTHOCHTe1l:bHO p:
Al + A2 1
p- -
- 2Ao - 1 _ 13 2 '
p2 (1 - (32) = 1,
13: Jt- 1 : 1- 4 .
p2 (A. I + A 2 ) 2
(35)
(36)
(37)
OTKY8
a. :; 4.
(38)
J];06ElBJISlSI ypaBHeHH.R (37) K yp8BHCHH.RM (16) II (19) HMeeM TPH
yp8.BHeHHSI KOTopLIe MO)l(BO pemHTh, uait,ZJ;a TPH HeH3BeCTBLIe: 13, q> II
R. HanpHMep, MomBO BLI'QHCJIHTh f3 H3 (37) nO,ZJ;CTaBHTh B (19) H pa3-
pemHTh OTHOCHTeJIbHO q>. IIocJIe STore 113 (16) MO)KBO n01lytIHTh pac-
CTOHHHe R. Cne,ZJ;oBRTeJIhBO, H3MepeHHe C,ZJ;BHHYTLIX no J];onJIepy AJIHH
B01lB .n;aeT OIteHKY paCCTO.KHH.s 1J.O HCTO'QHHKB, ecJlH H3BecTHhI co. H (1)2.
224

xxx MeJKAYHapOAHaR
cl»M3MllecKaR OnMMnMGAa
1999 r., MTanMI1
3aJJ;8"1a 1
BepTHKaJILHO pacnOJIOmeHHbIH IUIJIHH.APII"leC.KHH COCY.A COAepmHT
rR3 no,ll; nopmHeM B COCTOSlHIIH TepMo.AHH8MHYIeCKOrO paBHOBeCH.R.
lIopmeHL npe,n;CTftBJUleT C060H CTeKJI.RHHyIO nJI8CTHBKY II MOJReT CBO-
6o,ll;HO nepeMeIr(8TLCJl. IIpe,lJ,noJIomHM, qTO YTeqKH raaa BeT II YITO Tpe-
HIIe Me>KAY CTeKJIHHHLIM nOpmHeM II CTeHKaMII nHJIHHAPa ,n;OCT8TO"l-
HOe, 'iTo6Ll nOJ];aBJUITL KOJIe68HH.R, HO He BROCHT 3RMeTHhIX nOTepL
SBeprHH. IIepBOHaQaJILH8.SI TeMIIepaTypa ra3a paBHa TeMllepaType OK-
pymalOm;eii Cpe,ll;Ll. C xopomeH TOYIHOCThlO I'M MomBO C'iUTaT.b H,AeaJIL-
HLlM. BYJ];eM T8Kme npe,lJ,nOJlaraTh, 'tlTO CTeHKH q;HJIlIH,A"pa (BKJIIO'tJ.ag
nopmeHL II J];HO) HMelOT OqeHL HH3KyID TenJIOnpOBOJJ;HOCTL II TenJIOeM-
KOCTh, n03TOMY TeIDIOo6MeH MemJ];y ra30M II oKPym8IOIr(eH cpeAOH npo-
HCXO,lJ,HT O"leHL Me)J;JIeHHO H HN MO)KHO npeHe6pe'lh.
qepea CTeKJISlHHyIO nJIacTHHKY B q;IIJIHHAP HanpaBJI.HeTC.R nyqoK
CBeTa OT J1a3epa nOCTO.HHHOH NOHOCTH. 3TO H3JIyqeuHe pacnpocTpa-
HReTCH qepea B03,l1;YX H CTeKJIO 6e3 nOrJIoII(eHH.R, HO nOJIHOCTblO norJIO-
I4aeTcH r830M B cocy,n;e. B pe3ynhTaTe MOJIeKYJIhI raaa nepexO,lJ,ST B B03-
6Ym,AeHHLle COCTO.HHHB, 83 KOTOpbIX OUH 6bICTPO B03Bpam;8IOTCSI B OC-
HOBHoe COCTO.RHHe nYTeM cTyneH"l8TOrO HcnYCKaHH.R HHl}>paKpaCHOro
H3JIyqeHHH. STO H3JIyqeHHe nOrJIOII(ReTCSl ,lI;pyrHMH MOJIeKYJI8MH H OT-
pamaeTCH CTeHKftMH cocYJ];a H CTeKJIHHHhIM nopmHeM. TaKHM 06paaOM,
3BeprH.H JI8aepHOro H3.rryqeHHH, nOrJIOIIteHH8.R ra30M, TPaHC<pOpMHPY-
eTCH B TeYIeHHe OQeHh KOpoTKoro BpeMeHH B 3HeprHIO TenJIOBOrO J];BH-
.meHHS MOJIeKYJI.
B pe3YJILT8Te CTeKJI.JIHHLIH nopmeHL CMeII(ae-rcSI BBepx. CnYCTH He-
KOTopoe BpeMSI JI83ep BhlKJllOQaeTC.H H H3MepHeTCH CMem;eHHe nopm-
HH.
1. HCnOJIh3Y.R npHBeAeHHLIe HH»ce ,AaHHhIe, onpe.lJ;eJIHTe TeMnepa-
TYPY H .AaBJIeHne ra3a nOClle 06JIyqeHHH.
2. BhIQHCJIHTe MeX8HHQecKYIO pa6oTY, BhInOJIHeHHyJO r830M B pe-
3YJlbTRTe nornOII(eHHB H3JIyqeImH.
8 379
225

3. lIocqHTaATe norJI0n::teSHyro 3ueprHIO H3JIyqeBH.R.
4. PaCCQHTaATe MOUOCTh H3JIyqeHHSI JI&3ep8, nOrJIOrn;eHHOro ra-
30M, H COOTDeTCTByroee qHCJlO nOrJIOeHHhlX €'pOTOHOD (H, CJIODa-
TeJIbHO, qRCJIO 8JIeMeBTapUhIX npoxtecCOD nOrJIOIIteBJUI JI8.3epBhIX €.pO-
TOHOD) B eRHHxtY DpeMeHH.
5. Onpe,AeJIHTe OJIIO OnTJAecKoA 8seprHH, npeo6pa30DaHHoA D Me-
X8BHQeCKYJO nOTeU:Q,HaJIhBYJO 8uepruIO CTeKJISlBBOrO nOpmH.R.
6. ITocJIe BbIKJlIO'tJeHHB JIaaepa ItHJIBB,Zq> Me,ltJIeHHO nOBOpa'lHB8.IOT
Da 90. T8K, 'tJTO ero OCb npHHRN8.eT ropH30HT8.JIhHOe nOJIOHCeHne. Ten-
JIoo6MeHoM MroK,ltYraaOM H CTeHKaMH coeY,l];8no-npemHeNY MOmBO npe-
He6peTiL.
H3MeHHTCH JIH AaBJIeHHe H TeMIIepaTypa raaa B peaYJIbTaTe TRKoro
DODOpoT8? Oupe,n;eJIHTe HOBhIe 3BaQeUHJI TeMnepaTypbI H aBJIeHH.R.
qaCJIOBLle .n;8HHhle
,{t;aBJIeHHe oKpym8.IO:n::tero B03.l{yX8 Po = 101,3 KIIa
KONB8TB8JI TeMnep8Typa to;: 20°C
BHyrpeHHHA: ,AHaMeTP xtBJIRH)J;pa 2r;: 100 14M
Macea CTeKJUIHBOA DJIaCTRHhI m = 800 r
KOII1AecTBO ra38 B coeYAe v = 0,100 MOJIL
MOJIHpH8.R TenJIOeMKOCTL npH nOCTOHUHOM o6eMe C v =
;:; 20,8 ,{t;m / (MOIIL . K)
.l{JIHHa BOJIBLI R3JlyqeHHH JI8aepa A ;:; 514 UM
BpeMH 06JIeHH.R A 1"= 10,0 C
CMe:n::teuHe nopmH's Doene o6nyqeUH.R As ::;: 30,0 MM
IIoCTORBH8H IIJIB.HKa h  6,62 -10- 84 ,{t;m. c
r830Baa nOCTO.RHHas: R;: 8,314 )tm/(MOJIL. K)
Saqa 2
K nepBhlM YCDeX8MTeopHH MRrHHTHhlX aBJleHHA AMnepa OTHOCHT-
cs: Bh[tIHCJleHUe HH;:J;YKxtHH B M:arHHTBOro DOJla, C03,l];8.B8eMoro npo-
DOAHHKOM C 3J1eKTpHQeCKHM TOKON, H cpaBHeHHe C npoBeAeUHhlNH
paHee paC'tJeT8MH BIIO H CaBapa.
HHTepecHhlN 'lRCTUbIM CJIyqaeM .JIBJlBeTCH OqeUL MUBHWI nposo-
JlOKa C nOCTOSlHHbIM TOKON i, H30ruyT8.SI D BHAe6YKBhI V C yrJlOM paCTBO-
pa 2a. (B aTOH 3a"le yrJIbl Bhlpa.m8.1OTC1I B PH8Bax) (pac. 156). Co-
rJI8CHO BhIQnCJleUH1IM AMnepa, .MOYJIL MarHHTHOA HH)J;yKI\HH B B
TO'lKe P aa OCR V -o6pa3HoA npOBOJIOKH D8 paCCTOSlHHH dOT BepmHHhI
a
nponopI\HoBaJIeU tg -. Pa60TaAMnepa n03me BOInJIa B TeopHlO 3JIeK-
2
TpOHHaMHKH M8KCBeJlJIa H SlBJI.ReTCSI o6enpH3B8HHoA:.
226

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.
<
d
he. 156
1. HaH.n;UTe BanpUJIeHHe MarHBTBoA HHJJ;YKHB B B TO'tJKe P.
a
2. SHaB, 'tJTO MOAYJIL MarHHTHoA HH.IJ;YKItHH nponOpI:\HOH8.JIeH tg - ,
2
a
T.e. B(P) = ktg-, Bhf'IBCJIHTeKOaHItHeHTnponopI:\HOB8JlLHOCTHk.
2
3. Onpe,Q;eJIHTe MOAYJlb MarBHTHoA HH.ltYKItHH B B To'tJKe PI' CHM-
MeTpl{qHOA TO'lKe P OTHOCHTeJILHO BepmHHLI, T .e. ua OCB npoBOJIOKB,
H aa TONJlCe paCCTOHBHH d OT BepmHHhI, BO BRYTpH KOHTypa(pHc. 157).
4. BhI'tIHCJIHTe K8K fl>}'HKItHlO B nepHO,D; M8JlLlX KOJle68HUA DOMe-
eHHoA B TOl:lKY P M8rHHTuoii CTpeJIKR, eCJIU CTpeJIK8 HMeeT MOMeBT
HBepItHH I H MarHHTBLlH HnOJILHhIH MOMeHT p. CTpeJIK8 KgJIe6neTCH
B Tal: me nJIOCKOCTH, B KOTopoA ne)KHT BeKTOp HH,q;YKHH B.
5. lIpH Tex me YCJIOBHHX, 'tJTO H Y AMnepa, Buo H Cuap npe,q;nOJIa-
raJIH, '1TO MO,Q;YJI:b MRrHHTHOA HH.a;YKHH B B To'tJ:Ke P onpeJJ;eJI.aeTCH
ijJoa
BLIpIDKeHueM (RcnoJl:&3yeN COBpeMeBHYJO aanBC:b) B(P) =, r.r:te
7td
110 - MarHHTHSSI nOCTO.RHHasI. ORB nOllhIT8JIBCL C nOMOLIO 8Kcnepu-
MeBTsnpoBepHTb CnpUe,lt1lHBOCTL COOTBeTCTBYJOHX npep;nOJIOmeUUH
(AMnepa H CBOUX), H3MepHH nepHO KOJIe6SHuA MarHHTBoA CTpeJIKB
K8.K €'p}'HKItHIO yrJIS a. OnuaKO AJIH aeKOTophIX 3UaQeRUA a pa3Bu:qa
pe3YJIbT8TOB CTOJIL MaJla, TlTO ee TPYAHO H3MepHT:b.
Y C'l'aBOBHTe llpH6JIHmeHHO, KaRoA HHTepBaJI yrJIOB a CJIe;tYeT BbI-
6paTL, 'tJTo6L1 o6HapYmHT:b pa3JIHQUe MemAY BYMH npe,Q;CKa3aHUSlMH
,n;JIH nepHO,l{a T Kone6aHuA MarHHTHoA CTpeJIKH B TO'lKe P, KOTopoe B
p
.
<C
d
PRC. 157
227

IIX 3HaQeHHSlX COCT8BJUIJlO He MeHee 100/0, 8 HMeHHO Tl > 1,10T 2
(r,lJ,e T 1 COOTBeTcTByeT npe.n;nOJlOmeHHIO AMnepa, a T 2 - npe)];llOJlO>Ke-
HHIO BHO H CaBapa).
n p H Me Q a H He. B 3RBHCHMOCTH OT BhI6paBHOrO cnoco6a peme-
HRS! 3a,n8QH MomeT 6LITb nOJle3HOH l}>opMYJIa
t .!!. - sina
g 2 - 1 + cosa .
3aJJ;aWJa 3
PaCCMOTpHTe MeTOjJ., 'I8CTO npHMeH.HeMLIA ,lJ,JIH YCKopeHHH KOCMH-
'IeCKRX 30H)J;OB B HymHoM HanpaBJIeHHH. lIpoJleT8H B6JIII3H nJIaHeTLI
KOCMlPIeCKHH 30H)J; MomeT 3H8TlHTeJILHO YBeJIHTlHTL CBOIO CKOpOCTL H
cym;ecTBeHHO H3MeHHTb HanpaBJIeHHe DOJIeTa 38 C'IeT He3HaTIHTeJIhHOH
qaCTH 3HeprHH Op6HT8JILHOrO ,lJ,BHmeHH.R llJI8neTLI. lIpoaHaJIH3HpYH-
Te CJIyqaii KOCMH'IlecKoro 30HA8, npOJIeT81Om;ero B6JIH3H IOnHTep8.
IOnHTep BpaIIt8eTC.R BOKpyr COJIHa no 3JIJIHnTWleCKOH op6HTe,
KOTOpY1O MO.>KHO annpOKCHMHpOB8TL oKpymHocTLIO cpe.n;HHM p8JI.HY-
COM R. QT06bI npHCTyuHTL K aHaJIH3Y I}>H3H'IIecKOH npo6JIeMhI, npe,u;-
BapHTeJILHO pemHTe ,ABe aa,u;a'lln.
I. Onpe,u;eJIHTe CKOpoCTb v ,ABH:meHI:ISJ. IOnHTepa no op6HTe BOKpyr
COJIHn;a.
2. H8iI,AHTe paCCTO.RHHe OT IOuHTepa, aft KOTOpoM CHJIbI rpaBHTa-
HOHHoro B38HMO,lJ,eHCTBHH 30H.n;a C lOuHTepoM If COJIHn;eM paBHLI
(npep;llOJIaraeTC.R, 'tITO aOH)]; uaXOjJ;HTC.H B To'tJKe H8 OTpe3Ke, coejJ;HH.RIO-
eM COJIHe II IOnHTep).
3. KOCMH'IIecKHH 30H,u; M8CCOH m = 825 Kr npOJIeTaeT B6JIH3H IOnn-
Tepa. ,II;JI.H ynpom;eHHH npe,u;nOJIOmHTe, 'ITa TpaeKTopHH 30H)J;8 nOJIHO-
CTLIO JIemHT B nJIOCKOCTIf Op6HThI IOnHTep8, T.e. MomHO npeHe6pe'lb
CJIyqaHMH, Kor.n;a 30HjJ. nOKHJJ;aeT STY UJIOCKOCTb.
PaCCMOTpHM TY 06JIaCTL npOCTpaHCTBa, B KOTOPOH npHT.R)KeHHe
lOnHTepa 3HaTJHTeJlbHO npeBOCXO,lJ,HT Bce OCT8.JIbHLIe rpSBHT8n;nOBHbIe
CHJIhI .
B CHCTeMe OTCqeTa, CBH3aHHOH C eHTpOM COJIHIta Ha'll8JIbH8H CKO-
pOCTL KOCMHTJecKoro 30H)];a V o = 1OO.I04 M/C (B nOJIomHTeJIbHOM Ha-
npaBJIeHIlH OCII Y), B TO BpeMS .K8.K CKOpoCTb IOnHTepa HanpaBJIeHa B
OTpHItSTeJIbHOM HanpaBJIeHHH OCH X (pRC. 158). lIo)]; UaTIaJIbHOH CKO-
pOCTblO MbI nOHHMaeM C.KOpOCTb KOCMHqecKOI'O 30H)];ft B MemnJIaHeT-
BOM npocTpaHcTBe ,u;OCTaTO'llHO p;aJIeI<:O OT IOnHTepa, HO yme B o6JIac-
TH, l',lJ,e npHTHJKeHHe COJIHa npeBe6pemHMo MaJIO. ITpe,lJ,nOJIOJKHM, qTO
B3aHMO,lJ,eHCTBHe npOHCXO,D;HT aa ,lJ,OCTaTO':IHO KOPOTKHH npOMejKyroK
BpeMeBH, TaK 'iTO MomBO upeBe6pe'flb H3MeHeBHeM nanpaBJIeBH.H CKO-
228

IOnumep
-
v
.
(5
o (op6uma IOnumepa)
YL
x
e
PRC. 158
pOCTH ABHmeHH.H IOnuTepa no op6HTe. IIpeAllOJIOmHM TaKme, "ITO 30H,lJ,
npOXOAHT n038.)J;H IOllHTepa, T.e. ero KOOp,n;HHBTa Xl 60JIbme, TIeM KO-
op,n;HHaT8 x 2 10nHTepa B 'TOT MOMeHT, KorAa KOOPAHHB'TLI y paBHLl.
HaHJJ;HTe HanpaBJIeHHe CKOpoCTH ,n;BHmeHHJI KOCMUqeCKOrO 30H,u;a
(yrOJI eo Mem.u;y BeKTopoM CKOpOCTH 30H)'J;a II OCLIO X) H MO,n;YJIL ero CKO-
pOCTH v' B cnCTeMe OTC"CIeT8, CBSl38HHOH C IOnUTepOM, Kor,n;a 30HJJ; HBXO-
,lJ;IITC.R AMeRO 0"1' IOnHTepa.
4. HabHTe nOJIHyx> MeX8UHQecKyID 3HeprnIO E 30HJJ;8 B CHCTeMe
OTC"tleTB, CBH38HHOH C IOllHTepoM nOJIar8S1, KaK 06hNHO, ':ITO nOTeH-
n;H8JThB8.R sueprHSI DB OQeHL 60.m.WHX paCCTOHHH.HX paBH8 Hyn1O a
CROpOCTL 30H)J;8 MO)KBO C"CIHTBTIt nOCTOHHHOH H3-3a MaJIOCTH Bcex rpa-
BHTSn;HOUHLlX B38HMo,n;eHcTBHH.
5. Tp8eKTopneH KOCMH'IeCKOrO 30ll)];a B CHCTeMe OTCQeTa, CBSl3BH-
HOH C IOllHTepoM, HBJISleTCH rnnep60JIa, ypSBHeHne KOTOPOH B nOJI.Hp-
HLIX ROOp.u;HH8T8X HMeeT BII,n;
1 GM ( 2EU 2 b 2 )
- =:  1 + 1 + 2 2 case 
r v' b G M m
(1)
rAe b - paccToHHHe Mem,lJ,Y aCHMllTOTOH H neHTpOM IOnHTepa (TaR Ha-
3hlBaeMbIH npHneJIbHh£H napaMeTp), E - nOJIH8.H MeX8HHTIeCK8H 3Hep-
ru.H 30H.n;a B CHCTeMe OTCqeTB, CBH38HHOH C IOIIHTepOM, G - rp8BHTa-
:Q,HOHHWI IIOCTOSlHBa.H, M - Macca IOnHTep8, r n e - nOJI.RpHlde KOOp-
,lJ,HH8ThI (paCCTOIfHHe AO neHTpa IOTIRTepa n yrOJI).
Ha pRC. 159 nOKa3aHhI ,lJ;Be BeTBH rnnep60JILl, onHchIBaeMoit ypaB-
HeBneM (1), H ee 8CHMTITOTbI B nOJISlpHbIX Koop,lJ;nB8Tax. OTMeTHM, 'liTO
ypaBHeHHe (1) onHCLIB8eT rnnep60JIY, I}>OKYC RO'TOPOH HaXO,lJ,HTC.H B
.n;eHTpe npHTSI*eHH.R., T.e. B n;eHTpe IOnHTepa. TpaeKTOpH.H KOCMH-
"IeCKoro 30HJJ;B npe,n;cT8BJI.ReT C060H .BeTBb npHTHmeHHa:. R n306pa-
>KeR8 Ha pncYUKe cnnomHoA JIHHHeit.
HCnOJIL3ya: yp8.BHeBHe (1), OTIHCLlB8lOm;ee TpaeKTopHlO 30B,n;a, HaH-
,lJ,HTe nOJIHOe yrnoBoe OTKJIOHeHHe fl.8 TpaeKTopHH 30H)J;a B CHCTeMe OT-
229

KOC.MUlleC1CUU
30niJ
PHC. 159
C'Q:eTa, CB.H3aHHOA C lOuHTepoM (CM. pRC. 158), H BhIpa3HTe ero RaR
c}>yHR:qmo Ha-q8JIbHOH CKOpOCTH v' II npH:Q,eJIhHOrO napaMeTpa b.
6. IIoJIBras, 'tIT0 30HA He MOmeT npOHTH MHMO IOIlliTepa Ha paCCTO.
BURH OT ere qeBTpa, MeHLmeM, -qeM TPH IOIIBTepH8.HCKHX pa.AHyca, 8m-
JJ;HTe NHHHN8JIbHO B03MOBoe 3H8-qeHHe nplIt:I;eJILHOrO nSpSMeTpS H
MSKCHNaJILBO B03MOJKHOe 3HS'tJeHHe yrJIOBOrO OTKJIOHeHH.H /J.B.
7. IIo.1Iyqme BhIpR)KeHHe ,wI.H KOHe'1lHOA CKOpocTH 30H)];a v" B CRC.
TeMe OTC'1leT8, CB.H38BHOH C CoJIHII;eM, K8.K 4>YHK:Q,HIO TOJILKO CKOpOCTII
IOnuTepa v, Hs'tJ8.JThHOH CKOpocTH 30H VoH yrJIOBorO OTKJIOHeHH.R l:J..B.
8. HCnOJIb3YSl npe,ALI)];HHpe3YJIhTaT, BsA,AHTe 3HB.'1leHHe KOHe-q-
BOH CKOpoCTH 30BJJ;S v" B CHCTeMe OTC'1leTa, CBH38.HHOH C COJIHt:I;eM, npH
MaKCHM8JIbHO B03NomHOM 3H8'tIeHHH yrJIOBOrO OTKJIOHeBHH.
"IIHCJIOBhIe AaHHMe
Macca COJIHt:I;S Me = 1,991 - 10 30 KI'
Maccs IOuHTepa M = 1,901 - 10 27 HI'
Cpe.u;mdi pa.,u;HYC Op6HTLI IOnHTeps R = 7,783 · 10 11 M
9KBSTOpH8JIhHhIH p8.,lJ;HYC IOIIIITepaRjQ = 6,98 -10 7 M
IIepHo.u; o6pmqeHHJI lOuHTepa T 10 = 374,32 Mc
I'p8BHTHOHHaH nOCTOHHHaH G = 6,673 - 10- 11 M 3 /(Rr. c 2 )
PeweHMII
3a.u;8'Q:a 1
1. IIpH paBHOBeCHH JJ;aBJIeHHe p BBYTPH COCY,A8 .u;OJUKHO 6hITL PU-
HO CYMMe aTMoc4>epnoro ,P;8.BJIeUHSJ. Po II ,lJ;aBJIeIUlSl, npoH3BO,AHMOro no.u;-
BIIJKHhIN nopwueM:
mg
P=Po+.
7tT
230

STO BepHO KaK 11.0 06JIyqeUH.R, TaK H nOCJIe. Ha'tIaJIhHBSl TeNnepaTY-
pa ra38 paDHa KOMH8THOR. CornaCHO ypaBHeHHIO COCTO.HHH.R H)J;eaJIb a
Horo raaa, Ha'tI8JIbHhIH 06'heM raaa 1Ii ::;:: nR7Q (R - r8aOB8JI nOCTO$lH-
p
Hag), n03TONY BLICOT8 h 1 38JUITOrO raaOM :qHJIHHJij)R PUBR
 1Ii nR
= nr 2 ::;:: p o 1tr 2 + mg .
TIocne o6nyqeBHH BHCOTa CT8.HOBHTCR PUHOR h 2 ::;:: hI + 1:18, TaK "lTO
HOBaR TeMlIepaTypa PUH8
1'. - '7T (1 &l J _ '7T l1s(po1tr 2 + mg)
2 - ..( 0 + Ii;' - o + nR .
'tfHCJIOBhIe OTBeThI: p::; 102,3 KIIa, T 2 ::;:: 322 K IIJIII t 2 ::;:: 49°C.
2. COBepmeHHaH r8aOM MeXaHH'tIecK8$1 pa60Ta IIO nO)J;'heM}' nopm-
H.H paBH8 mgl:1s, 8 pa60T8 npoTHB CRJI 8TMocf}>epHoro AaBJIeHH$I p8BH8
p o 1trI:18, n03ToM}' IIOnH8JI pa6oT8
A = (mg + p o 7tr) 1:18 = 24,1 )];m.
3. lIa-38 H3MeHeHH$I TeMlIepaTyphI BHyrpeHH.R.$I3HeprH$I H3MeUJI-
eTCH Ha BeJIHtIHHY I:1U = nc v (T 2 - To ).
'tfTo6h1 npORaOmJIO T8.KOe H3MeHeHHe BHyrpeHHeli 3HeprHH H 6LIJIB
cOBepmeHa pa60Ta, CHCTeMB )J;OJImH8 nOJIyrIHTh KOJIHqecTBO TenJIOTLI
Q, pBBHoe
Q = MJ + A = nc v T6s + (mg + Po,.,.2)ill! = 6s(mg + Pollr2 {  + 1) .
IICTOQHHKOM 3Toro KOJIHQeCTBB TenJIOThI HBJIHeTCH OUTl[t{eCKOe
H3JIyqeHHe. CJIeAOBBTeJIbHO, nOrJIoxq;eHHB$I 3HeprH$I H3nyqeHH$I pu-
HB. KOnH'tIeCTBY TeUJIOThI Q. B&rtIHCJIeHHH )J;8IOT Q = 84 )];m.
TOT me peaYJIbT8T MO)KBO nOJIytlHTb, p8CCMBTpHB8.B. Hao6BpHLIR
npo:v;ecc H BCnOMHHB CB.RSh Nem.n;y Ten.noeMKOCTaMH:
[ 118(m g + po1tr 2 > ]
Q;; nc p (72 -70);;: n(c v + R) nR =
= 6s(mg+ Pollr2{  +1}
231

4. IIoCKOJILKY MOII:{HOCTh H3JIyqeHH.H lIS3epa nOCTOSlHH8 nOrJIOeH-
BaSI onTlf<lecKaH MOBOCTh p8BHa
Q ( Cy ) !!.s 2
P = T: = If + 1 /),:t (mg+ ponT) = 8,4 BT.
3HeprHH RWK.ZJ;oro I}>OToH8 paBHa he , noaToM}' '1HCJIO I}>OTOHOB, no-
A
rJIOeHHLIX B e.ZJ;HHHY BpeMeHH, paRHD
PJ.. = 22 .10 19 c- I .
he
5. H3MeBeHHe nOTeHIJ;H8JlbHOH aaepl'HH paRHO MeXaHH'IeCKOH pa-
60Te no nO"heMY nopmHH, n03ToMY K03I}>HHeHT npe06p830BaHHH
3HeprHH 11 paBeR
11 = m Q gM = ( )( r 2,8 .10- 3 '" 0,3%.
1 ponT 1 Cv
+- +-
mg R
6. KorAs U,HJIHHAP nOBOpa'lHBSeTCH TaK '1TO ero OCL CTaHOBHTC.H
ropH30HTWIhHOH, npOHCXO)];HT a;J;Ha6aTHLIH npo:Qecc, B KOTOpOM AaB.
JIeHHe H3MeHHeTCJl OT p 11.0 Po' a TeMuepaTypa npHHHMaeT HOBoe 3H8'1e-
BHe T 3' YpaBHeHHe a.zr;Ha68THoro npou;ecca pvY = const NomBO aanH-
C8Tb B BHe
( ) y-I
73=72  \
C c +R R
r,lJ,e y -=..L == y == 1 + - = 1,399. OKOH'IftTeJIbHO T a == 321 K HJ1H
Cy Cy C y
t a == 48 ac.
3a):\8'l& 2
1. KSJK)];aa'laCTL V -06P83HOii npoBOJIOKH COa,lJ,aeT M8rBHTHoe nOJIe,
BeKTOp HHJ1;YK.tI;HH KOToporo HMeeT TO me H8np8B.JIeHHe 'ITO II Y COOT-
BeTCTByroeH 6eCKOHe'lHOH npOBOJIOKH. IIoaToMY, ecJIH TOR HJJ;eT  KaK
nOK838ao H8 pRC. 160, BeKTOp HHAYKQ.HH MSrH:HTHOrO nOJIH opTorOH8-
JIeB nJIOCKOCTH npOBOJIOKH (BLI6p8.HHOA 3a nJIOCKOCTh XY). ECJIH HC-
nOJIh3yeTC.H np8BocTOpOHHH.H CHCTeMa KOOPAHH8T, IIOK838HH8sr H8
pllC. 160, TO BeKTOpHHAYKD;HH B(P) HanpaBJIeBBnOJIOmHTeJIbHOMH8-
np8.BJIeUHH OCR Z.
232

!z<
PRe. 160
no coo6pameHHHM CHMMeTpUH HH)];yKHSI nOJIHOro nOJI.H no MO,lW'-
JIIO B,ABoe 60JIbme HH.n;yKHH nOJIH, KOTopoe C03AaeTC.H KWKJI,OH no.no-
BnUOH V -o6pa3HoA npOBOJIOKH, H HanpaBJIeH8 B Ty:me CTOPOHY.
2A. ECJIH a = ; , TO V -06pa3HhIH H3ru6 npeBpaxq;aeTCH B np.HMyro
6eCKOHe'llHO ,lJ,JIHHHyH> npOBOJIOKY. H3BecTHO, 'IlTO B 3TOM CJIyqae MO-
,lJ,YJIb HHJJ;YKD;HH B paBeH
B= i io
21t£oc 2 d = 2nd '
8 TaR K8.K tg  = 1, TO MBO)KHTeJIb k = i:r .
2B. EcJIH ytIaID;HHCSI 3HaKOM C 4JOPMYJIOH
B = oi eose l - cose:!
47t h
AJIH nH,lJ,YXD;HH M8I'HHTHOro nOJI.H, COa;D;8BaeMOrO KOne'1lHbIM oTpe3KOM
npOBOJIOKH, JIemaIItHMaanp.HM:OH JIHHHH Us paCCTO.HHHH h OT TOQKH
P, nplftleM KOH:qbI oTpe3Ka BHAUb! H3 P no,n; yrJI8MH 8. H 6 2 , TO OB NO-
:meT y6e,n;HTLCSI qTO KBm)J;8JI 113 ,lJ,BYX nOJIOBHH V-06pa3Horo H3rH6a
.. .. J.Loi 1 - coso.
C03,AaeT nOJIe C HH,D;yKD;HeH, p8.BHOH - . . CJIeJJ;oB8TeJ.lhBO, HH-
47t dsma.
.u.YKHH B cYMMapBoro nOJIH paBH8
B - oi 1- coso. _ ).loi t a
- 21td sinn 2nd g"'2'
STO 60JIee nOJIBOe pemeHHe, TaR K8.K 3aO)];HO BbI.HCBHeTCSI 3aBHCH-
MOCTb HU;:J;YKD;UH nOJIS OT yr JIa.
3A. QT06bI BbPlHCJIHTb MO,lJ,YJIh HB,D;yKD;HH B(ll), MomBO paCCM8T-
pHB8Th V -06pft3HyIO npOBOJIOKY KftK CHCTeMY ,D;BYX 6ecKoue-qHbIX npo-
BonOK a H b (pRC. 16!) H APyroA V-06pa3BOH npOBOJIOKH, CHMMeTpnq-
233

v'
v

he. 161
HoA nepBoA, nOK8aaHHOH Ha pHCy:HKe K8.K V' , no KOTOpOH H)1;eT TOT )lte
TOR B npoTI!BOnOJIOmHOM Hanp8.BJIeHHH. Torp;a
B(.fl) == Bo(.fl) + Bb(l{) + .8y.(l{).
Or.n;eJILH1de BKJI8.)J;h1 paBHL!:
J.1o i
Bo() == L\();: . t
21tdsma
Hanp8.BJIeHHe - B OTpHD;aTeJlLHyro CTOPOHY OCI! Z;
r::l ( P. ) =- J.1 o i t a
L'V' 1 21td g 2 
Hanp8BJIeHHe - B nOJIOmHTeJILHy1O CTOPOUY OCH Z.
IIoSToMY
B() = J.1 o i (  _ tg ) == k ( l +.cosa ) = kctg
2nd sma 2 SIn a 2
BeKTop Hanp8.BJIeH B oTpHD;aTeJILHYJO CTOPOHY OCII Z.
B03NOJKHLI CJIeJtYIOIItHe pemeHH.R.
SB. TOqKY p. BH}'TpH V -o6pa3Horo HarH6a c nOJIOBHHBhIM yrJIOM
pacTBopa a. MomHO paccMaTpHB8TD RBR HaxoJJ:HYIOCJI CH8pymH OT
7t-a
V -06pa3Boro II3rH6a C nOJIOBHHHhIM yrJIOM paCTBoa  t no KOTO-
POMY Te'tJeT TOT me TOK HO B npOTHBOnOJIO)KHOM HanpaBJIeHHn. TIOSTO-
MY MarUHTH8H IIHJJ;YKD;HH p8.BHa
B() == ktg( 7t ;a) = kt g ( ; - ) = kctg  .
BeKTOp HHJJ:YKD;HH OCTaeTCH Hanp8.BJleHHLIM B.D;OJIL OCR Z t HO B OT-
pHn;aTeJILHOM HanpaBJIeHHH t 110CKOJILKY TOR" KaR OTMe1:IeBO SLIme,
Te1:IeT B npOTHBOnOJIO:HtHyK) CTOPOHY.
BB. ECJIH pemaTb 38,Ztaqy, CJIe,lJYa paCCeHHgM 2B, TO 6ea TPYJJ:8
TeN me MeTOP;OM MO)KHO BaATH H 3HaQ;eUUe HH,lJ;YKD;HH nOJIM B TOT:lKe p..
234

4. MexliHH'lecKllii MOMeHT AI, ,lJ,eHcTByrolIJ;uA Hft nOMem;eHHyro B
TOKY P MarHHTHyJO CTpeJIKY, paBeH 1M = [p, .8]. ECJ1H CTpeJlKft CMe-
m;aeTC.H 113 nOJIO)KeHH.H paBHOBeCH.H aa yrOJI 13, JJ;OCTftTO''IHO NaJlLIA 11.1111
TOro, -q-r06L1 MO)KBO 6hIJIO 3RNeHHTb sinl3 8ft 13, TO no TeopeNe 0 MONeu-
Te HMIIYJIhCa HMeeN
dL d 2 p
M = -pBI3 = -;; I-
dt dt 2
rp;e 3H8.K NHHYC CBH3aH C TeM, qTO MeX8HH"IecKHH NOMeRT CRn Bcer.n;a
Hanp8BJIeH npOTRBOnOJIOmHO OTKJIOHeHHIO OT paBHOBeCH.R.
I1epHOJJ; MaJlLIX KOJIe6RHHii onpe.n;eJI.fIeTCH c}>opMYJloii
T = 2n = 21t fT .
ro VPB
9TOT pe3YJIbTftT MomHO nOJI}"mTL, He pema.H JI,ucp<1>epeHD;RaJIbHOrO
ypaBHeHH.fI, ft HCnOJIh3YSI 8H8JIOrHIO c rapMOHlf'tleCKHM OCD;HJLJISlTOpON.
5. OTMeTHM BLIqHCJIeHRH, OCHOSftHHhte Hft HHTepnpeTarurH AMne-
pa, CHMBOJIOM A, 8 OCHOBaHHLIe aft HHTepnpeTan;HH Buo-CaB8pa -
CHMBOJIOM BS, Tor
B = floi t a n = J.1 o i a.
'A 2nd g 2'  2nd'
T A = 21t
27tld . T.. = 2"  I . T A = J 2", .
. t a floPW. TBS tg a.
J.loPI. g- 1t -
2 2
I1pu a = ; (M8KCHMaJIbHO B03MO)KHOe 3BftQeUue) T A = T BS' a npH
tg a
a -.0 T A -+ ...!-Tns = 1,128 TBS. TaKK8.KBHyrpH3Tofi:06JI8CTH 
 a
2
T A
.RBJI.HeTC.H NOHOTOHHO pacTym;eit <1>YHKItHeA a, TO T BS .RBJISIeTCa MOHO-
TOHHO y6hIB8lOeii c}>YUKItueii a. TI03TONY aa onbITe aeB03MO»CHO p83
JIHTlHTb .n;Be HHTepnperxan;HH, eCJIH 3HaqeHHe a 60JIhme, qeM TO, 1I.11SI KO-
Toporo T A = 1,10 T BS (pa3J1U'tlUe 8810 %) T.e.
a 4 a a
tg"2 = 1,217t '2" = 1,05 2 .
235

C nOMOm;blO TpHI'OaOMeTpHtleCKHX Ta6JIHU; HJIH K8JIbKYJI.RTOpa MO)K-
a
HO onpe,AeJIHTb , '1TO aTO YCJIOBHe xopomo BhlnOJIH.HeTC.H KOr,AR 2" =
= 0,38 pM. OrClO,lJ,a yroJI a ,AOJImeH 6h1Tb MeHbme qeM 0,77 p8.ZI; = 44 0 .
B03MO)KHO H rpa4>WlecKoe pemeHHe ypaBHeHH.R ,AJI.H a, XOTSI OBO
HeCKOJIbKO rpOM03,lJ,KO. B TO :me BpeM.R HeAonYCTHMO pemeHHe nyreM
pa3JIo:meHH.R B pH,lJ,.
3a,AaQa 3
lA. TIpe.nnOJIo:mHM (KaK 3TO YKa38HO B YCJIOBHH), qTO op6HTa IOnu-
v 2
Tepa - KpyrOB8.R. C'I:lHT8S1 t:lTO P8JJ;.HaJIbHOe YCKopeHHe Ii coo6aeT-
C.R rpaBHTaU;HOHHOH CRJIoii, .neiiCTBYIOeH co CTOpOH:hI COJIH:Q;a, T.e.
v 2 GM
Ii = R2C (Me - MaCC8 COJIHua), Haif):J;eM CKOpOCTh ,ABHmeHHH IOnH-
Tepa no op6HTe:
GM
v = If = 1,306.10. M/c.
(1)
IIpHeMJIeMO Ii .n:pyroe pemeHlie.
lB. TaR KRK ,ABHmeHHe IOnHTepa Ct:lHTaeTCH KpyroBbIM H C nOCTO-
 R 27tR T 6
HBHOH CKOpOCTbJO, TO V = ro = -  r,D;e IO - nepHO,A 0 paeHHSI
TIO
lOuHTepa, npHBe,AeHHhlH B cnHCKe nOCTOHHHbIX.
2. ,l!;Be CHJIhI THrOTeHHH, ,lJ,eiicTBYJQID;He Ha 30HA, pllBHhI ecJIH
GMm GMc m
-----p2 = (R - p)2
(2)
(rone p - paeCTOHHHe ,AO IODHTepa M - ero Macea, m - MaCCR 30H,n;a)
OTKY,Aa
.[M(R - p) = p .JMc :::) (3)
:::) p = JM R = O02997 R ::;:: 2,333.10 10 M.
.JMc + JM (4)
TaKHM o6pa30M, rpaBJIT8:Q;HOHHhle CHJIhI cpaBHHB81OTC.R aa paCCTO-
HHHR OKOJIO 23,3 MJIH KM OT IODHTepa (npHMepHo 3341ODHTepH8HcKHX
PMHyea).
236

3. C nOMObJO npeo6paaOBaHH.H r8JIHJIe.H HaXOHMt 'tITO KOMnOHeH-
Thl CKOpoCTH 30H)];a B CHCTeMe OTC'tIeT8, CB.H3aHHOH C lOuHTepoM t paBHLI
v = v, v; = vo
noaTOMY B STOll CHCTeMe OTC'tIeTa 30HJJ; JIeTHT no,lJ, yrJIOM 6 0 = arctg V o
v
no OTHomeHHIO KOCH X t a ero CKOpOCTh pftBH8 v' = .J vg + v 2 (38MeTHM
v v. Vo V o
T8K)f(e, qTO Cose o = J = -t Sin eo :; J = -).
V + v 2 v' vg + v 2 v'
llO.v;CTBBJI.H.H 'tIHCJIOBhIe 3HaqeHH.f.I, HaxO,AHM
6 0 = O653 pa.n;  374°; Vi = I t 65.10 4 MjC.
4. TpaeKTOpH.R aOH,lJ,a JIHllih npH6JIH:meHHo Mo:meT 6hITb onHC8HB
RBK peaYJIhTaT rpaBHTa:Q;HoHHoro B38HMO,AeHCTSH.R ,ABYX TeJI (cne)];YeT
TaK)f(e yqHThIBaTb npHT.RmeHHe co CTOpOHhI COJIHa H nJI8HeT). llpe.v;-
nOJIO»CHM, 'tITO 30H):1; HftXOAHTC.R Ha 60JILlliOM, HO KOHeqHOM paccToH-
BRH OT IOnHTepa. IIoJIH8H 3HeprH.f.I 30H,lJ,a B CHCTeMe OTCQeTa, CBB38H-
BOH C IOnHTepoM t npH6JIH>KeHHO p8BH8 KHHeTHqeCKOH aHeprHH 30H,l];8
H8 aTOM paCCTOHHHH:
mv,2
E-.
2
CooTBeTcTSYJOm;ee 'tIHCJIOBOe 3HBQeHHe E = 112 r.n;m.
5. Ha YPaBHeHHH (I) CJIe,lJ,YeT, 'tITO PB,ZJ;H8.JIhHOe paCCTO.RHHe cTpe-
MHTC.R K 6eCKOHeQHOCTH (8 BeJIHQHHB, 06PBTHB.R paCCTOHHHIO - K
HYJIJO), KOr)];a
(5)
2Ev,2 b 2
1 + 1 + 2 cos6 = 0 =>
G 2 Mm
(6)
=> cosS =
1
2Ev,2 b 2
1+
G 2 M 2 m
3aMeTHM 'liTO pa,zJ;Ha.IThHoe p8CCTO.RHHe He MO)Ke1' 6hITh OTpH:Q;8TeJlh-
HhIMt n03TOMY ,AOnYCTHMLIe 3HaQeHH.R yrJIOB Y,lJ,OBJIeTBOp.HIOT Hep8seH-
CTBY
(7)
2Ev ,2 b 2
1 + 1 + 2 cosS >: 0 =>
G 2 M m
(8)
=>cosS-
1
2Ev. 2 b 2
1+
C 2 M 2 m
(9)
237

PemeHHSI JIH npe,AeJIbHhIX 3Ha'leHHH t COOTBeTCTBYlORX 3HaKY
paBeHCTBa B BhIp8JKeHHH (9), T81(OBhI:
e=x[ 2V" b' ] =
1+
G 2 M 2 m
=I[n_M 2" b'J. (10)
1+
G 2 M 2 m
nOaTON}' yrOJI ft.8 (CN. pRC. 158) Memy YMg aCHNnTOTaMH Ie rH-
nep60JIe peeR
ft.8 = (8+ - 8_) - 7t = 7t - 2arccos
1
2Ev,2 b 2
1+
G 2 M 2 m
= 1t - 2arccos
1
(11)
v,4b 2
1+ 22
GM
B nOCJIeHeH ti>opMYJIe MhI HCnOJIh30BRJIH BhIpameHHe ,AJIH nOJIHOH
3HeprHH, BhI'tIHCJIeUHOe paHee (CM. n. 4).
6. YrJIOBOe OTKJIOHeHHe HBJIHeTCH MOHOTOHHO y6hIB81OeH YHK-
IUleA npHn;eJIbHOrO napaMeTpa CJIe,lJ,OBaTeJIhHO OTKJIOHeHHe MaKCH-
M8JIhHO t Kora npHn;eJIbHhIH nap8MeTp MHHHMaJIeH. 113 ti>OPMYJIbI (9)
CJIe.u;yeT t qTO TO'tIK8 M8KCHMaJIbHOrO C6JIHmeHHH ,AOCTHraeTCH npH 8 ==
= o. B 3TOM CJI}"Iae H3 (1) SaxO.u;HM MHHHM8.JIhHOe paCCTOHHRe Me:m,lJ;Y
30HJ];OM H I\eHTpON: nJI8HeThI:
v,2b 2 ( v,4b2 )
r_ = GM 1+ 1+ G'M' -:
(12)
OTCIO,Aa HaXOHM npHn;eJIhHhIH napaMeTp:
b=
2 20M
T mln +- 2 r mln .
v'
(13)
238

STOT pe3YJThT8T :MO)KBO nOJIytlHTI» HBS1Je. COrJIaCBO 38KOHY coxpa-
HeRHS M:OMeHTa HMIIyJILCa,
L = mv'b = mV:nin r min'
r"lJ;e BBe,qeHa CKOpOCTIa V:nin' COOTBeTCTBYJO8B TO'llKe M8.KCHM8JThROro
C6JIHmeSH.H.
KpONe Toro, H3 38KOH8 COXp8.HeHHJI 3HeprHH CJIeJ];YeT
mv,2 mv' GMm
E=-=--.
2 2 r min
KOM6HHHPYH 8TH J];Ba COOTBOmeBH.R: BROBL npHXOJ];HM K BbIpIDKe-
BHIO (13).
IT pHeJILHbIH napaMeTp .R:BJIHeTCSJ. pacTeii 4»YHKHeH paCCTOS-
HilS H8.HOOJIl»mero C6JIH>KeHH.H, n03TOMY, eCJIH 30H):t He Mo»<.eT npH6JIH-
3HTLCH K nOBepXHOCTH IOTIHTepa 6JIH)Ke 'lJeM Ha .D;B8 p8JJ;Hyca TIJI8HeTLI
(r min :::; 3RIO' rJJ.e RIO - p8JJ;HYC IOn:HTepa), MHHHMaJIbHOe JJ.onycTHMoe
3HaqeHHe npH£(eJILHOrO napaMeTpa paBHO
"- = J 9lij, + 6 R.o.
(14)
113 SToro yp8.BHeHHH nOJI}"IaeM M8.KCHMaJILHoe B03MomHoe OTKJIO-
HeHRe:
6.8 max ::: 1t - 2 arccos
1
v'4bln
1 + G2 M2
:::; 1t - 2 arccos
1
v'" ( 6GM )
1 + G2 M2 9R + ---;;;2 RIO
(15)
HCnOJIL3Y.H QHCJIOBbIe 3HaQeHHS, nonyqeHHhIe BhIme, H8XO"lJ;HM
b min == 4,90 . 10 8 M:::; 7 ,0 RIO
6. 8 mu. = 1,526 pa,zx  87,4 0 .
7. OKOH'l8TeJILHOe Banp8BJIeHHe )tBHmeHHH OTHOCHTeJIbHO OCR X B
CHCTeMe OTCqeTa CB.R:38.HHOH C IOnHTepoM onpeJ];eJISleTCSI C}'MMOR Ha-
qaJILHOrO yrJIa H yrJI8 OTKJIOHeHII.H 6 0 + .6.6, eCJIH 30H,lJ, npOXOJ];IlT 3a
239

DJIaHeTOH. TI03TOMY KoMnOHeHThI OKOlftlaTeJILHOH CKOpOCTH B CHCTe-
Me, CB.H3a_HHOH C IOnHTepoM
 paBHbI
v
 = v' cos (6 0 + ..16), v
 = v' sin (6 0 + ..16),


B TO BpeM.H K8.K B CHCTeMe, CBH38HHOH C COJIH:Q;eM

v;'= v' cos (6 0 + ..16) - v
 v;'= v' sin (6 0 + ..16).


TI08TOMY O:KOH"tJ8TeJILHa.H CKOpOCTL aOH,lJ,a B CHCTeMe OTCqeTa, CB.B:-
3aHHOH C COJIHD;eM, paBHa


v" = .J [V' cos(6 0 + ..16) - v]2 + [v' sin(6 0 + f16)]2 =


= .J v
 + 2v 2 - 2v'vcos(6 o + f16) =


= .J v
 + 2v 2 - 2v'v(cos6 0 cosf16 - sine o sinf19) =


= .J v
 + 2v 2 - 2v'(vcosf16 - Vo sin..16) =


= J vo(v o + 2vsinA9) + 2v 2 (1- cosf16).


8. llcnoJIL3YSl 3HaT:leHHe MaKCHMaJILHO B03MomHoro yrJIOBoro OT-
KJIOHeHHSI, HaXO,lJ,HM v"= 2
62 .10 4 M/C.






.




XXXI Me>KAYHapOAHaR

M3M'IeCKaR OnMMnMaAa


2000 r., AHrnMR


3a
aqa 1
(3Ta aa.ll:a'lla COCTOHT Ha 5 Be CB.IIaaBBI>IX Me3C.Il:Y co6oii qacTeii)
tIaCTI> 1. llpLIrYH npHB.IIaaH K KOH
Y ,nJIHHBOrO ynpyroro JKryTa.

yroH KOHe
 JKryTa npHKpenJIeH K BLICOKOMY MOCTY. llpLIrYB ,ne-
JIaeT mar H n
aeT C MOCTa BHHa K peKe C HYJIeBOH HaqaJILHOH CKO-
pOCTblO. OH He ,nOCTHraeT BO,nLI. Macca npLIryHa m. ,IJ;JIHHa HepaCT.II-
Hyroro JKryra L. JKecTKocTb JKryTa k. YCKopeHHe cB060,nHoro n
e-
HH.IIg.
BLI MOJKeTe npe,nnOJIOJKJiTb, qTO:
- npLIryJIa MOJKHO paCCMaTPHBaTb KaK MaTepHaJILHYIO TOqKY Mac-
COH m, npHB.II3aHHYIO K KOH
Y JKrYTa;
- Macca JKryTa npeHe6peJKHMO MaJIa no cpaBHeHHIO C m;
- B Te'lleHHe Bcero BpeMeHH nOJIeTa MOJKHO npeHe6peqb conpoTHB-
JIeHHeM Boa,nyxa.
llOJIyqHTe CJIe,nYIOlI.{He BLIpaJKeHH.II:
1. paCCTO.llHHe y, KOTopoe npLIryJI npOJIeTeJI K MOMeHTY nepBOH nOJI-
HOH OCTaHOBKH;
2. MaKCHMaJILHYIO CKOpOCTb npLIryHa v m ' ,nOCTHrHyrylO B npo
ec-
ce n
eHH.II;
3. BpeM.II t nOJIeTa npLIryaa,no nepBoH nOJIHOH OCTaHOBKH.
tIacTL 2. llpe,nnOJIaraeTC.II, 'liTO K MOMeHTY OKOH'IIaTeJIbHOrO cpop-
MHpOBaHH.II 3eMJIH B HeH npHcyrcTBOBaJIH HaOTonLI ypaHa 238U H 23SU,
HO He npo,nYKTLI HX pacna,na. Pacn
 2 38 U H 23SU HCnOJIbayeTC.II ,nJI.II
o
eHKH BoapaCT8 3eMJIH T.
1. llepHO,n nOJIypacna,na HaOTona 238U COCTaBJI.IIeT 4,5.10 9 JIeT.
I1epHo,nLI nOJIypacn
a npo,nYKToB pacna,na B nOJIyq8.1OIl\eHC.II p
HoaK-
THBHOH
enO'lKe HaMHoro MeHbme, H B nepBoM npH6JIHJKeHHH HMJi MOJK-
HO npeHe6pe'llb. lJ,enOqKa pacn
a aaKaHqHBaeTC.II Ha cTa6HJIbHOM Hao-
Tone CBHHI.l8 206Pb. Ha
HTe qHCJIO aTOMOB 206Pb, 06oaHaqaeMoe 206 n ,
KOTopoe nOJIyqaeTc.II B npo
ecce p
HoaKTHBHoro pacn
a aa BpeM.II T,


245


Il1o...-




KSK CPYHK
 tUICJIS 238N STONOB 238U. COXpaHHBmHXC.II K HaCTO.lllI.{e-
MY lIIOllleHTY, B nepBO,na nOJIypacn
a 238U. (ECJIH Balli y,n06HO. HCnOJIL-
3YHTe B KaqecTBe e,nHHHD.LI BpellleHH 10 9 JIeT).
2. AHaJIOrHqHo. 235U pacna,naeTC.II C nepHo,nOlll nOJIypacna,na
0.710. 10 9 JIeT qepea p;enOqKY KOpOTKOJKHBYID;HX npo,nyJ<:ToB. aaKaH-
qHB81OID;}'1OC.II CTa6HJILHLIM HaOTonOM 207Pb. llo.nyqliTe BLIpaJKeHHe ,nJI.II
207 n qepea 235 N H nepHO,n nOJIypacn8,l{a 235U.
3. YP8.ll0BaH Py.na. aarp.llaHeHHaH Py,nOH CBH
a. 8HaJIHaHpyeTCH npH
nOMOID;H MaccneKTPOMeTPa. I1aMepeHH.II OTHOCHTeJIbHLIX KOHD;eHTP8.D.IIH
HaOTonOB 204Pb, 206Pb H 207Pb ,nalOT COOTHOmeHH.II 1,00 : 29.6 : 22,6
COOTBeTCTBeHHO. I1aoTon 204Pb HCnOJILayeTc.II ,nJI.II KaJIH6pOBKH; OH He
.IIBJI.IIeTC.II npo,nyJ<:TOM p8,l{HOaKTHBHOro pacna,na. AH8JIHa qHCTOH CBHH-
D.OBOH py,nLI ,naeT cooTHomeHH.II1.00: 17,9: 15,5. 3Ha.ll, qTO OTHome-
HHe KOHD.eHTp8D.HH 238N : 235N paBHO 137 : 1, nOJIyqHTe BLIpaJKeHHe
AJI.II BoapaCTa 3eMJIH T.
4. llpe,nnoJI8raH, '!TO T MHoro 60JILme nepHo,noB nOJIypacna,na 060-
HX HaOTonOB ypana, paCCqHTaiiTe npH6JIHJKeHHOe aHaqeHHe BoapacTa
3eNJIH.
5. Ha caMOM ,neJIe 3TO npH6JIHJKeHHOe 3HaqeHHe He .IIBJI.IIeTC.II aHa-
qHTeJILHO 60JILmHM no CpaBHeHHIO C HaH60JILmHM nepHo,nOM nOJIypac-
na,na, HO OHO MOJKeT 6LITL HCnOJILaOBaHO ,nJI.II 60JIee TOqHOrO paCqeTa
BeJIWIHHLI T. llpOH3Be,nHTe TaKOH paCqeT H On;eHHTe BoapaCT 3eMJIH C
TOqHOCTblO 2% .
qaCTJ. 3. TenJIoBoH ,nBHraTeJIb pa60TaeT. HCnOJILaY.ll,nBa o,nHHaKO-
BLIX TeJIa. HMelOlI.{HX nepBOHaq8JILHO paaJIHQHLIe TeMnepaTypLI T A H
T B (T A > T B). KaJK,noe TeJIO HMeeT Maccy m H HeHaMeHHYIO y,neJILHYIO
TenJIoeMKocTL c. TeJIa no,n,nepJKHBaIOTC.II npH nOCTO.llHHOM ,naBJIeHHH H
He lIIeH.IIIOT CBoero cpaaOBoro COCTO.llHH.II.
1. llpe,nCTaBLTe nOAP06HLIH BLIBO,n BLIpaJKeHH.II,nJI.II KOHeQHOH TeM-
nepaTypLI To ,nBYX TeJI B npe,nnOJIOJKeHHH. QTO TenJIOBOH ,nBHraTeJIL
COBepmHJI MaKCHM8JILHYIO TeopeTHqeCKH BOaMOJKHYIO MeXaHHqeCKYIO
pa60TY.
2. llOJIyqHTe BLIpaJKeHHe ,nJI.II MaKCHMaJIbHO BOaMOJKHOH MexaHH-
QeCKOH pa60TLI Am.
3. TenJIoBa.ll MamHHa pa60TaeT MeJK,ny ,nBYM.II eMKOCT.IIMH C BO,nOH
06'beMOM 2,50 M 3 . TeMnepaTypa BO,nLI B nepBOH eMKOCTH 350 K, a BO
BTOpoH 300 K.
BLIQHCJIHTe no aTHM ,naHHLIM MaKCHM8JILHYIO MeXaHHQeCKYIO pa-
60TY.
qaCTL 4. 3ap.ll,n Q paBHoMepHo pacnpe,neJIeH B BaKYYMe no 06'beM}T
mapa p8,l{HYCOlll R.


246


,




,


1. llOJIyqHTe BLIpll.JKeHHe ,nJI.II Hanp.llJKeHHOCTH 3JIeKTpHQeCKOrO
nOJI.II Ha paCCTO.llHHH x OT
eHTpa lIIapa ,nJI.II x "" R H X > R.
2. llOJIyqHTe BLIpUJKeHHe ,nJI.II nOJIHOH 3JIeKTpUqeCKOH 3HeprHII,
CB.IIaaHHOH C 3THM pucnpe,neJIeHHeM aUPHAa.
qacTL 5. TOHKoe Me,nHoe KOJIbLlO BpalI.{aeTC.II OTHOCHTeJIbHO BepTH-
K8JILHOH OCH, npoxo,n.lllI.{eH qepea ero ,nHaMeTp, B MarHHTHOM nOJIe 3eM-
nH. BeJIHqHHa IIH,nYKLlHH MarHHTHoro nOJI.II 3eMnH B ,naHHOH TOqKe
B = 44,5 MKTJI. BeKTop HH,nYK
HH HUnpaBJIeH BHHa noon yrJIOM a = 64°
K ropHaOHTaJIH. Onpe,neJIHTe, aa KaKoe BpeM.II yrJIOBa.ll CKOpOCTL KOJIb-
D;a YMeHblIIlITcH B,nBoe, eCJIII HaBeCTHO, qTO nJIOTHOCTb Me,nH Po =
= 8,90. 10 3 Kr/M 3 , a eey,neJILHOe conpOTHBJIeHHe p = 1,70. 10- 8 OM . M.
31'0 BpeM.II MHOro 60JILme BpeMeHH o,nHoro 060pOTa. TpeHHeM B ono-
pax H conpOTHBJIeHHeM Boa,nyxa MOJKHO npeHe6peqb. llpH pemeHHH
,naHHOH aa,naqH BLI MOJKeTe He yqHTLIBaTL .IIBJIeHHe caMOHH,nYKD;HH,
XOT.II Ha caMOM ,neJIe OHO HrpaeT onpe,neJIeHHYIO pOJIb.


3aAa'la 2
1. KaTo,nHa.ll JIyqeBa.ll Tpy6Ka (KJIT), COCTO.lllI.{aH Ha 3JIeKTpOHHOH
nYlliKH H 3KpaHa, nOMelI.{eHa B O,nHopo,nHoe nOCTO.llHHOe MarHHTHoe
nOJIe C MarBHTHoH HH,nYK
HeH, PUBHOH B II nap8JIJIeJIbHOH OCH nym-
KH, KaK nOKaaaHO Ba plIcYHKe 162. :3JIeKTpOHHLIH nyqoK HaJIyqaeTc.II C
KaTo,na 3J"IeKTpOHHOH rrYlliKH B,nOJIb ee OCH, HO HMeeT pa36poc HunpaB-
JIeHHH,no 5° OT OCH. B 06lI.{eM CJIytlfle HU 3KpaHe 06paayeTc.II pa3MLIToe
n.llTHO, o,nHaKO npH orrpe,neJIeHHLIX 3HaqeHH.IIX MarHHTHOH HH,nYK
HH
OHO CTUHOBHTCH peaKO CCPOKYClIpOBaHHLIM.


B


PIIC. 162


PaCCMOTplITe ,nBIIJKeHHe 3JIeKTpOHa, KOTOpLIH rrepBOHUqaJIbHO 6LIJ"I
HaJIyqeH 3.iIeKTpOHHOH nYlliKoH noon HeKOTopbIM yrJIOM r3 K ee OCH
(0" r3 " 5°). PnaJIOJKHTe aTO ,nBHJKeHHe Hn KOMnOHeHTLI, napaJIJIeJIb-
HLIe H nepneH,nuKYJI.IIpHLIe OCH nymKH II BLIBe,nHTe BLIpaJKeHHe ,nJI.II
OTHOmeHH.II aap.ll,na 3JIeKTpOHn K ero Macce (e / m) qepea CJIe,nYIOlI.{He
BeJIHqHHLI: HaHMeHbmee 3HaqeHHe MarHIITHoH HH,nYKLlIlII B min , npH KO-
TOpOM nOJIyqaeTcH CCPOKYCHPoB8HHoe rr.llTHO, YCKOpHIOlI.{YIO paaHocTb


247




nOTeHU,HaJIOB B 8JIeK1'pOHHOH nym-
Ke U (npHMHTe BO BHHMlI.HHe. qTO
U « 2 KB). H paCCTO.llHHe Me)K,ny
KlI.TO,nOM H 8Kp8HOM D.
2. PaCCMOTpHM ,npyroH MeTO,n
onpe,neJIeHH.II OTHOmeHH.II aap$!AlI.
8JIeKTpoHa K ero Macce (ycTaHoliKa
nOKlI.all.HlI. HlI. pHC. 163). B o,nHOPOA-
HOM_ MarHHTHOM nOJle C HBAYKIUI-
eH B pacnOJlOmeHId ,nBe KpyrJILJe
JIaT}'HHLIe nJlaCTHHKH paAHYCOM 1(.
paa,neJIeHHlde OqeHb MaJIldM pac-
CTO.llHHeM t. Mem,ny nJlaCTHHKaMH
cYlI{eCTByeT pll.aHOCTb nOTeH
Ha-
JIOB U. llJIaCTHHKH napaJIJIeJlbHId
,npyr APyry H COOCHLI. npHtleM HX
061I{aH OCb nepneHAHKyJl.llpHlI. HB-
,nYK
HH MaI'HHTHOrO nOJI$!. q>OTO-
nJIeHKa nOKpLIBaeT BHyrpeHHlOIo
nOBepXHOCTb KpyrJIOrO
HJIHRAPlI.
pa,nHYCOM r + S. pll.cnOJIOmeBBoro
COOCHO nJIaCTHHlCaM. TaKHM 06pa-
aOM, paCCTO.llHHe MeJK,ny cpoTOnJIeHKOH H Kpa.llMH nJIlI.CTHHOK paBHO s.
Bc.II YCT8HOBKa HaxO,nHTC.II B BaK}'YMe. OTMeTHM, qTO t HaMHOrO MeHb-
me S H r.
ToqeqHLIH HCTOqHHK, HcnycKaIOIl\HH l3-qacTH
LI B HeKOTopOM HH-
TepBaJIe cKopocTeH paBHOMepHo BO Bcex HanpaBJIeHH.IIX. nOMelI{eH nocepe-
,nHHe MeJK,ny D;eHTpaMH nJIaCTHHOK. H o,nHa H TlI. me nJIeHKa 06JIyqaeTc.II
npH CJIe,nytOlI.{HX TpeX paaJIH'lHLIX YCJIOBH.IIX: B = 0 H U = 0, B = Bo H
U = U o ' B = -Bo H U = -U o . r,ne U o H Bo - nOJlOJKHTeJIbHLIe KOHCTaHTId.
llpHMHTe BO BHHMaHHe. qTO BepXH.II.II nJIaCTHHlCa all.p.llJKeHa nOJIOJKH-
TeJIbHO npH U> O. H qTO HH,nYK
H.II MaI'HHTHOrO nOJI.II HanpaBJIeHa, KaK
nOKaall.HO Ha pHCYHKe 163, npH B > O. B 3TOH qll.CTH aa,naqH paCCTO.ll-
HHe MeJK,ny nJIaCTHHKaMH MOJKHO
q> = 90 0 q> = 180 0
C'lHTaTb npeHe6peJKHMO MaJIbIM.
Ha pHcynKe 163 OTMeqeHLI ,nBe
06JIaCTH cpoTOnJIeHKH: A H B. O,nHa
Ha HHX nOCJIe aaCBeqHB8.HH.II H npo-
.IIBJIeHH.II nJIeBKH Ha06pWKeBa HlI.
pHcynKe 164. YKamHTe, KaKa.ll H3
06JIlI.CTeH (A HJIH B) Ha06pWKeBlI. Ha
8TOM pJicynKe.


BuiJ ceepxy:


IlJr.acmuHICu


tPomonJr.eHICa
8 t 0 6 Jr.acmbA
06Jr.acmb B


..


B
BuiJ c6oICY:
tPomOnJl.eHICa


tPomonJr.eHICa


+ IIJr.acmuHICu
t . .
t I ;
I I
I I
I I
". .....
: r : 8
.


B
PHC. 163


q>=0°


4Jomon.neHICG Y


«


..


q>


PHC. 164


248




3. PaCCTO.llBHe Me
 ABYM.II BHemBHMH CJIe,ltaMH H8 llJIeHKe HaMe-
p.lleTC.II C nOMOlI.{LIO MlUCpocKona. H8 pHcyaKe 164 n0K838HO 3TO paCCTO-
.IIHHe y ,nJI.II O,nHOro Ha 3HaqeHHH yrJIa cp. PeaynLT8TLI HaMepeHHH upHBe-
,neHLI B T86JIHD;e (cp onpe,neJIeH KaK yroJI MeJK,ny BeKTopoM l\I8.I'HHTHOH
HBAYKI1;HH H JIHHHeH, coe,nHH.IIIO
eH I1;eHTp llJI8CTHHOK H pacCM8TpHBa-
eMYIO TOqxy H8 cpoTOllJIeHKe). LJHCJIeHBLIe aH8qeHH.II nap8MeTpOB CHCTe-
MLI TaKOBLI: Bo = 6,91 MTJI, U o = 580 B, t = 0,80 MM, s = 41,0 MM. KpOMe
TOro, BLI MOJKeTe HCnOJILaOB8TL aHaqeHH.II CKOpOCTH CBeTa B BaKYYMe:
3,00. 10 8 M/C H MaCCLI aneKTpoHa: 9,11 . 10- 31 Kr.


TatiJtuqa


YroJI cp, rpa,nYCLI 90 60 50
PaCCTO.llHHe y, MM 17,4 12,7 9,7
Y roJI cp, rp
YCLI 40 30 23
P8CCTO.llHHe y, MM 6,4 3,3 KOHeI1;
CJIeAa


Onpe,neJIHTe MaKCHMaJILHytO Ha6JIIO,naeMytO KHHeTlP'lecKytO aHep-
rHIO J3-qaCTH
. BLIp8aHTe QHCJIeHHLIH OTBeT B 3JIeKTpoH-BOJIbTax (aB).
4. I1cnOJlL3Y.ll HHcpopM8.IU:I1O, ,naHHytO B nyHKTe 3), Haii:Awre aH8QeHHe
OTHOmeHH.IIaapH,na 3JIeKTpoHa K ero Macce nOKM. YqTHTe, 'ITO no.nyqeH-
Hoe BaMB 3HaQeHHe MOJKeT He COrJIaCOBaTbC.II C H3BecTHLIM Ha JIwrepaTY-
pLI aHaQeHHeM H3-38 CHCTel\laTH'IecKoH norpeIIIBOCTH upH Ha6Jl1OAeHH.IIX.


3aAaQa 3
1JaCTL 1. 3Ta QaCTL a
8QH nOCB.II
eHa Tpy,nHOCT.IIM aKcnepHMeH-
TaJILHOrO 06HapYJKeHH.II rpaBHTaI1;HoHHLIX BOJIH, reHepHpyeMLIx npH
onpe,neJIeHHLIX aCTpOHOMH'IecKHX .IIBJIeHH.IIX. 3aMeTHM B aTOH CB.II3H, 'ITO
B3pLIB y,naJIeHHOH cBepxHoBoH aBea,nLI MOJKeT BLIaBaTb cpJIYKTyaI1;HIO
rpaBHT8IJ;HOHHOro nOJI.II H8 nOBepxHocTH 3eMJIH nOpH,nKa 10- 19 H/lCr.
Mo,neJIL ,neTeKTopa rpaBHTaI1;HOHHLIX BOJIH (pHC. 165) COCTOHT Ha
,nBYX MeTaJIJIlP'leCKHX CTepJKHeH
,nJIHHOH 1,0 M KaJK,nLIH, pacnOJIO-
JKeHHLIX noon np.llMLIM yrJIOM
,npyr K ,npyry. O,nHH Ha TOpI1;OB
K8»<,noro CTepJKH.IIaepKaJILHO OT-
nOJIHpOBaH, a ,npyroH JKeCTKO
aaKpenJIeH. llOJIOJKeHHe o,nHoro
113 CTepJKHeH BLI6paHO T8K, QTO-
6LI CHrHaJI, npHHHMaeMLIH <l><>TO-
aJIeMeHTOM, 6LIJI MHHHMaJIbHLIM.
nbeaOaJIeKTpHqeCKHe YCTpoii-


II OJtynpo3pa'l.Hble 3epICaJta


PeZYJtup080'l.HblU
8UHm cmep:JIC1Ul


PHc.165


249




CTBa C006lI.{aIOT CTepmH.IIM pe3KHe npo,nOJILHlde HMynLcld, B peaynL-
TaTe qero cBo6o,nHLIe KOHLI CTepJKHeH KOJIe6JIIOTC.II C npo,nOJILHLIM CMe-
ID;eHHeM Ax = ae- I1 ' cos(rot + a), r,ne a, Jl, ro H a - KOHCTaHTId.
1. Onpe,neJIHTe 3HaqeHHe Jl, eCJIH HaBeCTHO, qTO aMnJIHTy,na KOJIe-
6aHHH YMeHblIIueTC.II Ha 20% B TeqeHHe 50 c.
2. 3Ha.ll, qTO CKOpOCTL npo,nOJIbHLIX BOJIH onpe,neJIJleTCJI no p-
MYJIe v = Jf ' BLIqHCJIHTe HaHMeHblIIee aHaqeHHe qaCTOTLI roo' ec-
JIH HaBeCTHO, qTO CTepJKHH c,neJIaHLI Ha aJIIOMHHH.II C nJIOTHOCTLIO
p = 2700 Kr/M 3 H Mo,nYJIeM IOHra E = 7,1 . 10 10 lla.
3. HeBOaMOJKHO HarOTOBHTb CTepJKHH a6COJIIOTHO o,nHHaKoBoH ,nJIH-
HLI, a Ha-3a paaHocTH ,nJIHH CTepJKHeH BoaHHKalOT 6HeHH.II CHrH8JIa
cpoT03JIeMeHTa, qaCTOTa KOTOpLlX COCTaBJI.IIeT 0,005 . LJeMY paBHa
pa3HOCTL ,nJIHH CTepJKHeH ol?
4. CqliTa.ll,nJIHHY CTepmH.II paBHOH I, nOJIyqHTe 8JIre6p8.H'tIecKoe BId-
paJKeHHe ,nJI.II HaMeHeHH.II ,nJIHHLI CTepJKH.II AI, nO.llBJI.IIIOlI.{eroC.II BCJIe,n-
CTBHe H3MeHeHH.II YCKopeHH.II cB060,nHoro neHH.II g Ha BeJIHtlHHY Ag.
HaMeHeHHe BeKTopa Ai npoHcxo,nHT B HanpaBJIeHHH TOJILKO o,nHoro
Ha CTepJKHeH. BLIpa3HTe Bam OTBeT qepea I H napaMeTpLI l\IaTepHaJIa
CTepJKH.II.
5. JIaaep HcnYCKaeT MOHoxpOMaTHqeCKHH CBeT C ,nJIHHOH BOJIHLI
A = 656 HM. llpe,nnOJIOJKHM, qTO MHHHM8JILHIdH c,nBHr Topa, Kr>TOPLlH
MOJKeT 6LITL 3aperHCTpHpOBaH, paBeH ,nJIHHe BOJIHLI H.-;:yqeHH.IIna3e-
pa. Onpe,neJIHTe, KaKa.ll MHHHM8JIbHa.ll ,nJIHHa CTepJKH.II He06xo,nHMa,
qT06LI YCTaHOBKa CMOrJIa aaperHCTpllpOBaTb lIaMeHeHHe rpaBIIT8IJ;HOH-
Horo nOJI.II Ha BeJIHqHHY nOp.ll,nKa 10- 19 H/Kr.
tIacT!. 2. B 3TOH qaCTH 38,AaqH HayqalOTc.II a<}>c}>eKTLI, CB.IIa8HHLIe C
BJIH.IIHHeM rpaBHTaD;HoHHoro nOJI.II Ha pll.cnpocTPaHeHHe CBeTa B npo-
cTpaHcTBe.
1. <l>OTOH, H3JIyqaeMbIH C nOBepxHocTH COJIHD;a (Macca COJIHl\a M,
PHYC R), HcnLlTLIBlI.eT KpacHoe CMelI.{eHHe. llocTaBHB B COOTBeTCTBHe
aHeprHH cpOTOHU aKBHB8JIeHTHYIO Maccy nOKO.ll, BOCnOJILayHTeCL Teo-
pHeH T.IIrOTeHHH HLIOTOHU H nOKaJKHTe, qTO Ha6JIIO,naeMa.ll Ha 6ecKo-
HeqHOCTH qaCTOTa cpoToHa YMeHbmaeTC.II B (1- GM / Rc 2 ) paa (KpacHoe
CMelI.{eHHe ).
2. YMeHblIIeHHe qaCTOTLI cpoToHa aKBHBaJIeHTHO YBeJIHqeHHIO ero
nepHo,na KOJIe6aHHH. EC.iIH HCnOJIbaOBaTL <l><>TOH B KaqeCTBe aTanOHHLrX
qaCOB, TO YMeHLmeHHe qaCTOTLI aKBHB8JIeHTHO aaMe,nJIeHHIO BpeMeHH.
,IJ;8JIee MOJKHO nOKa3aTL, qTO 3aMe,nJIeHHe BpeMeHH Bcer,na conpoBOJK-
,naeTC.II COKpalI.{eHHeM e,nHHHLlLI ,nJIHHLI BO CTOJIbKO JKe paa. PaCCMOT-
pHM BJIH.IIHlIe 3Toro 3cpcpeKTa HU pacnpOCTpaHeHHe CBeTa B6JIHaH COJIH-
250

Lla. BHaqane onpe,nenHM a4>cpeKTHBHLIH nOK8a8TeJIb npeJIOMJIeHH.II
B TOqKe, Haxo,nJllI{eHC.II aa paCCTOJlHHH r OT eHTP8 ConHD;a. nYCTb
n r = c/C', r,ne c - CKOpoCTL CBeT8, HaMepeHHaH B CHCTeMe KOOp,nHB8T,
,nOCTaroqHO y,naJIeHHOH OT rpaBHT8.IUl0HHOro BnH.IIBH.II CoJIB8 (r 00),
c' - CKOpOCTb CBeTa, HaMepeaHaH B CHCTe1\le KOOp,nHH8T H8 P8CCTO.ll-
HHH r OT eHTp8 COJIHa. llOK8JKHTe, 'liTO ,nJI.II GM/Rc2« 1 DOK83a-
TeJIL npeJIOMJIeHH.II MOJKHO 8nnpoKCHMHpOBaTL BLIp8»<eBHeM n r = 1 +
+ aGM /rc 2 , r,ne a - KoaclKPHHeHT, KOTOpLIH BLI ,nOJIJKBLI onpeAeJIHTb.
3. I1cnOJILay.II BLIpameHHe ,nJI.II n r , B}dqHCJIHTe B P8AH8Bax yroJI e
OTKJIOHeHH.II JIyqa CBeTa OT np.llMOJIHHeHHOro nyrH np" ero npoXOJK-
,neHHH B6JIH3H Kpa.ll CO.ilHa. HCnOJILaYHTe CJIe,nYIOII{He AaHHLIe:
G = 6,67.10- 11 H 1\I2/ Kr 2, M = 1,99.10 30 Kr, R = 6,95. 10 8 M,
C = 3,00 10 8 M/C. BaM TaKJKe MOJKeT npHro,nHTLC.II TaKoH HHTerpaJI:
+co dx 2
l (x 2 + a 2 )3!2 - a 2 .
PeweHMR
3aAaqa 1
1JaCTL 1. 1. llOJIeT npLIryBa MOJKHO pa3,neJIHTb Ha ,nBe 4»8aLl. llep-
Ba.ll - ,nBHJKeHHe C nOCTO.llHHLIM YCKopeHHeM (cB060,nHoe n8AeHHe) -
nOK8 npLIryH He y,n8JIHTC.II OT MOCTa Ha p8CCTO.llHHe, paBHOe ,nJIDBe »<ry-
Ta L. BTopa.ll - ,nBHJKeHHe no rapMOHH'lecKOMY aaKOHY - ,no 1\I0MeH-
Ta, Kor,na CKOpOCTL npLIryBa 06paTHTC.II B HOJIL. B 060HX CJIyqaHX 1\10»<-
HO npHMeHHTL 3aKOH coxpaHeHH.II MeXaHH'lecKoH 3HeprHH.
Tor,n8 mgy = lk(y - L)2,
2
a HCKOMoe paCCTO.llHHe y = L + :g + J( L + :g J - L 2
2. CKOPOCTL npLIrYHa MaKCHM8JILHa B M01\leHT, Kor,na YCKopeHHe
MeH.IIeT CBoe HanpaBJIeHHe, TO eCTL npH npoXOJK,neHHH npLIryaoM no-
JIOJKeHH.II paBHOBeCH.II. BeJIHqHHy,necpopM8D;HH mHypa !!.LB 3TOT MO-
MeHT onpe,neJI.IIeM H3 YCJIOBH.II paBHOBeCH.II: !!.L = mg . CHOBa npHMe-
k
H.IIeM aaKOH coxpaHeHH.II MeXaHHqeCKOH 3HeprHH:
2 k ( M. ) 2
mm = mg(L + !!.L) -  .
251

I1cKoMa.ll CKOpOCTb V m =  2gL + m: 2
3. BpeM.II nOJIeTU npLIrYHa C YCKopeHHeM cB060,nHoro na,neHH.II
 = Pf .
lleplIo,n KOJIe6aHHH JKryra C npbIrYHoM T = 2n Jii. .
BpeMH ,nBHJKeHH.II OT nOJIOJKeHH.II paBHOBeCH.II ,no KpaHHero nOJIO-
T 1t r;;;
JKeHH.II t 3 = 4" = "2f'k .
OCTaeTcSI HaHTH BpeM.II t 2 , KOTopoe npOlUJIO OT Haqana ,neHcTBH.II
CHJILI ynpyrocTH ,no npOXOJK,neHHH nOJIOJKeHH.II paBHOBeCH.II.
llYCTb BO BTOpOH cpaae ,nBHJKeHH.II CKOpOCTb H3MeH.IIeTC.II no 3a-
KOHY v = v m sin(mt + CPo). Kor,na JKryr HUqHHaeT ,necpopMHpoBaTLcH,
v = V o = .j2gL npH t = to = o. Tor,na sin CPo = V o . llpH npOXOJKAe-
V m
HHII
nOJIOJKeHH.II
paBHOBeCH.II
v = vm(t = t 2 ),
3HaqHT,
sin(ml 2 + CPo> = 1.
TaKHM
06paaOM,
1t
mt 2 = "2 - CPo '
H
t 2 :::: r;; (  - arcsin ) .
Vk 2 v m
MCKoMoe BpeM.II
"t = I) + t 2 + t 3
= Pf + Jii. ( n  aresm(  )]
qacTb 2. 1. COrJIaCHO aaKOHY PMHoaKTHBRoro pacna,na, KOH:n;eHT-
paLlHH aTOMOB U 238 , coxpaHHBmHXC.II K HllCTO.lllI.{eMY MOMeHTY, N238 =
= No . 2 -... , r,ne No - Hcxo,nHa.ll KOHD;eHTpULlUH aTOMOB, T - BoapaCT
3eMJIli (B MJIp,n JIeT).
KOHD;eHTpaLlII.II aTOMOB Pb 206
n 206 = No - N238 = N o (l - 2-7..",,) = N 238 (27.:oo - 1) .
2. AHfi."IOrHtJHo, ,nJI.II KOHLleHTpaLlll1I aTOMOB Pb 207
n 207 = N23S(2Yo.lIo - 1) .
252

3. OTHomeHlIe KOHLleHTpuLlIlH aTOMOB CBIIHLla, BoaHIIKlIIlIX B pe-
aYJIbTaTe pucrra
a ypaHu:


n206
20;
n


N 238 (2Y-."'" - 1)
T/ _
N235(2/0..10 - 1)


137. 27."", - 1 = (29,6 - 17,9) == 164-
2%0710 _ 1 (22,6 - 15,5) ,;)


I1cKoMoe BLIpUJKeHlle ,nJ"IH HaXOJK,neHH.II T: 0,0120 - (2%.110 - 1) =
(27."", - 1) .
4. OLleHKY BoapacTa 3eMJIU MOJKHO npolIaBecTU c."Ie,nYIOlI.{IIM 06pa-
aOM.
llpe,nnOJIOJKIIM, qTO 2Y-."0 » 1, Tor,nn Hame ypaBHeHHe npllo6peTa-
eT 60JIee npocToH BII,n:


27."",
--rr- = 0,0120.
2/ 0 . 710


PelIIa.ll ero UHaJ"IHTHqeCKII, nOJIyqlIM T:::: 5,38 (MJIp,n J"IeT).
5. HCHo, qTO HCnOJIbaOBUHHoe HaMU rrplI6JIIIJKeHIIe He BnOJIHe onpaB-
5.3y.
,nUHHO ( 2 "'0 == 2,3). TeM He MeHee, OHO nOaBO.iI.IIeT nOJIyqllTb II 60JIee


TOqHYIO OLleHKY.
AJIH aToro no,nCT8BIIM nplI6JIIIJKeHHOe aHaqeHlIe T B rrpuBYIO qUCTb
IICxo,nHoro ypaBHeHIIH II HUH,neM HOBoe (YTOqHeHHoe) aHaqeHHe T, pe-
lIIa.ll aTO ypaBHeHlIe OTHOCUTe:IbHO HelIaBecTHoH B ero JIeBoII qaCTII.
By,neM rrOBTOp.llTb IITepaLluoHHLIH LlIlK."I, nOKa Be.1IIq1lHa rronpaBKII!!J.T
He CTaHeT MeHbme Tpe6yeMoH norpelliHOCTII B onpe,neJ"IeHlUI BoapaCTa
3eMJIII.
BOT KaK 6y,nYT BLIrJI.II,neTb paCqeTLI B cJIyqae BLIqllCJIeHlI.II BoapacTa
3eMJIlI C TOqHOCTblO,nO 0,01 MJIp,n ."IeT:


Ta6J1u14a


M mara 1 2 3 4 5 6
T 5,38 4,80 4,63 4,58 4,57 4,56
.C:l.OAM

T 4,80 4,63 4,58 4,57 4,56 4,56
YTO'IH
KH

!!J.T 0,58 0,17 0,05 0,01 0,01 0,00


TaKuM 06paaoM, MLI MomeM OLleHIITb BoapflCT 3eMJIH KaK 4,5 7 4,6
!'>Llp,n JIeT.


253




PHCYHOK 166 HJIJIIOCTpHpyeT npHMeHeHHLIH npH pemeHHH aa,naqu
MeTO,n IITepuHH.
f(T)
o
T yr"'"
T HC '
T
PRC. 166
qacTb 3. 1. KOJIUqeCTBO TenJIOTbI, OT,nUHHoe HarpeBaTeJIeM (A):
QA = mc(T A - ) .
KOJIHqeCTBO TenJIOTLI, nOJIyqeHHoe XOJIO,nHJIbHUKOM (B):
QB = mc(T o - Tn).
Pa60Ta, npoHaBe,neHHU.II ,nBHraTeJIeM:
A = QA - Qn = mc(T A + Tn - 2).
BLIpaallM uaMeHeHHe aHTporrHH HarpeBaTeJI.II:
To To
ASA = J  = J mT = mcln( : ) .
T A T A
AHaJIOrlIqHO, HaMeHeHHe 3HTponHII XOJIO,nHJIbHHKa:
To
AS n :::: J BQ = mcln ( 7;, )
T Tn .
Ts
Y CJIOBUe cOBeplIIeHuH MUKCHMaJIbHO BOaMOJKHOH pa60TLI:
!!J.S A + !!J.S n = o.
I1JIII, In( : ) = -In( ).
254

I1cKoMa.ll TeMnep8Typ8
 = .JTATR = 324 K .
2. MaKcHMnJIbHO BOaMOJKHR.II MeXaHJ:ltIecKRH pa60T8:


A.n.x = me( T A + T B - 2 .JTA T R ) = mc( JT: -.ji;r.


3. C
eToM ,n8HHbIX a
8qHAmax:::: 20 MAJK.
qacTL 4. 1. I1cnOJIbayeM TeopeMY f8YCC8.
Q- E4 2 E Q
llpH X > R EO - . nx
 = 41tEoX2 .
llpH X
 R .!..Q (
 ) 3 = E . 4nx 2
 E =
X.
EO R 41tE o R


2. llJIOTHOCTb aHeprHH 3JIeKTpOCT8THqeCKOrO nOJI.II
dW EoE 2
w= dV =""2'
llOJIH8.11 aHepm.ll nOJI.II:


w = J R Eo (
 ) 2 . 4nx2dx + J - EO ( -5L ) 2 .4nx2dx =
2 41tE o R 3 2 41tE0%2
o R


Q2 ( 1 ) 3 Q2
= 81tE o R '5 + 1 = 20 1tE o R .


qacTL 4. llpH Bpall.leHHH KOJIb
a B MarHHTHOM nOJIe BoaHHK8eT TOp-
MOa.lllI.{HH MOMeHT M. 06YCJIOBJIeHHbIH Ba8HMoAeHcTBHeM HH,nyK
H-
OHHoro TOK8 I C MRrHHTHbIM nOJIeM B. BeJIHqHHa 3Toro MOMeHT8
M = I BrS sin q> , r,ne Br - ropHaOHT8JIbH8.11 COCT8BJI.IIIOlI.{a.ll BeKTopa
M8rHHTHoH H

HH, S - nJIOlI.{
b KOJIb
a, q> - yroJI Mem.ny BeKTO-
pOM HOpMaJIH ii H Br (pIlC. 167).


-
n


PHc.167


255








3anHlIIeM ,nJI.II KOJIbLla OCHOBHOe ypaBHeHHe ,nHHaMliKH BpalI.{aTeJIb-
Horo ,nBliJKeHH.II TBep,noro TeJIa (B CKaJISIpHOM BH,ne) C yqeToM Toro, qTO
MOMeHT CHJI - TOpM03SIm;HH: M = _J dco ,r,ne J - MOMeHT HHepLlHH
dt
. KOJIbIJ;a OTHOCHTeJIbHO ,naHHOH OCR, ill - yrJIOBa.ll CKOpOCTb KOJIbIJ;a.
I1H,nYKUHOHHbIH TOK B KOJIbIJ;e I =  , r,ne £i - 3AC HH,nYKIJ;HH,
R - conpOTHBJ"IeHHe KOJIbLla.
del>
dt '
r,ne  = Be-S cos q> - MarHHTHbIH rrOTOK qepe3 KOJIbIJ;O.
llo 3aKOHY aJIeKTpOMarHHTHoH HH,nYKIJ;HH £,
T KHM 06 p a30M J de)) = _ (Bx.8 sin cp)2 dcp
a 'dt R dt .
llpe,nnOJIOJKHM, qTO aa BpeM.II o,nHoro 060poTa yrJIOBa.ll CKOpOCTb
KOJIbIJ;a HaMeH.IIeTC.II He3HaqHTeJIbHO ( ':: :::: const). Tor,na MOJKHO BbI-
d 1 Ifi.S2 1
paaHTb cpe,nHee aa nepHo,n YCKopeHHe: d = -'2 JR ill. ( '2 - cpe,nHee
. 2 v v
aa rrepHo,n 3HaqeHHe SIn q> rrpH Bpam;eHHH C rrOCTO.llHHOH yrJIOBOH CKO-
pOCTbIO ).
CqHTaSI nO-rrpeJKHeMY MaJIbIM H3MeHeHHe yrJ"IOBOH CKOpOCTIi aa rre-
pHO,n, pelIIHM nOJIyqeHHoe ypaBHeHlie:
dw Ifi.S2 ( (oJ ) B 2 8 2
- = --dt => In _0_ = --X-!:!.t
(oJ 2JR (oJ 2JR .
CKOpOCTb BpaU.leHHSI KOJIbIJ;a YMeHbmHTC.II B,nBoe aa BpeM.II
M = 2JR In 2
BxS2 .
YIHTblBa.ll, qTO Be- = B cos a, S = nr 2 , R = P 2;r , J = m;2 ,
m = Po . 2nr 0 S, r,ne r - pa,nHYC, m - Macca, as - llJIOU.la,nb uouepeI-
Horo CeqeHHSI KOJ'IbLla
M = 4(ln 2)po :::: 1,1 . 10 6 C
(B cos a)
IIoJIyqeHHbIH peaYJIbTaT orrpaB,nbIBaeT Hallie npe,nnOJIOJKeHHe 0
rrOCTO.llHCTBe yrJIOBOH CKOpOCTH Ha rrpOMejKYTKe BpeMeHH, 3a KOTOpbIH
256

COBepmaeTCSI O,nHH 060POT, ,naJKe npH ,nOCTaTOqHO Me,nJIeHHOM BpalI.{e-
HHH KOJIbn;a.
Apyrou cnoc06 (aHepreTHlIeCKHu)
IIpH BpalI.{eHHH KOJIbLla B MarHHTHOM nOJIe ero MeXaHHqeCKa.ll aHep-
dE
rH.II npeBpaw;aeTcSI B aJIeKTpHqecKYIO, UpH aTOM - ill ex . = P3JIeKTP.;
_ J",2 dE -J..,
E ......   = J(jJ.
Nexo - 2 dt dt '
P3JIeKTP. = 1 2 R.
3TO npHBo,nHT c yqeTOM BbIpaJKeHH.II ,nJI.II CHJIbI HH,nYKn;HOHHoro
TOKa K YJKe aHaKOMOMY HaM ,nHcpcpepeHLlHaJIbHOMY ypaBHeHHIO:
J dw (BrSsincp)2 d<p
dt = - R dt .
3a,naqa 2
1. )J;BHJKeHHe aJIeKTpOHa B MarHHTHOM n6JIe MOJKHO npe,nCTaBHTb
KaK KOM6HHaLlHIO paBHOMepHoro np.llMOJIHHeHHOro ,nBHJKeHH.II B,nOJIl,
CHJIOBbIX JIHHHH nOJI.II H paBHOMepHoro ,nBHJKeHH.II uo OKPYJKHOCTH B
rrJIOCKOCTH, rrepueH,nHKYJI.IIpHOH CHJIOBbIM JIHHH.IIM (pHC. 168). <l>oKY-
CHpOBKa rrpOHcxo,nHT, Kor,na 3JIeKTpOHbI COBeplIIalOT n;eJIoe qHCJIO 060-
pOTOB. BpeM.II o,nHoro 060poTa T = 2;: He aaBHCHT OT BeJIHqHHbI CO-
CTaBJI.IIIOlI.{eH CKOpOCTH Vl.. LJT06bI nOJIyqHT:b CCPOKYCHpOBaHHoe n.llT-
HO, 3KpaH HYJKHO paCnOJIOJKHTb Ha paCCTO.llHHH D = NvnT OT aHo,na
(N - qHCJIO 060pOTOB). Tal< KaK 13 < 5°, paaJIHqHeM MeJK,ny MO,nyJI.IIMH
CKOpOCTH V H ee COCTaBJI.IIIOlI.{eH un MOJKHO npeHe6peqb. MOAYJI:b CKO-
pOCTH U = 2 .
TaKHM 06paaoM, YCJIOBHe CPOKYCHPOBKH
D = N 2eU 21tm = N81t2U 
m eB B 2 'Ie.
V l_
Un
13
..
PHC. 168
9 <l>H3H"" 10-11 ,,-,aeebl
257

TaK KaK CPOKYCHPOBKft npOHaBo,nHTC.II llpH MHHHMaJILHO BOaMOJK-
HOM aHftqeHHH B, TO N = 1.


e _ 8n 2 U
Y,neJIbHbIH aap.ll,n 3JIeKTpOHa - - 2---;::2.
m BminLr


IIpuMe'laHue. YCJIOBHe U« 2 KB nOaBOJI.IIeT B aTOH qaCTH aftAaqH
nOJIbaOBaTbC.II HepeJI.IITHBHCTCKHMH cpopMYJIaMH.
2. llPH pemeHHH aTOH qftCTH aa,naqH 6y,neM nOJIbaOBaTbC.II CJIe,nyw-
lI.{eH Mo,neJIblO. TaK KnK paCCTO.llHHe MeJK,ny nJIftCTHIIKaMH t OqeHb MaJIO,
TO aJIeKTpOH CMOJKeT ,nOCTHrHYTb cpOTOnJIeHKH, eCJIH He 6y,neT OTKJIO-
H.IITbC.II aJIeKTpH'leCKHM H MarHHTHLIM nOJI.IIMH K nJIaCTHHKaM. :3TO Boa-
MOJKHO TOJIbKO B TOM CJIyqae, Kor,nn peaYJIbTHPYIOlI.{a.ll CHJIa, ,neitcTBY-
lOlI.{a.ll Ha qaCTHLlY, p8.BHft HYJIIO.
llPH U> 0 (pHC. 169), 3JIeKTpHqeCKn.ll COCTaBJI.IIIOlI.{ft.ll CHJIbI 6y,neT
HftnpaBJIeHa K BepxHeH nJIaCTHHKe. MarHHTHnH COCTaBJI.IIIOlI.{a.ll6y,neT
HanpaBJIeHft BBepx ,nJI.II aJIeKTpOHoB, ,nBHJKylI.{HXC.II K aOHe A (pHC. 170),
BHHa - ,nJI.II 3JIeKTpOHOB, ,nBHJKYlI.{HXC.II K aOHe B. TaKHM 06pftaoM, aJIeK-
TPOHbI HH npH KaKHX YCJIOBH.IIX He MoryT ,nOCTHqb aOHLI A. K aHano-
rH'lHOMY BbIBO,ny npHxo,nHM npH HaMeHeHHH HanpaBJIeHHH 060HX no-
JIeH Ha npOTHBOnOJIOJKHbIe.
3HnqHT, aaCBeqeHHOH MOJKeT OKaaaTbC.II TOJIbKO aOHa B.


.
13


H anpaBJleHUe 3JleKmpU'leCKOZO
nOAJl B CJlY'lae U > 0 (BUiJ C60l<Y)


H anpaBJleHUe MaZHUmHOu. CO-
CmaBJl.RHJ
eu. CUJlbl cOBnaiJaem C
HanpaBJleHUeM 3JleKmpU'leCKOu.
COCmaBJl.RHJ
eu. npu B > 0 U CKO-
pocmu, HanpaBJleHHOu. B 30HY A
(BUiJ cBepxy)


PHC. 169


PHc.170


258




3. CliJIa, ,neHcTByrorn;a.a Ha aJIeKTpOH, ,nBHJKYlI.{HHC.II B cKpelI.{eHHbIX
aJIeKTpHqeCKOM H MarHliTHOM nOJI.IIX, F = e(.E + [v x 81).
AJI.II tIaCTliD;bI, ,nBHJKYlI.{eHC.II B aOHY B noon yrJIOM cp K CliJIOBbIM JIIi-
HH.IIM MarHliTHoro nOJI.II,
F = lel(E - vBsincp) = 0 npH v = E
B sin cP
CJIe,n Ha cpOTOnJIeHKe 06pbIBaeTCSI npli CPo = 23°. 3T0 OaHaqaeT, qTO
MaKCHMaJIbHa.ll Ha6JIIO,naeMaH B onbITe CKOpOCTb
v rnax =  = U/t) = 2,685. HIM/C.
B Slll CPo sm CPo
3Ta CKOpOCTb 6JIliaKa K CKOpOCTH cBeTa, noaToMy,nJI.II paCqeTa aHep-
I'HIi aJIeKTpOHa BOCnOJIbayeMc.II peJI.IITHBliCTCKOH CPOPMYJIOH:
2
w  1-T,;:' r - m"c' , 1.02 .1O-"i\m' 6.0KaB.
4. By,neM CqliTaTb, JTO ropliaOHTaJIbHa.ll COCTaBJI.IIIOlI.{a.ll CKOpOCTH
aJIeKTpOHa B npoLlecce ,nBHJKeHH.II OCTaeTC.II HeHaMeHHoH H MHoro 60JIb-
meH ee BepTliKaJIbHOH COCTaBJI.IIIOlI.{eH, T. e. YCKopeHHe Bce BpeM.II no-
CTO.llHHO H HanpaBJIeHO BHlia liJIIi BBepx B aaBHCliMOCTIi OT HanpaBJIe-
HH.II BeKTopa MaI'HliTHOH HH,nYKLlHH (B > 0 liJIH B < 0).
aT 2
Tor,na paCCTO.llHHe MeJK,ny nOJIOCaMIi Ha rrJIeHKe y = 2 2 ,
r,ne 1" = .!
v
sB sin cp
E
tsB sin cp
U o
evB
m
= eV o m (  ) 2
mot V -l c J
a =
eV o J l _ ( E ) 2
mot cB sin cp .
TaKHM 06paaoM,
= eU o 1 1 _ (  ) 2. ( tsB sin cp ) 2 =
y mot V cBsmcp U o
=  . ts 2 B 2 sin 2 cp 1 _ ( Uo ) 2
mo Un tcB sin cp
sB sin cp
y.
e
H-
mo
J( tsB ;n cp r - (  r
259
.

BOCnOJIbayeMc.II nOJI}"IeHHOH CPOPMYJIOH H ,naHHLlMH, npHBe,neHHLI-
MH B Ta6JIHIte, ,nJI.II BLIqHCJIeHH.II y,neJIbHOrO aap.ll,na aJIeKTpOHa.
Tal5Jruqa
q>, rpycbI 90 60 50 40 30
y,MM 17,4 12,7 9,7 6,4 3,3
e/m o ' 1,678 1,672 1,678 1,668 1,668
1011 KJI/Kr
Cpe,nHee aHaqeHHe  :::: 1,673 . lOll KJI/Kr.
fflo
3aa'la 3
':IacTI,l.
1. AMnJIHTy,na KOJIe6aHHH MeH.IIeTC.II no aaKOHY A = ae -P' . OrKy,na
J.L = .; l{ : ) :::: 4,46. 1O-3 C -l.
2. CKOpOCTb pacnpOCTpaHeHH.II npo,nOJIbHLIX BOJIH B aJIlOMHHHH
V = ../E/p :::: 5,13 KM/C. ro = 21tf, r,ne A - ,nJIHHa BOJIHLI. ':IacToTa
KOJIe6aHHH MIIHDM8JIbHa llpH MaKCHMaJIbHOH ,nJIDHe BOJIHLI
Amax = 41 = 4,OM  roo = 21t :::: 8,05 . 10 3 pa.n/c .
max
3. ':IacTOTa 6HeHHH paBHa paaHocTH qaCTOT KOJIe6aaHH CTepJKHeH &0.
Tax KaK &0 « roo' TO ,nJI.II paaHOCTH ,nJIHH CTepJKHeH MO»CHO aanHcaTb
' =  = -   01 :::: 3,9 MKM.
4. I1aMeHeHHe rpaBHTaD;HoHHoH CHJILI, ,neHcTBYIOlI.{eH Ha aJIeMeHT
CTepJKH.II ,nJIHHOH dx, pacnOJIOJKeHHLlH Ha paCCTO.llHHH x OT ero cB060,n-
Horo KOHLla, M = m6.g, r,ne m = p8x (8 - nJIOlI.{a,nb nOnepeqHOrO ce-
qeHH.II CTepJKH.II).
MaMeHeHHe ,naBJIeHH.II !!.p = t1F = px!!.g.
S
I1aMeHeHHe BeJIHqHHLI OTHOCHTeJIbHOH ,necpopMaD;HH
ilp = pilg x.
E E
I 2
I1aMeHeHlIe ,nJIHHLI CTepJKH.II !!.l =  !!.(dx) = J pilg xdx = pi ilg .
L..J E 2E
o
!!.e = t,(dx)
dx
260

5. MHHHMaJILH8.JI ,nJIHHB CTepJKH.II
'min = 2EA '" 18,6 . 10 9 M'" 20 MJIH KM!
pt:.g
qaCTL 2.
M "VO
1. :3HeprH.II cpoToHa y nOBepXHOCTH COJIHD;a Eo = hv o - G It .
:3HeprH.II cpOTOHB Ha 6ecKoHeHocTH E = hv. TaK KBK E = Eo. TO
v = V ( 1 _ GM )
o &2.
2. 3aMe,nJIeHHe BpeMeHH H COKpalI.{eHHe ,nJIHHLI npHBo,nHT K YMeHb-
meHHIO CKOpOCTH CBeTB B (1 - -:: r paa HB paCCTO.llHHH r OT D;eHTPa
GM
COJIHa. TorAa c yqeToM &2 «1 n,
cpHHeHT a. = 2.
= 1 '" 1 + 2 GM . Ko34>-
( 1 _ GM ) 2 re2
re 2
3. 3anHmeM aaKOH npeJIOMJIeHH.II CBeTa (pHC. 171):
n(r + dr) sin q> = n(r) sin (q> - de)
......
......
q>
!R - dO
..................
r ................
......
,
,
,
,
,
,
,
,
,
,
--::[e------_.
,
,
,
:0..

PRC. 171
[n(r) + :; dr ]sin q> = n(r)[sin q> cos de - sin de cos q>]
cos de '" 1 , sin de :::: de :::) '! sin q>dr = -n(r)cos q>de
dr
261

de = _-.L dn tgq>dr
n(r) dr
dn = _2 GM . n :::: 1
dr r 2 e 2 ' .
dS = 2 GM tgq> dr
e 2 r 2
GM
TaK KaK &2 « 1, TO nOK83aTeJIb npeJIOMJIeHR.II HRqTOmHO MaJIO
OTJIHqaeTC.II OT 1, CJIe,nOB8TeJIbHO, yroJI OTKJIOHeHH.II MomHO CqHTaTb
MaJIbIM.
 "' ..) q>
.........
...
...
r ......
x
.
PRC. 172
Tor,na tg q> = R ; r 2 = R 2 + x 2 R rdr = xdx (pHC. 172).
x
+00
e = J de = 2GM J tg cp rdr = 2GM J R dx 3
c 2 r 3 c 2 ( 2 2 ) %
-00 R + x
= 4GM :::: 85.10- 6 P ",rr
Rc 2 ' ...,...,

XXXII Me>KAYHapOAHaSi
cl>M3M'IeCKaSi OnMMnMaAa
2001 r., TYP....MR
3aaqa 1
(3Ta 3ap;aqa COCTOHT H34 He CB.IIaaHHLlX Me)ICY C060H qaCTeii:)
qacTh 1. KJIHCTpOHbI - aTO YCTpoHcTBa, HCnOJIb3yeMble ,nJI.II YCH-
JIeHH.II CHrHMOB OqeHb BbICOKOH qaCTOTbI. KJIHCTpOH COCTOHT Ha ,nBYX
o,nHHaKOBbIX nap uapaJIJIeJIbHbIX nJIaCTHH (nOJIOCTeH), pacnOJIOJKeH-
HbIX ,ZJ;pyr OT ,ZJ;Pyra Ha paCCTO.llHHH b, KaK nOKaaaHO Ha pHCYHKe 173.
3JIeKTpoHHbIH nyqoK C HaqaJIbHOH CKOpOCTblO V o nepeceKaeT BCIO CHC-
TeMy, npoxo,D;.11 qepea MaJIeHbKHe oTBepCTH.II B nJIaCTHHax. BbICOKO-
qaCTOTHoe HaUp.llJKeHHe, KOTopoe ,nOJIJKHO 6bITb yCHJIeHO, no,naeTC.II K
06eHM napaM UJIaCTHH C onpe,neJIeHHOH pa3HocTbIO cpaa MeJK,ny HHMH
(nepHo,n T COOTBeTcTByeT HaMeHeHHIO cpa3bI Ha 2n), nopoJK,na.ll MeJK,ny
nJIaCTHHaMH (B nOJIOCT.IIX) B ropH30HTaJIbHOM HaUpaBJIeHHH nepeMeH-
HbIe aJIeKTpHqeCKHe nOJI.II. Kor,na aJIeKTpHqeCKOe nOJIe HanpaBJIeHO
BnpaBo, aJIeKTpOHbI, npOJIeTalOU.lHe qepe3 BXO,nHYIO nOJIOCTb, aaMe,nJI.II-
IOTC.II, a Kor,na BJIeBO - YCKOp.llIOTC.II, TaK qTO BbIJIeTalOlI.{He l:Ia BXO,n-
HOH nOJIOCTH aJIeKTpOHbI Ha orrpe,neJIeHHOM paCCTO.llHHH OT Hee 06paay.
lOT crYCTKH (rpynubI). ECJIH BbIXo,nHa.ll rrOJIOCTb nOMelI.{eHa B TOJKY 06-
paaOBaHH.II crycTKa, aJIeKTpHqeCKOe nOJIe B aToH nOJIOCTH 6y,neT
nOrJIoU.laTb 3HeprHIO Ha crycTKa npH YCJIOBHH, qTO paaHocTb cpaa (Bpe-
MeHHa.ll aa,ZJ;epJKKa) BbI6paHa cooTBeTcTBYIOU.lHM 06paaoM.
, a I b
PHc.173
IIycTb CHrHaJI (Hanp.llJKeHHe, rro,naBaeMoe Ha nJIaCTHHbI) HMeeT np.ll-
MOyrOJIbHyro CPOPMY C uepHo,noM T = 1,0 . 10- 9 C. HanpSIJKeHHe MeH.IIeTCH
B HHTepBaJIe U = IO,5 B, HaqaJIbHa.ll CKOpOCTb aJIeKTpOHoB
263

v o = 2,0. 10 6 M/C, OTHomeHHe aap.ll,na K Macce elm = 1,76. 1011 KJI/Kr.
PaCCTO.llHHe a MeJKAY nJIaCTHHaMH HaCTOJIbKO MaJIO, qTO BpeMeHeM npo-
xo.m,neHH.II aJIeKTpOHa qepea rrOJIOCTb MOJKHO npeHe6peqb. C TO'lHOCTblO
,no qeTbIpex aHaqalI.{HX D;Hcpp BbIqHCJIHTe CJIe,nYIOlI.{He BeJIHqHHbI:
1. paccToSIHHe b, Ha KOTOpOM aJIeKTpOHbI 06paaYIOT crYCTOK;
2. He06xo,nHMYIO paaHOCTb cpa3, KOTOpylO ,nOJIJKeH 06eCneqHTb cpa-
aOBbIH CMeCHTeJIb.
qacTL 2. llYCTb d l H d 2 rrpe,ncTaBJISIIOT C060H cpe,nHHe paCCTO.llHH.II
MeJK,ny LleHTpaMH MOJIeKyJI BO,nbI B JKH,nKOH H raao06pa3HoH cpaaax co-
OTBeTCTBeHHO. llpe,nnOJIOJKHM, qTO 06e cpaabI Haxo,n.llTC.II npH 100°C H
aTMoccpepHoM ,naBJIeHHH, a uap Be,neT ce6.11, KaK H,neaJIbHbIH ra3. I1c-
rrOJIbay.ll cJIe,nYIOlI.{He ,naHHbIe: nJIOTHOCTb BO,nLI B JKH,nKOH cpa3e
PI = 1,0. 10 3 Kr/M 3 , MOJI.IIpHa.llMaCcaBo,nbIM = 1,8. 10- 2 Kr/MOJIb, aTMO-
ccpepHoe ,naBJIeHHe P a = 1,0 . 10 5 H/M2, yHHBepCaJIbHa.ll ra30Ba.ll nOCTO-
.IIHHa.llR=8,31AJK/(MOJIb. K), qHcJIoABora,npoN A =6,0.10 23 MOJIb -I,
no,nC'lHTaHTe OTHomeHHe d 2 /d l .
qacTL 3. Hanp.llJKeHHe U 0 nHJIO-
06paaHOH cpOpMbI Mo.meT 6bITb no-
JIyqeHo MeJK,ny nJIaCTHHaMH KOH-
,neHcaTopa C, nOKaaaHHoro Ha PH-
CYHKe 174. PeaHCTOp R npe,nCTaB-
JISIeT C060H nepeMeHHoe conpOTHB-
+ JIeHHe, U j - Hnnp.llJKeHHe H,neaJIb-
HOH 6aTapeH, a SO - HCKpOBOH
npOMeJKYToK, COCTO.lllI.{HH Ha ,nBYX
aJIeKTpo,noB C perYJIHpyeMbIM pac-
CTO.llHHeM MeJK,ny HHMH. Kor,na Hanp.llJKeHHe, no,naBaeMoe Ha aJIeKTpO-
,D;bI, rrpeBocxo,nHT Hanp.llJKeHHe np060.ll Un' BoaJJ;YX MeJK,nyaJIeKTpo,na-
MH HOHHaHpyeTC.II, B rrpOMeJKYTKe npOHCXO,D;HT KopOTKoe aaMbIKaHHe,
H OH OCTaeTC.II B TaKOM COCTO.llHHH ,no Tex rrop, nOKa Hanp.llJKeHHe Ha
npOMeJKYTKe He CTaHeT OqeHb MaJIeHbKHM.
1. Ma06paaHTe rpacpHK aaBHCHMOCTH Hanp.llJKeHH.II U 0 Ha KOH,neH-
caTope OT BpeMeHH t nOCJIe Toro, KaK KJIlOq aaMKHYT.
2. KaKoe YCJIOBHe ,D;OJIJKHO BbInOJIH.IITbC.II, qT06bI nOJIyqHJIOCb no-
qTH JIHHeHHO HaMeH.IIIOlI.{eeC.II nHJIo06paaHoe Hanp.llJKeHHe U 01
3. B CJIyqae, Kor,na YCJIOBHe 2 BbIrrOJIHeHO, nOJIyqHTe ynpOlI.{eHHoe
BbIpaJKeHHe ,nJI.II nepHo,na T nHJIo06pa3Horo Hanp.llJKeHH.II.
4. qTO BbI ,nOJIJKHbI MeH.IITb (R HJIH SO HJIH TO H ,npyroe), qT06bI Ha-
MeHHTb TOJIbKO nepHo,n1
5. qTO BbI ,nOJIJKHbI MeHSITb (R HJIH SO HJIH TO H ,npyroe), qT06bI
H3MeHHTb TOJIbKO aMnJIHTy,ny1
SG
C
U o
PHc.174
264

 : - 
t
6. BaM ,naH ,nonOJIHHTeJIbHbIH
HCTOqHHK nOCTO.llHHOrO TOKa, Ha-
np.llJKeHHe KOToporo MOJKHO H3Me-
H.IITb. llpH,nYMaHTe H HapHcYHTe
HOBYIO D;enb, npH nOMom:H KOTOpOH
MomHO nOJIyqHTb nHJIo06paaHoe Ha-
np.llJKeHHe U, Ha06pameHHoe Ha
pHCYHKe 175. YKaJKHTe nOJI.IIpHOCTb BKJIlOqeHH.II aToro HCTOqHHKa H
ero Hanp.ll.meHHe.
PHc.175
'IacTL 4. YaKHH nyqoK aTOMOB MOJKHO nOJIyqHTb, HarpeBa.ll COBO-
KynHocTb aTOMOB B neqH ,no HeKoTopoH TeMnepaTypbI T H n03BOJI.II.II
aTOMaM BbIJIeTaTb ropHaOHTaJIbHO qepea MaJIeHbKOe (aToMHbIx paaMe-
pOB) OTBepCTHe ,nHaMeTpOM D Ha O,nHOH CTopOHe neqH. OD;eHHTe ,nHa-
MeTp nyqKa nOCJIe Toro, KaK OH npoH,neT B ropH30HTaJIbHOM HanpaBJIe-
HHH paCCTO.IIHHe L. Macca aTOMa paBHa M.
3a,na'la 2
1. Xopomo HaBeCTHO, qTO 60JIbmHHCTBo aBe3,n 06paaYIOT ,nBoHHbIe
CHCTeMbI. TaKa.ll CHCTeMa COCTOHT H3 06bI'tJHOH aBea,nbI C MaccoH mo H
pa.n;HYCOM R H 60JIee MaccHBHoH H nJIOTHOH HeHTpoHHOH aBe3,nbI C Mac-
COH M, BpalI.{alOlI.{HXC.II ,nPyr BOKpyr ,npyra. Ha6JIIO,neHH.II TaKoH ,nBOH-
HOH CHCTeMbI C 3eMJIH (,nBHJKeHHeM KOTOpOH MOJKHO npeHe6peqb) ,nalOT
CJIe,nywll.lYIO HHcpopMaD;JIIO. MaKCHMaJIbHOe yrJIOBOe CMelI.{eHHe 06bPl-
HOH aBea,nLI paBHO LlS, a MaKCHMaJIbHOe yrJIOBOe CMelI.{eHHe HeHTpoH-
HOH aBea,nbI paBHO Llcp (pHC. 176); BpeM.II, Tpe6yeMoe Ha TaKHe CMelI.{e-
HH.II, paBHO t; xapaKTepHcTHKH H3JIyqeHH.II 06bIqHOH 3Bea,nbI YKa3bIBa-
lOT, qTO TeMnepaTypa ee nOBepXHOCTH paBHa T, a aHeprH.II HaJIyqeHH.II,
na.n;arolI.{ero Ha e,nHHD;y nJIOlI.{a,nH nOBepXHOCTH 3eMJIH B e,nHHHD;y Bpe-
MeHH, paBHa P; JIHHH.II KaJIbD;H.II B aTOM H3JIyqeHHH CMelI.{eHa OT CBoeH
HOpMaJIbHOH ,nJIHHbI BOJIHbI "Ao Ha Be- I II I II
JIJItlHHY!:!J.. TOJIbKO H3-aa rpaBHTaD;H-  - -1- - - - - -r- - ,
OHHoro nOJI.II 06bI'tJHOH 3Bea,nbI (,nJI.II \ \ , I
\ \ I I
aToro paCqeTa acpcpeKTHBHa.ll Macca \ \ , I
cpOToHanpHHHMaeTc.IIpaBHoHhjcA). \ \ 6cp : "
BbIpaaHTe paCCTO.llHHe I OT 3eMJIH,n0 \  I
aToH CHCTeMbI TOJIbKO qepea aa.n;aH- \ / 06bL'lHaJl.
HbIe Ha6JIIO,naeMbIe BeJIJItlHHbI H yaH- {" \ ,- I 38e30a
\ \ , I
BepCaJIbHbIe KOHCTaHTbI. \ \ , I H eitmpoHHaJl.
2. I1pe,nnOJIOJKHM, qTO M» mo. \\,, · 38e30a

TaK qTO 06bPlHa.ll 3Bea,na BpalI.{aeTC.II T TTeJleC1COn
BOKpyr HeHTpoHHOH 3Bea,nbI no KPy-
rOBoH op6HTe pa,nHycoM r O. I1YCTb Pnc. 176
265

06LIqHn.ll 3Bea,na HaqHHaeT HaJIytlaTL raa no HanpaBJIeHHIO K HeHTpOH-
HOH aBea,ne co CKOpOCTblO V o OTHOCHTeJILHO 06LIqHOH aBea,nLl. CqHT8.11
rpaBHTaUHOHHoe Boa,neHCTBHe HeHTpOHHOH aBea,nLI Ha ,nBH)KeHHe HC-
neHHoro r8an aHaQHTeJILHO 60JILmHM, qeM BJIH.IIHHe 06L1QHOH aBea-
,nLl, II npeHe6pern.ll HaMeHeHH.IIMH Op6HTLI 06bI'lHOH aBea,nLI. H8H,nHTe
paCCTO.llHHe r max M8KCHM8JILHOrO C6JIH)KeHH.II aToro raaa H HeHTPOHHOH
aBea,nLI.
3a,naQa 3
fopHaOHTaJIbHn.ll np.llMoyrOJIbHa.ll nJIaCTHKOBa.ll Tpy6a mHpHHoH d
H BLICOTOH h (pHC. 177), aaMKHYT8.11 Hn ce6.11, a8nOJIHeH8 PTYTLIO, y,neJIL-
Hoe aJIeKTpHQeCKOe conpOTHBJIeHHe KOTOpOH p.
3mcopa'tuealO14uu
npoeoOHUK
Zy
x
tB
II .-
L
PHC. 177
I1a6LITOqHOe ,naBJIeHHe p B Tpy6e coa,naeTCH Typ6HHOH, KOTopnH
npOKaqHBaeT )KH,nKOCTL no Tpy6e C nOCTO.llHHOH CKOpOCTblO v o . ABe
npOTHBOnOJIOJKHLIe BepTHKaJILHLIe CTeHKII ytlaCTKn Tpy6LI ,nJIHHOH L
HarOTOBJIeHLI Ha Me,nH. 3TH nJIaCTHHKH CHapYJKH coe,nHHeHLI npOBO,n-
HHKOM. ABHJKeHHe peaJIbHOH JKH,nKOCTH .IIBJI.IIeTC.II CJ"IOJKHLIM. AJIH
ynpom:eHHH onHCaHH.II aTOro ,nBHJKeHHH npe,nnOJIOJKHM CJIe,nYlOU.lee: He-
CMOTp.ll Ha TO, qTO JKH,nKOCTb BHaKaH, ee CKOpOCTL nOCTOHHHa no BceMY
CeqeHHIO; CKOpOCTb JKH,nKOCTH Bcer,nn nponopD;HOHaJIbHa peaYJIbTHpy.
lOU.leH BHemHeH CHJIe, ,neHCTBYIOU.leH Ha Hee; JKH,nKOCTL .IIBJIHeTC.II He-
CJKHMaeMOH. O,nHopo,nHoe BepTHKnJIbHoe MarHHTHoe nOJIe C HH,nYKD;HeH
B ,neHcTByeT TOJIbKO Hn ytlnCTKe MeJK,ny Me,nHLIMH nJInCTHHaMH.
1. HnH,nme €HJIY, ,neflcTBYIOlI.{YW Ha JKH,nKOCTb co CTOpOHLI MnrHHT-
Horo nOJI.II (BLIpaaHTe ary CH.'IY qepea BeJIHqHHLI L, B, h, d, P H CKOpOCTb
,nBHJKeHH.II JKH,nKOCTH v npH HaJIHqHH MarHHTHoro nOJIH).
2. llO.iIyqHTe BbIpaJKeHHe ,nJIH CKOpOCTH ,nBHJKeHHH )KH,nKOCTH v
(BLIpaaHB ee qepea Be;mtIHHLI v o ' p, L, B H p).
266

3. BLIBe,nHTe BLIpaJKeHHe ,nJIH ,nonOJIHHTeJIbHOH MOU.lHOCTH Typ6H-
HLI, KOTopaH OJIJKHa 6LITL npHJIOJKeHa, qT06bI 06eCneqHTL YBeJIHqe-
HHe CKOpoCTH ,no HaqaJILHOrO 3HaqeHH.II v o .
4. Tenepb MarHHTHoe nOJIe BLIKJIlOqeHO, a pTyrb 3aMeHeHa BO,nOH,
TeKeH no TPy6e C nocToHHHOii CKOpoCTblO vO. MOHoxpOMaTHqeCKaH
aJIeKTpOMarHHTHa.ll BOJIHa qaCTOTOH f pacnpocTPaHHeTcH B,nOJIL Tpy6LI
no HanpaBJIeHHIO nOTOKa H npoxo,nHT yqaCTOK ,nJIHHOH L. I1oKa3aTeJIb
npeJIOMJIeHHH BOLI - n, a v o « c (r,ne c - CKOpOCTb CBeTa B BaKYYMe).
llOJIytlHTe BLIpameHHe ,nJIH ,nonOJIHHTeJIbHOH pa3HOCTH cpa3 BOJIHLI Ha
Bxo,ne H BLIxo,ne 113 paCCMaTpllBlIeMoro ytlaCTKa, KOTOpa.ll B03HHKaeT
BCJIe,nCTBHe ,nBHJKeHH.II BO,nLI.
PeweHMR
3a,na 1 Ia 1
tIacTb 1. 1. :3JIeKTpOHbI, YCKopeHHbIe aJIeKTpHqeCKHM nOJIeM
BO BXO,nHOH nOJIOCTII (p HC. 179), npH06peTalOT CKOpOCTb
VI = J v + 2:U :::: 2,0435 - 10 6 M/C,
aaMe,nJIeHHbIe - CKOpOCTb v 2 :::: I v - 2eU "" 1,9555 . 10 6 M/C.
V m
OTHOCHTeJIbHaH CKOpOCTb v OTR _ = VI - v 2 :::: 0,0880 . 10 6 Mjc.
CrycToK 06pa3yeTcH, Kor,na YCKopeHHLIe 11 3aMe,nJIeHHLIe B Teqe-
HHe o,nHoro nepHo,na H3MeHeHHH HarrpHmeHHH aJIeKTpOHLI OKa3LIBaIOT-
CSI B O,nHOH 06JIaCTH npOCTpaHCTBfl (pIlC. 178). LJT06LI crYCTOK 06pa-
T
V o .-
I . 2 I
: . ;r

3JleKmpOitbl. .WMeOJleHHble
60 exooltou nOJlocmu
T
v --
J 2
3JleKmpOHbl. YCKopeHHbie
eo exooHou nOJlocmll
06pa.JoeaHlLe
czycmKa
...
PHC. 178
267


30BLIBflJICH B BLIXO,nHOH nOJIOCTH, oua .D;OJI)KHa Haxo,nHTbC.II un PflC-
CTOHHHH b = v\'t OT BXO,nHOH, r,ne 't = V2() . I1cKoMoe paCCTOHHHe
V OTBo
b - 1 v\v 2 T - 2 270
- 2 ( ) -, CM .
v\ - v 2
2. 3aMe,nJIeUUbIe aJIeKTpOUbI ,nBIlJKYTC.II OT nOJIOCTH K nOJIOCTH B
v
TeqeHHe BpeMeuH ('t + %) = _ 2 IT:::: 11,61 T. LJT06LI aUeprH.II
vOl'll_
Ha crycTKa norJIOU.laJIaCb, aJIeKTpoHbI B BbIXO,nuoi'1 nOJIOCTH TnK)Ke ,nOJI-
JKULI aflMe,nJIHTbCH. :3To B03MOJKUO, eCJIH KOJIe6aaH.II uanp.llJKeUH.II B BbI-
XO,nUOH nOJIOCTH OTCTalOT OT KOJIe6aUHH BO BXO,nHOH nOJIOCTH ua pna-
HocTbcpaallcp = 0,61- 2n = 1,22n (HJIH,qTOTOJKeCaMOe,OnepeJKflIOT
HX Ha llq> 0,39. 2n = 0,78n) (pHC. 179, 180).
u u
r---1 r-- 
I I I I I
I I I I I
0 T/21 IT t 0 I I t
I I I I I I
L.....-.J L--..J L....-
H 3MeHeHue Hanp.R.:HCeHU.R. 60
eXOOHOU nOJlocmu. B me'leHUe
nep60U nOJl06UHbl nepuoeJa
(om 0 00 T/2) npoucxooum
3ailfeOJleHUe 3J1eKmpOH06, 6
me'leHUe emopou nOJl06UHbl
(om T /200 T) - YCKopeHue.
PHC. 179
H 3MeHeHue HanpJl:HCeHUJl6 6bl-
XOOHOU nOJ/.ocmu. £Jmo6bl 6
nOJ/.ocmu nozJ/.oU{aJ/.acb 3Hep-
ZU.R. U3 czycmKa, KOJ/.e6aHU.R.
Hanp.R.:HCeHU.R. OOJ/.:HCHbl 6blmb
c06uHymbl no l]Ja3e no omHo-
w.eHUlo K 6XOOHOU nOJ/.ocmu.
PHc.180
qacTI. 2. llJIOTHOCTb Bo,n.llUOro napa P2 = p;:: :::: 0,58 Kr/M 3 .
OTuoIlleuHe 06'beMOB, npHxo,nHU.lHXCH un O,nHY MOJIeKYJIY,
(  ) = (  ).
268

OTHOlIIeHHe Cpe,nHHX pnCCTOHHHH MeJK,ny D;eHTpaMIi MOJIeKYJI
(  ) = ro = rn :::: 12
'IaCTb 3. 1. I10cJIe aaMhIKaHH.II KJIlOqa, nOKa HanpHJKeHHe ua KOU-
,neucaTope U o < U", Kou,neHcnTop aapHJKaeTCH qepea peaHcTop. 3aBH-
CHMOCTh HanpHJKeUHH OT BpeMeHH npH aTOM U o = u. h - e--/i:). llPH
OCTH)KeHHH Hanp.ll)KeHHH np060H HCKpOBoro npOMeJKYTKa KOH,neHca-
TOp MruOBeuuo paapH)KaeTC.II, H Bce npOD;eCCLI nOBTOpHIOTC.II. fpacpHK
aaBHCHMOCTHUo(t) HMeeT TaKOH BH,n (pHC. 181):
U o
U,
U.
PHC. 181
t
2. 'ITo6LI uanp.ll)KeUHe ua Kou,neHcaTope MeH.IIJIOCh nOqTH JIHHeH-
L
HO, ,nOJIJKUO BLInOJIH.IITbCH YCJIOBHe: e ,., «1 HJIH T « RC. Tor,na
U" ::::u,L HUn «U , .
RC
3. I1epno,n nHJIo06pa3Horo uanp.ll)KeHH.II T :::: Rc Un .
U,
4. I1epHO,n aaBHCHT KaK OT R,
TaK H OT SG (U II)' aMnJIHTy,na -
TOJIhKO OT SG. lloaToMy,nJI.II HaMe-
HeUHH TOJIhKO nepHo,na ,nOCTflTOqUO
HaMeHHTb R.
5. )J;JI.II H3MeueHHH TOJIbKO flMn-
JIHTy,nLI UYJKUO npOliaBeCTH COOT-
BeTCTBYWlI.{J:le HaMeueHHH H R, H SG.
6. O,nHU Ha BOaMOJKHLIX BapH-
aHTOB TaKOH D;enH Ha06paJKeH Hn
pHCYHKe 182:
SG C J:
U.
PHC. 182
269

y
J_
D



PRC. 183
x
tIacTb 4. 1. Heonpe,neJIeHHOCTL
Koop,nHHaTLI qaCTHD;LI npH BLIJIeTe H3
OT BepCTH.II l1y = D (pHC. 183). 113 co-
OTHOmeHH.II Heonpe,neJIeHHOCTeH CJIe-
,nyeT, qTO pa36poe 3HaqeHHU npoeK-
D;HH HMnYJIbCa I1py pnBeH
h h
I1p ::::-::::-
1/ l1y D
CKOpOCTb TenJIOBOrO ,nBH)KeHHH
aTOMOB npH TeMnepnType T:
V :::
(kT =:v
VM r.
3a BpeM.II t, B TeqeHHe KOTOporo aTOMbI CMelI.{aIOTC.II B rOpH30HTaJIb-
HOM nanpaBJIeHHH Hn I1x = vt = L, IIllIpHHa llyqKa yseJIHql:lBaeTC.II Ha
I1PII " L 1IL
M) = I1v"t = M t :::: DM N :::: D MkT .
I1cKOMLIH ,nHaMeTp nyqKa DO = D + M).
3a,n;aQy MOJKHO pemHTL TaKJKe Ha OCHOBaHHH BOJIHOBLIX npe,nCTaB-
JIeHHH 0 ,nHcppaKD;HH aTOMOB Ha MHKpOCKOnH'leCKOM OTBepCTHH. YroJI
pnCXOJK,neHH.II nyqKa 8 :::: , r,ne A :::  - ,nJIHHa BOJIHLI one EpoHml.
D p
lloJIara.ll yrOJI e MaJILIM, orrpe,ne-
JI.IIeM paCXOJK,neHPe uytIKil Ha pac-
CTO.llHHH L OT OTBepCTH.II (pHC. 184)
M) :::: LO :::: LJ!, qTO e yqeTOM BLIpa-
Dp
JKeHHH ,nJIH HMnYJILCa aTOMa. a-
CTByJOU.lerO B TenJIOBOM ,nBHJKeHHH,
npHBo,nHT K nOJIyqeHHOMY BLIme pe-
3y JILTnTY.
npUMe'laHue. TaK KaK mHpHHY nyqKa HYJKHO TOJIbKO OD;eHHTL no
nopH,nKY BeJIHqHHLI, p.ll,n CPOPMYJI B pemeHHH 3anHCllH C TOqHOCTblO,nO
qHCJIOBOrO KoacpcpHD;HeHTa nopH,nKa 1.
.-::If[-------- ] M)
I ---_
, -----
,
,. -!
L
PRC. 184
3flJJ,aQa 2
1. PaccMoTpllM BpaI.D;eHHe ,nBofIHOH 3Be3,nLI BOKpyr 06lI.{ero D;eHTpa
Mace (pHC. 185).
PaccToHHHe MeJK,ny aJIeMeHTaMH ,nBOHHOH 3Be3,nLI (X M + X m) =
(l18 + l1cp)  _ '0
= 1- - . OTHomeHHe paccToHHHH OT aJIeMeHTOB ,nBOHHOH 3Be3,nLI
2
X m
,no ee D;eHTpa Mncc X
M
l11:J
l1cp .
270

x.y X..
M -1 1 mo
l.(eHmp Mace
cuemeMbl
PHc. 185
MaKCHM8JILHoe CMelI.{eHHe npOHCXOAHT aa nOJIOBHHY nepHo,na 06pa-
eHH.II ,nBOHHOH aBe3,nLI, T. e. yrJIOBWI CKOpOCTL BpaU.leHH.II CHCTeMLI
ro = .
't
CHJIa B38.HMo,neHcTBH.II MeJK,ny aBea,naMH
MTTIo 2
F = G 2 = Mro X N .
(X N + XIII)
_ w 2 ) 3 (j) = £1 3 (9 + )2
OrKy,na fflo - O(X M + XIII (e + M) rG 8 (1)
PaCCMOTpHM TenepL cMeeHHe JIHHHH K8JILqH.II B cneKTpe H3JIyqe-
HH.II 06L1'tJHOH aBea,nLI noon BJIH.IIHHeM ee rpasHTaD;HoHHoro nOJI.II:
"'0 ( J!.. ')
he _ G  = he (MLI yqJIH, qTO R « I).
1..0 R I..o+
TenepL MO)KHO BLlpa3HTb pa,D;uyc aBea,nLI
R = G1TIo ( ..!.. _ 1 ) -1 = G1TIo(1..0 + )
cIA.o 1..0 1..0 +  Cl.
C ,npyrOH: CTOpOHLI, npH TeMnepaType T aHeprH.II, HaJIyqaeMaH C
e,nHHHD;bI nJIOH nOBepxHocTH aBea,nLI B e,nHHHD;y BpeMeHH, paB-
Ha crT', a co Bcefl nOBepxHocTH - 41tlf . o'r. :3HeprHH HaJIy'tJe-
HHH na,nalOlI.{ero Ha e,nHHH:n;y nJIOU.lH nOBepxHocTH 3eMJIH B e,nH-
271

Puc.186
1 3e .M.JlR I
3ee30a -
UCmO'lHWC
U3JlY'leHUR
47tR 2 aT" ( R )
HHY BpeMeHH P = = - l 2 aT 4 (pHC. 186). Pa,nHYc aBea,nbI
4nl 2
R = l  P
aT 4 .
I1pHpaBH.IIeM IIOJIyqeHHbIe BbIpaJKeHH.II ,nnsI R:
Gmo(/...o + f1A) = l  P
e 2 f1A crT" .
C yqeTOM CPOPMYJIbI (1): (/...o + 6./...:::,o + I'1CP)2. l3 = l  cr" .
I1cKoMoe paCCTO.llHHe l = 2J2CT. f1A fE .
n(6.e + I'1cp)T (/...o + f1A)6.cp Va
2. Ao HaJIyqeHH.II qaCTHqKH raaa ,nBHJKYTC.II BMeCTe C 06LIqHOH aBea-
,nOH co CKOpOCTblO v = GM . PaccMoTpliM ,nBHJKeHHe qaCTHqKH raaa
ro
MaCCOH 11m. 3a CqeT HMnYJIbCa, IIOJIY-
qeHHOrO npH HaJIyqeHHH, OHa rrepe-
XO,nHT Ha aJIJIHnTHqeCKYIO op6liTY.
3anHlIIeM aaKOH coxpaHeHH.II aHep-
rHH ,nJI.II qaCTHqKH raaa IIPH ,nBHJKe-
HHH OT HaqaJIbHOrO rrOJIOJKeHH.II ,no
MOMeHTa MaKCHMaJIbHOrO c6JIliJKe-
HH.II C HeHTpoHHOH aBea,noH:
6.m( v + v 2 ) 6.mM _
-G--
2 ro
..'........
.
.
.
,
.
.
.
.
.
.
,
,
.
.
.
.
.
.
................--......#'#
PIIC. 187
272
6.mv.. _ G 6.mM
2 'inax

TaKJKe 6y,neT BbIIIOJIHHThC.II H 3aKOH COXpaaeHHH MOMeHTa HMnYJIbca:
t.mvr o = t.mVmnxr.nax.
I10,nCTaBJI.IIeM BbIpaJKeHH.II ,nJIH v H V mox B cl>opMYJIY aaKOHa coxpa-
HeHH.II aHeprHH:
v2 _ GM = GMro _ 2 GM OTKy,na r: = ro
o r. o 2 r.. max 1 + l1 o .Jr. o/ GM .
r max l1lax
3aMeTHM, qTO KBa.n;paTHoe ypaBHeHHe HMeeT BTOpOH KopeHh. llPH
V o < v aTOT KopeHb ,neHcTBHTeJIbHbIH H cooTBeTcTByeT MaKCHMaJIbHoMY
y,naJIeHHIO OT HeHTpoHHOH aBea,nLI rrpH ,nBHJKeHHH no 3aMKHYToH Op6H-
Te. llpH V o  v HaqaJIbHOH aHeprHH qaCTHqKH ,nocTaTOqHO, qT06LI no-
KHayTb nOJIe T.IIrOTeHH.II HeHTpoHHOH aBe3,nLI, o,nHaKo npeJK,ne npoHao:ii-
,neT HX MaKCHMaJIbHOe C6JIHJKeHHe.
3a;:J;a'la 3
1.I1pH ,nBHJKeHHH npoBo,n.lllI.{e:ii JKH,nKOCTH B MarHHTHOM nOJIe
MeJK,ny Me,nHbIMH nJIaCTHHKaMH BoaHHKaeT paaHOCTb nOTeHD;lIaJIOB
t.cp = Bvd. TaK KaK rmaCTHHKH coe,nHHeHLI npOBo,nHHKOM, TO B D;enH
TeqeT nOCTO.llHHbIH TOK. :3KBHBaJIeHTHa.ll cxeMa aJIeKTpHqeCKOH D;e-
nH Ha06paJKeHa Ha pHcYHKe 188. CH;Ja TOKa B D;enH I =  ' r,ne
iff = t.cp = Bvd , a R = p L d - - conpOTHBJIeHHe yqacTKa JKH,nKOCTH
h
MeJK,ny Me,nHbIMH nJIaCTHHKaMH. TaKHM 06paaoM, I :::: BvLh . CHJIa,
p
,neHcTBYIOU.la.ll co CTOpOHbI MarHHTHoro nOJI.II Ha npOBo,nHHK C TOKOM,
F = IBd = B 2 vLhd
P
MeOubte
n.nacmUHKU
BueuLHuu
np0800ltUK
PHC. 188
10 C%>H3HKQ 10.11 KJlQCCIol
273

2. lloKn MarHIITHoe nOJIe He OKa3LIBneT conpOTHBJIeHH.II TeqeHHIO
JKH,nKOCTII, CKOpOCTb rrOTOKn no,n,nepJKIIBaeTC.II Ha6LITOtlHLIM ,naBJIe-
HlleM: V o = kpdh. llpH BKJIlOqeHlII:I MnrHHTHoro nOJI.II CKOpOCTb no-
TOKa rrn,naeT ,no BeJIHqllHLI v = k( P - B:L )dh . TaKHM 06pnaOM,
v = v o ( 1 - B p L ) , H, OKOHqaTeJIbHO, v = V o
r 1 + B 2 v o Lfpp .
3. MOlI.{HOCTb, pa3BHBaeMa.ll Typ6HHOH,n0 nOHBJIeHHH MarHHTHoro
nOJI.II, No = pdhv o . LJT06bI 06ecne t IHTb YBeJIHqeHHe CKOpOCTH nOTOKa
,no HaqaJIbHOrO aHaqeHH.II, HYJKHO YBeJIHqHTb Ha6LITOqHOe aBJIeHHe,no
B 2 v. L
BeJIHqHHLI PI = P + . Tor,na pnaBHBaeMnH Typ6HHOH MOII\HOCTb
P
( B2v.2L )
6y,neT paBHa N = Pldhv o = pv o + T dh.
B 2 v. 2 Ldh
AOnOJIHHTeJIbHa.ll MOU.lHOCTb W = N - No = 0
p
3Ta MOU.lHOCTb npeBpalI.{neTC.II B aJIeKTpHqeCKYIO, noaTOMY ee MOJK-
2 2
 2 ( BVoLh ) d B 2 v. Lhd
HO HaHTH H TaKIIM 06paaoM: W :::: I R = --p P Lh =  .
4. B Heno,nBHJKHOH cpe,ne aJIeKTpOMarHHTHa.ll BOJIHa pacnpocTpa-
e
HHeTC.II co CKOpOCTblO VI = -. :3To H 6y,neT ee CKOpOCTb OTHOCHTeJIbHO
n
,nBHJKYU.lei'lcH BO,nbI. Tor,na OTHOCHTeJIbHO .JIa60paTopHoH. CHCTeMLI OT-
CqeTa BOJIHa B TeKYU.leH no TPy6e Bo,ne 6y,neT pacnpOCTpaH.IITbC.II co CKO-
pOCTblO v 2 =
eln + V o
1 voeln
+-
e 2
= c 1 + nVo/e
n + vole .
PaaHOCTb cpaa Hn Bxo,ne H BbIxo,ne Ha yqacTKn ,nJIliHOH L: llcp
2n b
A '
r,ne /...
v
f - ,nJInHa BOJIHLI.
274

TaKHM 06paaoM, ,nonOJIHHTeJIbHa.ll pa3HOCTb CP83, B03HHK81OlI.{aH
BCJIe,nCTBHe ,nBIIJKeHH.II BO,nLI
o = L\q>2 - L\q>J = 21tL f ( l - 1.. ) =
V2 VI
= 2nLf ( n + vole _ n ) = 2nLf (1 - n 2 )v o le
c 1 + nVo/e e 1 + nVole .
2nLfv 2
TaK KaK V o « c , TO 5 :::: (1 - n ) .
e
n pU.M.e'laHue. llpII pelIIeHHII 38J1:aqH MLI yqJIH, qTO qaCTOTa f II ,nJIIIHa
yqacTKa L 38J1:aHbI OTHOCIITeJIbHO .JIa60paTOpHoih cnCTeMLI OTCqeTa.
10.

XXXIII Me>KAYHapOAHaSi
cl»M3M'IeCKaSi OnMMnMaAa
2002 r., MHAoHe3MH
3a,.'J;aqa 1
llo,n3eMHLIH pa,nap IICnOJIb3yeTcH ,nJI.II 06HapYJKeHII.II 06'beKTOB noon
3eMHoH nOBepXHOCTblO II onpe,neJIeHII.II nx MeCTOnOJIOJKeHH.II nYTeM IIa-
JIY<IeHII.II 3JIeKTpOMarHIITHbIX BOJIH B aeMJIIO II perllCTpaLlll1l BOJIH, OT-
paJKeHHbIX OT 3TIIX 06'beKToB. AHTeHHa lI,neTeKTOp paCnOJIar81OTC.II He-
nocpe,ncTBeHHo Ha aeMJIe 11 npaKTlIqeCKH B O,nHOM H TOM JKe MeCTe.
JIIIHeHHo nOJI.IIplI30BaHHU.II nJIOCKaH aJIeKTpOM8rHIITHU.II BOJIHa C
KpyroBoH qaCTOTOH W pacnpOCTpaH.IIeTC.II B HanpaBJIeHlI1I Z 11 onlICbIBa-
eTCH CJIe,nYIOU.ll:lM BbIpaJKeHlIeM (1) ,nJI.II Hanp.llJKeHHOCTII 3JIeKTplIqeC-
Koro nOJIH:
E = Eoe- uz cos (wt -l3z) (1)
r,ne Eo - nOCTO.llHHaH BeJIlIqllHa, a - Koacpq>IILlHeHT aaTyxaHl:I.II 11
13 - BOJ"IHOBOe ql:lCJ"IO, ,nJI.II KOTOpbIX cnp8BeJIHBbI q>OPMYJIbI (2):
a = w[  [ ( 1 + ,:, )t - 1 ] r
=w[  [(l+ ,:, l +l)r
(2)
3,nCCb 11, E H cr - COOTBeTCTBeHHO fl6coJIIOTHa.ll MarHHTH8.11 npoHuD;a-
eMOCTb, a6coJllOTHa.ll,nH3JleKTplIqeCKaH npOHIILlaeMOCTb II 3JleKTpllqe-
CKa.ll npOBo,nIlMOCTb (11 = lr 110; E = Er Eo, 11,. - OTHOCHTeJIbHaH MflrHIIT-
HUH rrpOHIII.laeMOCTb, E,. - OTHOCIITeJlbHa.ll ,n1l3JIeKTplIqeCK8.11 npOHII-
LlaeMOCTb).
OTpUJKeHHbIH CHrHaJI He perHcTpHpyeTcH, eCJIH aMrrJIHTy,na pa,nap-
Horo CHrHaJIa, ,nOCTlIralOU.lero 06'beKTfl, YMeHbIlIaeTC.II B e paa (,=37%)
OT CBoero HaqnJ"ILHOrO aHnqeHUH. B ,nUHHOM pa,nnpe 06LIqHO ncrrOJlbay-
IOTCH 3JIeKTpOMarHUTHbIe BOJIHbI C pa3J"IlI t 1HbIMII qUCTOTaMH B ,nlIarra-
aOHe OT 10 ,no 1000 MfLl.
276

KaqecTBo pa60TLI pa,napa aaBIICliT OT ero paapellieHlI.II - MIIHIIMaJIb-
Horo pnCCTO.llHlI.II Mem,ny ,nBYM.II paaJIliqneMbIMH pn,napOM 06'beKTaMH,
KOTopoe COOTBeTcTByeT MHHHMaJIbHOH paaHOCTII cpa3 B 180 0 Mem,ny
,nBYM.II OTpaJKeHHLIMII BOJIHaMH, rrplIH.IITLIMII ,neTeKTopoM.
AaJIee npe,nnoJIaraiiTe, qTO 3eMJI.II .IIBJI.IIeTC.II HeMnrHIITHoH cpe,noH
(T.e. J.l = J.lo)' y,noBJIeTBOp.lllOU.leH YCJIOBIIIO ( E7n J « 1.
MCrrOJIbayHTe CJIe,nYlOU.llIe aHaqellli.ll KOHCTaHT: J.lo = 4n. 10- 7 fH/M II
Eo= 8,85 .10- 12 <I>/M.
1. BbIBe,nliTe BbIpaJKeHlIe ,nJI.II CKOpOCTII pacrrpOCTpaHeHH.II ClIrHa-
JIa V qepea J.l Ii E, IICnOJIbay.ll ypaBHeHlI.II (1) II (2).
2. Onpe,neJIHTe MaKCIIMaJIbHyro rny6IIHY 06HapymeHII.II OO'beKTa B aeM-
JIe, eCJIII aeMJI.IIIIMeeT aJIeKTpJ:ltlecKYIO llpOBO,nIIMOCTb f! = 10- 3 OM-I. M- 1
11 a6COJIIOTHYlO ,nlIaJIeKTpliqeCKYIO rrpOHHLlaeMOCTb E = 9£0.
3. PaccMoTpHTe ,nBa napaJIJIeJIbHbIX npo-
BO,n.lllI.{IIX CTepJKH.II, aaKorraHHbIX B aeMJIIO ro- [JOJl.O:HCeHUe iJemeKmopa
plIaOHTaJIbHO Ha rJIy6l1He 4 M. CqHTaHTe, qTO
pa,nap Haxo,nHTC.II Ha,n O,nHIIM IIa CTepJKHeH, a
,neTeKTop TOqeqHbIH.
Onpe,neJIIITe MHHHMnJIbHYIO qnCToTY, He-
06XO,nIlMYlO, qT06bI nOJIyqlITb 6oKoBoe (no OT-
HomeHlI1O K CTepJKH.IIM) paapemeHlIe B 50 CM.
4. AJI.II onpe,neJIeHII.II r JIy6l1HbI d pacno""Io-
JKeHH.II OT,neJIbHOrO CTepJKH.II, aaKorraHHoro B
TOH JKe aeMJIe, paccMoTplIM peaYJlbTaTbI IIaMe-
peHIIH, rrpoBe,neHHbIx npII nepeMelI.{eHIIH pa-
,napa B,nOJIb JIHHlm, neprreH,nIIKYJI.IIpHOH cTep-
JKHIO. PeaYJIbTaTbI KaqeCTBeHHO nOKaaaHLI Ha
plIcyaKe 189. BbIBe,nIITe BbIpnJKeHHe ,nJI.II Bpe-
MeHII t KaK CPYHKLlIIII x II orrpe,neJIIITe BeJIII-
qlIHY d.
o
x
t
PliC. 189
rpafjJuK .Jaeucwllocmu
epeMeHU t npOXO:HCiJeHUR
CUZHaJl.a iJo o(J7JeKma
U o(JpamHO iJo iJemeKmo-
pa om KoopiJuHambl x
iJemeKmopa;
t mlll = 100 HC
3a,/J,aqa 2
HeKoTopbIe MopCKHe JKIIBOTHbIe cnoc06HbI 06HapYJKIlBaTb ,npyrlle
cylI.{ecTBa Ha paCCTO.llHlIn6."Iaro,nap.ll 3J"IeKTpnqeCKIIM TOKnM, TeKYU.lHM
B TeJIe 3TIIX CYU.leCTB B npOLlecce ,nbIxaHH.II IIJIII B ,npyrlIx npoLleccax,
CBHaaHHbIX C COKpaILleHHeM MbImLl. HeKoTopbIe XIIU.lHIIKII IICrrOJIbay-
lOT aTOT aJIeKTplIqeCKIIH ClIrHaJI ,nJIH 06HapYJKeHIIH CBOIlX JKepTB, ,naJKe
Kor,na nOCJIe,nHlIe np.llqYTC.II B neCOK.
<l>lIalI'leCKllil: MexaHllaM, JIemaU.lHH B OCHOBe BoaHHKHOBeHlI.II TOKa
B JKepTBe H ee 06HapYJKeHH.II XIIlI.{HIIKOM, MOJKHO CMo,ne."IlIpOBaTb, KaK
277

nOKaaaHO Ha plIC. 190. TOK, reHepH-
PyeMbIH JKepTBOH, TeqeT Mem,ny ,nBY-
M.II pacnOJIOJKeHHbIMH B TeJIe JKepTBbI
ccpepaMII, IIMelOII.{HMII nOJIOJKIITeJIb-
HbIM H oTplID;aTeJIbHbIH rrOTeHD;lIaJILI.
PaCCTO.llHlle MeJK,ny LleHTpaMII 3TlIX
,nBYX ccpep II = Is, KaJK,na.ll ccpepa IIMe-
eT pa,nIlYc r 1 = r., aHaqllTeJIbHO MeHb-
lliHH, qeM I,. Y,neJIbHOe conpOTIIBJIe-
HlIe MOpCKOH BO,nbI p. llpe,nrroJIo-
JKIIM, 'lTO y,neJIbHOe conpOTHBJIeHlIe
TeJIa JKepTBbI TaKoe JKe, KaK II OKPy-
JKalOII.{eH MOpCKOH BO,nbI, TaK qTO
rpaHIID;bI MeJK,ny TeJIOM JKepTBbI II OK-
pYJKalOII.{eH cpe,noH HeT.
AJI.II onlICaHH.II Toro, KaK XIIU.lHIIK
nplIHIIMaeT aJIeKTplIqeCKHH CHrHaJI,
HCXO,n.lllI.{HH OT JKepTBbI, CMo,neJIHpyeM
B TeJIe XllU.lHIIKa ,neTeKTOp B BII,ne ,nBYX ccpep, Haxo,n.llII.{IIXC.II B KOHTaKTe C
oKpYJKalOU.leH MOpCKOH BO,nOH II pacnOJIOJKeHHLIX napaJIJIeJIbHO aHaJIO-
rllqHOH nape c<pep B TeJIe JKepTBbI. CcpepbI B TeJIe XIIII.{HIIKa HaxO,n.llTCH Ha
paCCTO.llHIIH 1 2 = Id ,npyr OT ,npyra II IIMelOT pa,n:HYC r 2 = r d , aHaqHTeJIbHO
MeHbllilIH, qeM 1 2 . lJ,eHTP ,neTeKTopa Haxo,nIITCH Ha paCCTO.llHIIH Y BbIllie
JKepTBbI, a JIUHII.II, coe,nIIHHIOI.IJ;a.a,nBe CcpepbI ,neTeKTopa, oplIeHTHpoBaHa
B,nOJIb J"IIIHIIH 3JIeKTpllqecKOro nOJI.II, coa,naBaeMoro JKepTBOH. PaCCTO.llHlI.II
I, II 1 2 MHoro MeHbme y. Hanp.llJKeHHOCTb 3JIeKTPIIqeCKOrO nOJI.II MeJK)J;y
ccpepaMR XHII.{HHKa MomHO CqliTaTb nOCTO.llHHOH rro MO,nYJIIO.
TaKIIM 06paaOM, XHII.{HIIK II OK-
pYJKalOII.{a.ll ero cpe,na 06paaYlOT
aaMKHYTbIH KOHTYP, BKJIlOqalOU.lIIH
aa,naHHoeBHYTpeHHeeconpOTHBJIe-
HlIe R 2 = Rd XIIII.{HIIKa II conpoTIIB-
U 2 JIeHlIe OKpYJKalOU.leH cpe)J;bI, KaK
nOKa3aHO Ha plICYHKe 191. 3,necb
U - Hanp.llJKeHlIe MeJK,ny ccpepaMII
,neTeKTopa,06YCJlOB.iIeHHOeaJIeKTpH-
qeCKIIM rrOJIeM JKepTBbI (B OTCYTCT-
BlIe R 2 ), U 2 = U d - Harrp.llJKeHHe
MeJK,ny ccpepaMII C yqeToM BHYTpeH-
Hero conpOTIIBJIeHlI.II XIIII.{HUKa.
1. Onpe,neJIIITe BeKTOp nJIOTHO-
CTII TOKa ] (TOK Ha e,nHHIILlY nJIO-
<BHS"
,
I
:y
I
,
.
PIIC. 190
MoiJeJlb. onUCbt6alOaR npueM
tUelcmpU'lecICOZO CUZHUJla.
npuxoiJRezo
om :HCepm6bt IC XUHUICY
U
Pllc.191
3U.'If1'HymbtU ICoHmyp.
6IC.11O'laHJUli 6HympeHHee
conpOmU6J1eHUe R 2 XUHUICU
U conpomUfueHue oICpY:HCaHJeu
cpeiJbt Rg
278

II.{a,nH), 06YCJIOBJIeHHbIH TOqeqHbIM IICTOqHHKOM TOKa l. Ha paCCTO.llHlI1I
rOT Hero B HeOrpaHII'leHHOH O,nHOpO,nHOH cpe,ne.
2. AJI.II aa,naHHoH CHJILI TOKa l., TeKYII.{erO MeJK,ny ccpepaMH JKepT-
BLI B HeOrpaHlIqeHHOH cpe,ne, onpe,neJIIITe, onHpaHCb Ha aaKOH E = PJ,
HanpHJKeHHOCTb aJIeKTpHqeCKOrO nOJI.II rrocepe,nHHe MeJK,ny ccpepaMII ,ne-
TeKTopa (TOqKa P Ha plIC. 190).
3. AJI.II TOH me CIIJIbI TOKa l. onpe,neJIHTe Hanp.llJKeHHe U. MeJK,ny
ccpepaMII B JKepTBe.
Orrpe,neJIIITe conpOTHBJIeHlIe R. MeJK,ny aTHMH ccpepaMH.
Onpe,neJIIITe TaKJKe MOU.lHOCTb p., BbI,neJI.IIIOU.lYlOc.II rrpH aTOM B OK-
pYJKalOU.leH cpe,ne.
4. Onpe,neJIHTe conpOTHBJIeHHe cpe,nLI R3 = Rm MeJK,ny Cc}>ePaMH XIIII.{-
HHKa.
Onpe,neJIIITe Hanp.llJKeHlle U d Mem,ny aTHMH ccpepaMII.
llOJIyqHTe BbIpaJKeHHe ,nJI.II MOII.{HOCTH P d' nepe,naBaeMoH OT JKepT-
BbI K XIIII.{HHKY.
5. Onpe,neJIHTe OnTHMaJIbHOe aHaqeHHe Rd' rrpH KOTOpOM ,neTeKTII-
pyeMa.ll MOII.{HOCTb P d MaKCI:IMaJIbHa.
Onpe,neJ"IHTe TaKJKe ,ny MaKCHMaJIbHyIO MOII.{HOCTb.
3a..aqa 3
Ha pllcYHKe 192 IIa06paJKeHa ynpOU.leHHa.ll Mo,neJIb MaccHBHoH Te-
JIeJKKH C O,nHIIM nepe,nHHM H O,nHIIM aa,nHIIM D;IIJIIIH,npOM B KaqeCTBe
KOJIe"C. TeJIeJKKa MOJKeT ,nBlIraTbC.II no HaKJIOHHOH ,nopore C yrJIOM Ha-
KJIOHa e. KaJK,nbIH D;IIJIIIH,np IIMeeT Maccy M H COCTOHT lIa HJIHH)I;pH-
qeCKOrO CJIOH C BHelIIHIIM pa,nIIYCOM Ro II BHyrpeHHIIM pa,nHYCOM 0,8Ro
11 HeCKOJIbKIIX cnHD;, CYMMupHa.ll Macca KOTOpbIX 0,2M. CXeMaTlIqeC-
Koe Ha06paJKeHIIe LlIIJIHH,npOB nOKaaaHO Hn pHCYHKe 193.
PIIC. 192
Ynpow,eHHaR HodeJlb tCaCCII6HOU
meJleJKKIl, d61l:Jf(yw,eucR no
HaKJlOHHOU dopoze
PHC. 193
Ynpow,eHHaR Mode b
qllJlUHdp06
279

Maccoii CTepJKHen, rrO,n,nepJKIIBalOU.lIlX nJI8TcpOpMY TeJIeJKKlI, MOJK-
HO npeHe6peqb. TeJIeJKKa ,nBIIJKeTC.II BHlla noon ,neHCTBHeM CIIJl TH>KeCTH
11 TpeHII.II. llepe,nH.IIH 11 aa,nH.IIH qaCTII TeJIeJKKH CHMMeTpllqHbI. CTaTII-
qeCKIIU 11 KHHeTHqeCKlIii KoacpcpllLlHeHTbI TpeHHH (KoacpcpllLllleHTLI Tpe-
HIIH rrOKOH II TpeHII.II CKOJIbJKeHII.II) MeJK,ny LlIlJIHH,npflMII H ,noporOH paB-
HbI 11. II lk COOTBeTCTBeHHO. OJJ:Hopo,nHa.ll nJIaTcpopMa TeJIeJKKII IIMeeT
MflCCY 5f1,l. ,nJIIIHY L II TO."IU.lIlHY d. PaCCTO.llHlIe MeJK,ny OC.IIMII nepe,nHe-
ro 11 aa,nHero Lll:lJIIIH,npOB - 21. a paCCTOHHlIe OT D;eHTpn KaJK,noro LlII-
JIIIH.1;pa,no HIIJKHeH qaCTII TeJ"IeJKKII - h. TpeHlIeM KaqeHlI.II1I TpeHlIeM
B OC.IIX LlIl."'IIIH,npOB rrpeHe6peqb.
1. BbIqllc."IIITe MOMeHT IIHepLll:I1l Ka>K,noro D;II.lI1H,npa.
2. 11306paal:lTe Bce CIIJIbI, n;eiicTBYIOU.llle Ha nJIaTcpopMY TeJIeJKKlI,
Ha ee nepe,nHllii U an,nHIIH LlIIJIHH,npbI. 3anlllIIHTe ypflBHeHIIH ,nBIIJKe-
HUH KaJK,nOH qaCTII TeJIeJKKH.
3. CqlITa.ll, qTO TeJleJKKa HaqIlHaeT ,nBlIraTbC.IIlIa COCTO.llHII.II nOKO.ll
rro,n ,neHcTBlIeM CIIJIbI T.IIJKeCTH, YCTaHoBIITe Bce BOaMO>KHbIe TlIrrbI ,nBH-
JKeHIIH CIICTeMLI II HaH,nIlTe ee YCKopeHlI.II BO Bcex 3TlIX CJIyqa.llx. BbI-
paallTe YCKopeHlI.II qepea Cpll3l1qeCKIIe napaMeTpbI, an,naHHbIe B YCJIO-
BUll aa,naqll.
4. HaqaB ,nBIIraTbCH Ha COCTOHHU.II nOKO.ll, TeJ"IeJKKfl npOlliJIa rrYTb
so' ,nBlIra.llCb 6ea npOCKaJIbabIBaHII.II, a aaTeM nonaJIa Ha yqacToK ,nopo-
rlI, Ha KOTOpOM CTaTlIqeCKIIH U KIIHeTl:lqeCKIIi'1 KoaCPCPuLllIeHTbI TpeHII.II
YMeHbIlIaIOTC.II,no BeJIlIqlIH 11: 11 11 , TflK qTO ,na.'IblIIe 06a D;IIJIIIH,npa Hfl-
qUH810T rrpocKa..lbabIB8Tb. BLIqnCJIIITe JIIIHei':lHYIO CKOpOCTb TeJIeJKKII
1:1 yr:IOBLIe CKOpOCTII BpaU.leHII.IILlII."IIIH,npOB nOCJIe Toro, KaK Te."IeJKKa
rrpOlliJIa rrOJIHbIH rrYTb s. CqIlTaHTe, tlTO BeJ"IHtIlIHbI So 11 s aHaqIITe."IbHO
60JIblIIe paaMepOB TeJIeJKKII.
PeweHMSI
3a..1a'Ia 1
1. B ,nflHHOM CJIyqae peqb lI,neT 0 cpa30Boii CKOpOCTII, T. e. 0 CKOPOC-
TII, C KOTopOi'1 rrepe:\teU.laeTCH B rrpocTpaHcTBe 4>aaa q> rr.locKoii BO.1HbI.
B MOMeHT BpeMeHII t B TOqKaX C Koop,nHHaToH Z cpaaa KOJIe6aHIIH
q>=!Ot-l3z.
nYCTL 3a BpeM.II dt cpaaa q> CMeCTII.laCb Ha dz, Tor,na
dlj) = wdt -Bd: = O.
dz It)
dt =  - II eCTL cpa30BaH CKOpOCTL.
I1crro;'IbayeM BbIpaJKCHlle (2) n;.l.11!3
280

v =  = [ ": [ (I + ,, f + 1 W
2
C yqeToM ( ..Q... ) «1:
(lIE
v = 1
.s;E '
r,ne IlII E - a6COJIIOTHa.ll MarHIITHaH 1I,nllaJIeKTplIqeCKS.II npoHIILlaeMo.
CTII.
MCrrOJIbay.II CB.IIab MeJK,ny U6COJIIOTHbIMII II OTHOCIITeJIbHbIMII npo-
HHD;aeMocTHMII, nOJIyqIIM:
1 1 1
v = - . - . llOCKOJ"IbKY - = C - CKOpOCTb CBeTa B Ba-
.JlloEo .JllrEr .JlloE o
KYYMe, TO
v =
C
.J1l.E. .
llo YCJIOBIIIO, aeMJI.II - HeMarHIITHa.ll cpe,na (Ilr = 1), CJIe,nOBaTeJIb-
c
HO, ,nJI.II paccMaTpllBaeMbIx B aa,naqe BO.1H: v =  .
v Er
2. LJT06bI OTpSJKeHHLIH ClIrHaJI 6bIJI 3flperHCTplIpOBaH, aMrrJIIITy,na
Eo
pa,napHoro ClIrHaJIa, ,nOCTnrruero 06'beKTa, ,nOJIJKHa 6bITb He MeHbrue - .
e
AMn.1IITy,na CnrHaJI8 MeHHeTC.II no aaKOHY E = Eoe -aZ.
E
LJT06bI Eoe - aZ   ,3.0J1JKHO C06JIIO,naTbC.II YCJIOBlIe
e
1
aZ"; L,T.e.Z max =-
u
I1cnoJIbayeM BbIpaJKeHlIe (2) ,nJIH a:
1
Zmax =
1
W [ ": [ [I + , r - I ] r
2
C yqeToM ( ..Q... ) «1:
WE
[(1+ ;7.  l-IJ=rI+
1 cr 2
2 E2(1)2
- 1]
1 cr 2
2 E20
281

Zmax
1
1
{  . ! } 
2 2 E2{if
=
(,>
C yqeToM qlICJIOBLIX ,naHHbIx (E = 9Eo; J.! = J.!o):
Zmax = 15,9 M.
3. LJ T06bI pnaJIHqHTb ,nBa CTepJKH.II, MIIHIIMaJIbHa.ll paaHOCTb cpaa OT-
paJKeHHLIX OT HIIX ClIrHaJIOB Y ,neTeKTopa ,nOJIJKHa 6bITb 180 0 . :3TO COOT-
A
"2.
BeTcTByeT paaHocTII xo,na t1
COrJIaCHO PHCYHKY 194:
6. = 2( ./ d 2 + r 2 - d).
TaKIIM 06paaoM, MHHIIMaJIbHOH qaCTOTe 6y,neT COOTBeTCTBOBaTb
MaKCHMaJIbHaH ,nJIIIHa BOJ"IHLI Amax = 0,125 M.
t. - v - c 800 1\KT"
min -  - In, = UB l.l
....x ,,91\.max
( Er
= ...£. = 9Eo = 9 )
Eo Eo
ART. (,neT.)
.,
I
I
I
o A
d [ ", /"
x
.
,
" ./ d 2 + r 2
,
d I ,
I ,
I "
I ,
I r '
. - - - - - - -.
CTepmeHb
CTepmeHb
CTepmeHb
PHc.194
PHc.195
4. llYCTbA - rrOJIOJKeHlIe ,neTeKTopa, onllCLIBaeMoe Koop,nIlHaToH
X (pIlC. 195).
Tor,nfl BpeM.II npOXOJK,neHII.II ClIrH aJIa ,no C TepJKH.II II 06paTHo:
_ 21 _ 2 .j d l + x 2
t - --
v v
MIIHIIMaJ"IbHOe BpeM.II t min COOTBeTcTByeT x = O. OTclO,na CJIe,nyeT, qTO
d = vt = 5M
2
282

3a.a:aqa 2
1. B O,nHOpO,nHOH cpe,ne TOK OT TOqeqHOrO HC-
TOqHIIK8 pacxo,nIITCH paBHoMepHo no BceM HanpaB-
JIeHI:IHM.
PaccMaTpllBaH ccpepy pa,nIlYCOM r, rrOJIyqlIM B1oI-
paJKeHHe ,nJI.II j B TOqKe, rrOJIOJKeHHe KOTOpOH OT-
HOCHTeJIbHO IICToqHIIKa TOKa onpe,neJI.IIeTC.II pa,nlI-
YC-BeKTopoM r :
j
I. r
-.-
4 nr 2 r
,.
,
I
I
I
,
\
,
"
"
,
,
\
I
I
I
,
---"
PHc.196
r
(r,ne - e,nIlHlIqHblH BeKTop, onpe,neJISIIOlI.{HH HarrpaBJIeHlIe ] ).
r
3,neCb . CqllTaeTCH rrOJIOJKIITeJI1oH1oIM, eCJIH TOK H,neT OT HCTOqHH-
Ka, oTplILlaTeJI1oH1oIM - K HCTOqHIIKY.
2. llo npllHLlHny cyrrepnoalILlIIH E p = E; + E;, r,ne E; II E; co-
OTBeTCTBeHHO Hanp.llJKeHHOCTH nOJIeH, coa,naBaeM1oIx B T. P nOJIOJKH-
TeJI1oHOH (Is > 0) II oTpllLlaTeJIbHOH (Is < 0) ccpepaMII JKepTBbI (plIC. 197).
MCnOJIbayeM aaKOH E = pj 11 BbIpa-
meHHe IIa n. 1:
E p
E p = pU; +}p) = ( :. -  ) =
=
pl.
- r )
. (r
4n[ +(HJ
C yqeToM y» is 11 (r+ - j-) = ii,
r,ne e - e,nIIHlIqH1oIH BeKTOp, aa,nalO-
lI.{I:IH HarrpaBJIeHHe E p .
PHc.197
- - - --.
is
+
E p = pI.I. e
4ny3
3. Onpe,ne.'IIIM Harrp.llJKeHHOCT1o nOJIH B npOlIaBOJI1oHOH TOqKeA MeJK-
,ny ccpepaMII >f<epTBbI
E - E+ E- _ C' -:_) - P I. ( r; _ i; )
A - A + A - P JA + JA - 4n (r;)3 (r )3
283

E, E A
.till . e+
E A A
BbI6epeM OCb x, rrapaJIJIeJIbHYIO OCH
JKepTBbI TaK, IT06bI KOOp,nIIHaTa LleHTpa
OTpHn;aTeJIbHOH CCpepbI X- = 0 (pIIC. 198).
Tor,na Hnrrp.llJKeHlIe MeJK,ny ccpepaMH B
JKepTBe
-.
I
I
,
I
.
o

x
l!t-r.
U. = - J EAxdx = cp+ - cp- ,
r.
PHC. 198
pl. ( 1 1 )
r,ne E Ax = -- 4 2 +"2 .
n (ls - X) X
TaKI:IM 06pa30M,
I . [ ]
U. =  J 1 + ...!.- dx =
4n r. (i. - xt x 2
_ pl. [[ 1 _ 1 ] + [ 1 + 1 ]] _
- 4n (i. - (L. - r.)) (L. - rs) (L. - r.) r. -
=  ( :. - is r. - i. r. + :. )
T // 1 U "" pl. . !
.K. r............. .' TO .
4n r.
U = pl.
· 2nrs.
ConpOTHBJIeHlIe MeJK,ny ccpepaMH onpe,neJIHeM 113 aaKOHa OMa:
R :::: Us = .....e.-
· Is 2nr. .
MOlI.{HOCTb, BbI,neJI.IIIOlI.{a.llCH B oKpYJKalOU.leH cpe,ne:
_ _ pI;
P. - U.I. - _ 2 .
nrs
4. ConpOTHBJIeHlIe cpe,nbI MeJK,ny cq>epaMH XHlI.{HIIKa no aHaJIOrIIII
C peaYJIbTaTOM n. 3:
R =-L
m 2nrd .
CqlITa.ll HanpHJKeHHOCTb nOJI.II Ha OTpeaKe MeJK,ny ccpepaMH XI:IlI.{-
HHKa rrocToHHHOH, HCnOJIbayeM BbIpnJKeHII.II,nJI.II E p IIa n.2 H YCJIOBlIe
r d « Id ,nJI.II HaXOJK,neHH.II U (Harrp.ll>f<eHIIH MeJK)J;y ccpepaMH B OTCYTcTBlIe
R d ) (pIlC. 199).
U = E p .ld - pI.l.l d
- 4 ny 3
284

p
. .-. .
I Ep
I
... '
ld
Pllc.199
Rm
PIIC. 200
PaCCMOTpHM npe,nJIOJKeHHYIO B YCJIOBHH Mo,neJIb aJIeKTPH'leCKOH IJ;errH
(pHC. 200), rro KoropoH rrepe,nalOTC.II CHrHaJIbi OT JKepTBLI K XHlI.{HHKY:
U Rd
U d = 0 R R
d + m
_ pI,lid . Rd
U d - 4nl Rd + 
21trd
MOlI.{HOCTb, nepe,naBaeMa.ll OT JKepTBLI K XHU.lHHKy:
P, =  = ( : r ( 11, p r
R d +-
2n: r d
5. 'tlT06LI orrpe,neJIHTb OnTHMaJIbHOe aHaqeHHe R/nT o aanHlIIeM BbI-
paJKeHHe ,nJI.II P dB TaKOM BH,ne:
1>.J =( R d Rm J oRd
t. BTO BLIpaJKeHHe ,nOCTHraeT MaKCHMYMa npH Rd = Rm.
ponT. _ P
TaKHM 06pa30M, . 'd - 2n: r d
MaKCHMaJIbHa.ll,neTeKTHPyeMa.ll MOlI.{HOCTb:
1>.J rnax = U 2 = ! ( pIsl.Ld ) 2 . 21tr d = ....E!!L(I Ild)2
4Rm 4 4nl P 321tl s 8
3ap;aqa 3
1.MoMeHT HHepIJ;nH IJ;HJIIIH,npa 10 OTHOCHTeJIbHO
ero OCH CKJIa,nbIBaeTC.II Ha MOMeHTOB HHepIJ;HH IJ;HJIHH-
,npllIeCKoro CJIOH Ie H MOMeHTOB IIHepIJ;HH cnnIJ; I,.:
Io=Ie + 81,.
Macca IJ;HJIHH,npHqeCKOrO CJIOH meH KaJK,nOH cnH-
PHc.201 IJ;bI m,. COOTBeTCTBeHHO me = 0,8M H m,. = 0,025M.
MOMeHT HHepIJ;HH IJ;HJIHH,npHqeCKOrO CJIO.ll:
110
J 2 - me - 0,8M' _ 20 lv!
Ie = r pdV, r,ne p - V - ( 2 _ 2 ) - 9n 1)2 1 - n.iIOTHOCTb
e It RQ R; le "Oe
11,
MaTepHaJIa LlHJIHH.n;pHqeCKOrO CJIO.ll, V e - 06'beM CJIO.ll, le - ,nJIHHa IJ;H-
JIIIH,npa.
285

dV = Ie . dS = Ie 0 2nr"dr - 06'beM 6eCKOHeqHO TOHKoro D;HJIIIH,npHqe-
CKoro CJIOH pa,nHYCOM r H TOJIlI.{HHOH dr.
CJIe,noBaTeJIbHO,
110 110
I = J r2 . 20 M .1 .2nrdr = 40 M J r3dr = 0,656M
e 9n 1)2 , e 9 1)2
 'e . 

MOMeHT HHepnll1l cnHnLI I, = J r 2 ' dr ,
o
r,ne I, = Rj = 0,8 Ro - ,nJIIIHa cnHnbI,
m, _ O,025M
t; - O,8Ro - Macca eon. ,nJIHHLI cnHnbI.
0 0 8 110
J 0 025M 2 2
TaKIIM 06P830M, I, = O,8Ro r dr = 0,005(3)MR.;
o
10 = lc + 81, :::: 0,7M.
2. 060aHaIHM Maccy TeJIeJKKH m (m = 5M), a Maccy KaJK,noro H3
nIlJIHH,npOB m" m, (m, = m,= M)-=-. Ha T!:JIeJKKY ,neHcTByeT CHJIa T.IIJKeCH
mg (pHC. 202), a TaKJKe CHJIbI Nl 11 t, - co CTOpOHLI rrepe,nHero H N,
11 l - co CTOpoHLI 38)1;HerO nHJIHH.npa (.frollh - nepe,nHHH, .rear. -
a8)l;HHH).
il
PHC. 202
BTOpOH 3aKOH HblOToHa ,nJIH TeJIeJKKII:
mci = mg + N , + 1, + N , + l
B rrpoeKnHII Ha OCb x (HunpaBJIeHlIe ,nBHJKeHHH):
ma = mg sill 0 - t, - t
B rrpoeKnHH Ha rreprreH,nHKYJ"I.IIPHYIO eH OCb y:
mgcosO = N l + N,
(1)
(2)
286

\ Y CJIOBHe OTCYTCTBH.II BpalI.{8TeJIbHOrO ,nBHJKeBHH aanHmeM OTBOCH-
TeJIbBO OCH, rrpoxo,nHU.leH qepea D;eHTp T.IIJKeCTH TeJIeJKKH:
(N, - N,)l = (I, + f ,) ,
(3)
d
re  = h + -.
2
Ha KaJK,nLIH lIaLlIlJIHB,npOB ,neHcTByeT CHJI8 T.IIJKeCTH ( m,e H m,e),
a TaKJKe
- N ,; -1, - Ha nepe,nHHH co CTOpOBbI TeJIeJKKH;
- N ,; -1. - Ha 3a,nBHH co CTOpOBbI TeJIeJKKH;
Q,; F, - Ha nepe,nHIIH co CTOpOHLI ,noporH;
Q,; F, - Ha 38,nHHH co CTOpOHbI ,nopurH (pHC. 203).
e!
PHC. 203
II 38KOH HblOToHa ,n."I.IILlIlJIHH,npOB:
_ I!epe,nl!ero: _
mIa, m,i + (-Nt> + (-t,) + Ql + F,
a8)l;Hero:
m,a, = m,e + (- N ,) + (-1.) + Q, + F,
B rrpoeKLlll1l Ha OCb x:
m,a, = m,g sin 0 + I, - F, m,.a, = m,g sin 8 + I, - F, (4)
B npOeK1I11 Ha OCb y:
m,gcos8=Q,-N, m,gcos8=Q,-N, (5)
3aMeTHM, ITO YCKopeHlI.II rrocTynaTeJIbHOrO ,nBIIJKeHHH Bcex TpeX
IacTeH CHCTeMbI o,nIlHaKOBbI: a = a, = a,.
Bpfl..U.leHHe LlHJIHHpoB BOKpyr CBoeH OCH onHCbIBaeTCH ypaBHeHIIHMII:
,nJI.II rrepe,nHero ,nJIH aa,nHero
IoE,=F,Ro IoE,=F,Ro (6)
3. CJ/.Y'lau 1. l.fllcmoe lca'leHlle.
HII TOT BH ,npyroH LlIlJIHH,np He npOCKaJIbabIBfilOT, rroaTOMY F, 11
F, - CHJIbI TpeHIIH nOKOH.
287

I
B aTOM CJIrae MeJK,ny yr JIOBblMH 11 JIHHeHHbIMH YCKopeHlI.IIMH C f -
lI.{eCTByeT C.iIe,nYIOlI.{a.ll CB.II3b:
a, a r a
El =  11 Er = Ro ,TaKHM 06pa30M E, = Er - Ro .
I.a
113 ypaBHeHlI.II (6): F,. = F, =  = 0,7 Ma.

ll3 ypaBHeHH.II (4): I, = i = M(I,7a - g sin 8).
llo,ncTaBJI.IIeM aTOT pe3YJIbTaT B ypaBHeHHe (1).
5Ma = 5Mgsin8 - 2M(1,7a - gsin8)
a = O,833g sin 0
BbIHCHHM, rrpll KaKHX YCJIOBH.IIX peaJIH3yeTc.II aTOT CJIyqaH.
TaK KaK F, H Fr - CHJIbI TPeHH.II nOKO.ll, TO F,  J..l.Q, 11 F,.  J..lsQr.
HarpY3Ka Ha nepe,nHee KOJIeCO 60JIblIIe, nOaTOMY Q, > Qr (TOT JKe
pe3YJIbTaT MOJKHO nOJIIlITb, paCCMOTpeB ypaBHeHH.II (3) H (5». 3Ha-
lHT, ,nOCTUTOIHO, qT06LI BbIIIOJIH.IIJIOCb YCJIOBHe F,.  J..lsQr (MLI yqJII:I
TaKJKe, qTO npH IHCTOM KaleHHH F,= Fr).
ll3 ypaBHeHHH (2) H (3):
N r = ( 5 Mg cos 8 - (I, + Ir) . '; ) = Mg(  cos 8 - 1 5 2 sin 8 i )-
ll3 (5): Q,. = g ( 7 cos 8 -  sin 8 i )
F,. = 0,7 Ma = .!..... Mg sin 8.
12
CJIe,noBaTeJIbHO, IHCToe KaleHHe B03MOJKHO npH
.!..... Mg sin 8  J..l Mg ( 7 cos 0 -  sin 8!1. )
12 s 2 6 I
 7sin8
HJIII J..l. :? .
42 cos 0 - 5 sin 8 i
CJlY'laiJ. 2. C08JKecmHoe npOCKaJlb.Jbt8QHue.
06a LlHJ"IHH,npa rrpOCKaJIb3bIBaIOT, noaTOMY F, H F,. - CHJILI TpeHH.II
CKOJIbJKeHH.II. B aTOM CJIYlae ,nJI.II KaJK,noro lIa I:(HJIHH,npOB MOJKHO npH-
MCHHTb 3aKOH cyxoro TpeHH.II: F, = J..lI1QI H F,. = JlkQr. Pe3YJIbTaT pe-
meHIIH COCTaBJIeHHOH B rrYHKTe 2 CIICTeMbI ypaBHeHIIH HHqeM He 6y,neT
OT JIHqaTbC.II OT HaBeCTHoro CJIlaH COCKaJIb3LIBaHH.II,nOCKII no HaKJIOH-
HOH nOBepXHOCTH:
a = g(sin 8 - Jlk COS 8)!
288

'\ :3TOT CJIIaH B03MOJKeH, eCJIH CHJIbI F, H Fr' He06xo,nIlMbIe ,nJI.II CO-
OdII.{eHH.II yrJIOBbIX YCKopeHHH E, = lOr = .!!... (ypaBHeHlIe (6» LlHJIHH-
I Ro
\
,npaM, 60JIblIIe MaKCHMaJIbHOrO aHaIeHH.II CHJI TpeHlI.II rrOKO.ll MeJK,ny H-
JIHH,npaMH H ,noporoH - flsQs H flsQr COOTBeTCTBeHHO.
IIepBLIM HaqHHaeT npOCKaJIb3LIBaTb aa,n;HIIH LlIlJIIIH,np (CM. paccYJK-
,neHH.II B n. 3 CJIyqaH 1).
IIoaToMY YCJIOBHe nOJIHOrO npOCKaJIb3LIBaHH.II,nocTaToIHo paccMoT-
loa Q
peTb Ha rrpHMepe nepe,nHero LlHJIIIH,npa:  > fl. " r,ne a - YCKope-
HHe CIICTeMLI npll YCJIOBHH, Kor,na 3a,nHHH LlIlJIHH,np YJKe npOCKaJIb3LI-
BaeT.
HaH,neM aTO YCKopeHHe.
CJlY'lail 3. If aCmU'lHOe npOCKaJlb3blBaHue.
3a,nHHH HJIHH,np npOCKaJIh3LIBaeT (F,. = flkQr - CHJIa TpeHH.II
CKOJIbJKeHII.II), a nepe,nHHH - HeT (F, = Io = 0,7 M a - CHJIa TpeHH.II
RO
nOKO.ll). AJI.II ,nBHJKeHH.II BceH CHCTeMLI B,nOJIb OCH X CJIOJKHM ypaBHe-
HII.II (1) H (4) ,nJI.II 060HX LlHJIHH,npOB.
7 Ma = 7 Mgsin8 - F, - F,. = 7 Mg sin 8 - 0,7 Ma - flhQ,
- 5 f,+f. h
l1a ypaBHeHHH (2) 11 (3): N, = "2Mg cos 8 -  . -to
M3 ypaBHeHH.II (1): t, + t = 5 Mg sin 8 - 5 Ma.
M3 ypaBHeHlI.II (5):
Qr = Mg cos 8 + N, = Mg cos8 - Mg sin 8 ; + Ma ; .
CJIe,nOBaTeJIbHO,
7,7 Ma = 7 Mgsin8 - lk(MgCOS8 - iMgSins }; + Ma }; )
7 . 8 7 8 5 . O 
a = g sm - '2llk cos + "2llk S1I1 [
5 
7,7 + '2 Ilk [
:3TOT CJIyqaH B03MOJKeH, Kor,na
Io " fl.Q, (CM. n. 3 CJIIaH 2)
RO
7 sin 8
fl. > (CM. n. 3 CJIraH 1)
42 cos 8 - 5 sin 8 i
289

PHC. 204
4. llpH BHJKeHII 6e3 npOCKaJIb3bI i -
HII.II:  = O,833g Slll e .
CKOpOCTb, KOTOpylO rrpH06peJIa TeJIeJK-
Ka, ,nBHraHCb 113 COCTOHH IIH nOKO.ll (pHC . 204):
VI = .J2so = .J l,666s o g sin 8
YrJIOBaH CKOpOCTb Bpam;eHHH IJ;HJIIIH,npOB
K aToMY MOMeHTY BpeMeHH:
\.)
 =  .
TIpH ,nBHJKeHlIll rro BTOPOMY IaCTKY: a 2 = g(sin 8 - Jl cos 0)
V 2 _ v 2
2 2 1 = (s - so) ,CJIe,nOBaTeJIbHO, V 2 =  V + 2a 2 (s - so) .
a 2
BpeM.II,nBHJKeHII.II Ha BTOpOM yqacTKe:
tJ.t = \.)2 - \.)1 = J1) + 2a 2 (s - So) - \.)1
a 2 a 2
YrJIOBaH CKOpOCTb IJ;HJIHH,npOB B KOHLle 2-ro yqaCTKa
{ W r = WI + E/.;.t
W, = W 2 + E,tJ.t
YrJIOBbIe YCKopeHlI.II LlIlJIHH,npOB E, H Er orrpe,neJI.IIIOTCH ypaBHe-
HH.IIMH (6)
j E = F,.Ro = IlkQrRo
r 10 10
E = F,Ro = IlkQ,Ro
, 10 10
BbIpaJKeHIIH ,nJIH CIIJI Qr II Q, MOJKHO rroJ"IIlITb, HCnOJIb3Y.II ypaB-
HeHHH H3 n.2 ,nJI.II CJIyqaH 2, onHcaHHoro B n. 3.

XXXIV Me>KAYHapOAHaSi
M3M...eCKaSi OnMMnMaAa
2003 r. , TOMBOHb
3ap;aQa 1
TBep,nbIH LlHJIHH,npHIecKIIH CTepJKeHb pa,nIlYCOM R 3aKpenJIeH ro-
pH30HTa.iIbHO Ha,n 3eMJIeH. fpyallK MaccoH m npHKpenHJIH HeBecoMoH
HepaCT.IIJKHMOH HIITblO ,nJIHHLI L(L > 2nR) K BepxHeH TOqKe CTepJKH.II
A, KaK nOKaaaHO Ha pHcYHKe 205. fpY3HK rro,nHHJIH,nO ypOBH.II TOqKH
A TaK, ITO HHTb paCnO.iIOJKHJIaCb ropH30HTaJIbHO, H OTrrYCTIIJIII. fpy-
allK, KOTOpbIH MOJKHO CqHTfiTb MaTepllaJIbHOH TOIKOH, ,nBHJKeTC.II TOJIb-
KO B BepTIIKaJIbHOH nJIOCKOCTII, neprreH,nllKYJIHPHOH OCH CTepJKH.II.
YCKopeHlIe cB060,nHoro rra,neHHH - g. llpHMeM TOIKy 0 3a HaqaJIO OT-
CIeTa CHCTeMbI Koop,nHHaT.
m
g
p.
PHC. 205
llYCTb rpyaHK Haxo,nHTC.II B TOqKe P, a HHTb HanpaBJIeHa rro Kaca-
TeJIbHOH K nOBepXHOCTH CTCpJKH.II H KaCaeTCH ee B TOqKe Q. AJIHHY OT-
pe3Ka QP 0603IIaIHM s. KaCaTeJIbHbIH (TaHreHLlHaJIbHbIH) H HOpMaJIb-
HLIH (pa,nHaJIbHbIH) e,nHHHqIIble BeKTopbI 0603HaqHM €, H €r COOTBeT-
CTBeHIIO. YroJI OTKJ"IOHeHHH 8 OTpe3Ka OQ OT BepTHKaJIbHOH OCH x,
291

(COBna,nalOU.leH no HarrpaBJIeHlllO C OA) 6y,neM 0 rCqUTbIBaTb npOTHB qa-
COBOH CTpeJIKI1. npll e = 0 HaqaJIbHa.ll ,n."IIIHfi S = L, rpElBHTfiLlHOHHYIO
nOTeHLlH8."ILHYIO 3HepniIO B aTOM nOJ"IOJKeUHn rrpnMIITe pfiBHOH HYJIIO.
0603Ha t ll:lM CKOpOCTH 1I3MeHeHH.II Be."IH<IHH ellS rrpu ,nBIIJKeHIIH rpY3H-
Ka d8/dt II ds dt COOTBeTCTBeHHO.
ECJIU He orOBopeHO HHoe, TO Bce BeKTopbI CKopOCTeH H HX BeJ"IIIIII-
HbI paCCMaTpHBElIOTCH B CIICTeMe OTCqeTa, CBH3aHHOH C Heno,nBHJKHOi'l
TOqKOH O.
LJacTb 1. B aToH qaCTII 38,nfi1I1 paccMfiTpllBaeTcH ,nBHJKeHHe rpyallKEl
rrpll rrOCTOHHHO H8TRUYTOH HHTII. BbIpElallTe <Iepea BBe,neHHLIe pflHee Be-
JIlIqllHLI (T. e. s, 0, ds/dt, d8/dt, R, L, g, i, II e,) cJIe,nYIOlI.{ee:
L CBH3b MeJKAY Be."IHqIlHaMI:I dO/dt H ds/dt.
2. CKOpOCTb v Q )J:BIIJKeHIIH TOlKH KaCaHH.II Q OTHOCHTeJIbHO TOIKII o.
3. CKOpOCTb rpY3l1Ka V OTHOCIITeJIbHO ,nBIIJKYU.leHcH TOqKH Q B MO-
MeHT, Kor,na OH H8XO,nHTCH B TOqKe P.
4. CKOpOCTb rpY3HKa U OTHOCIITeJIbHO TOqKH 0 B MOMeHT, Kor,nfi
OH Haxo,nIlTCH B TOqKe P.
5. llpoeKLlll1O YCKopeHH.II rpyaHKa Ha OCJ;., 3a,naBaeMylO BeKTopoM
e, , OTHOCIITeJIbHO TOGKII 0 B MOMeHT, Kor,na OH Haxo,nIlTC.II B TOIKe P.
6. lloTeHLlllaJILHYIO aHeprHlO rpY3l1Ka U B MOMeHT, Kor,na OH Haxo-
,nHTCR B TOtlKe P.
7. CKOpOCTb rpyaHKu v rn B HIIJKHeH TOtlKe ero TpaeKTopHH.
qacTb 2. OTHorueHlIe L K R HMeeT c."Ie,nYIOU.lee 3HfiqeHue:
L 9n 2 n 
- = - + - ct g - = 3 534 + 3 3<>2 = 6 886.
R 8 3 16' , ,
1. LJeMY paBHa CKOpOCTb rpY31:1Ka v.. B MOMeHT, Kor,na Y<IaCTOK HIITII
QP eU.le OCTaeTCH OTpe3KOM np.llMOIl H HMeeT MIIHIIMfiJIbHYlO ,nJIHHY?
(BbIpa3HTe OTBeT tlepea g 11 R.)
2. LJeMY paBHa CKOpOCTb rpY3l1Ka VI/' Kor,na OH Haxo,nHTCH B BbIC-
lIIel'i TO<IKe CBoell TpaeKTopHII C ,npyroH CTOpOHbI CTepJKH.II? (BbIpa3HTe
OTBeT tlepe3 g H R.)
L
m
PIIC. 206
292
,\[
'tIacTb a. Ha O,nHOM KOHLle HIITII
rrO-rrpeJKHeMY Hfixo,nnTCH rpY3HK
MaCCbI m. fi K ,npyrOMY KOHLlY HIITH,
ncpe6pOlIIeHHOJI <Iepe3 CTepJKeHb,
rrpHKpen."IeH 60Jlee THJKeJIbIH rPY3
r.mCCOll ,'\-1 , "aK rrOKfia8HO Ha PHCYH-
Ke 206. rpY3 M TflK)Ke MOJKHO CqIl-
TlITb MfiTepna."IbllOl'i TOqKOH.

llepBOHaIaJIbHO rpY3HK m no,nHHJIII ,no ypOBHH TOqKH A TfiK, tlTO .
yqaCTOK HHTII ,nJIHHOH L paCnOJIOJKHJICH ropH30HTaJIbHo. fpY3HK m
oTnycKalOT 6e3 H8qaJIbHOH CKOpOCTlI, H CHCTeMfi HaqHHfieT ,nBHraTbCH
B BepTHKaJ"IbHOH nJIOCKOCTH, rrpH aTOM rpyabI He CTa."IKHB8IOTCH. Cqll-
TaHTe, qTO npH CKOJIbJKeHHH HHTII no nOBepXHOCTH CTepJKHH CHJIa Tpe-
HH.II npeHe6peJKHMO MaJIfi, O,l1HaKO nOCJIe OCTfiHOBKII 6JIaro,nap.ll CIIJIe
TpeBlI.II nOKO.ll rpY3 ,naJIee He ,nBHJKeTCH.
By,neM CqllTaTb, ITO rpya M ,neHCTBHTeJIbHO onYCKaeTC.II Ha paCCTO-
.IIHHe D (npHqeM (L - D)>> R) H OCTaHaBJIHBaeTC.II. LJT06LI rpY3HK m
c,neJIaJI rrOJIHLIH 060POT BOKpyr CTepJKH.II (,no 3HaqeHH.II 8 = 2n) H npll
aTOM 06a yqucTKa HIITlI, He KacalOU.llleC.II CTepJKH.II, OCTaBaJIIICb HaT.II-
HYTLIMH, He06xo,nHMO, qT06LI OTHOlIIeHlIe Ct = D / L npeBLICHJIO HeKOTO-
poe KpHTHqeCKOe 3HaqeBHe Ct c . llpeHe6pera.ll MaJIbIMH BeJIHqUHaMH no-
p.ll,nK8 R/ L (H MeHbme), OLleHHTe BeJIlIqHHY Ct c Iepe3 OTHomeHHe M / m.
3a,l1aqa 2
PfiCCMOTpHM O,nHOpO,nHbIH CTep-
JKeHb ,nJIHHOH 1 (B HeHanp.llJKeHHOM
COCTOHHHH) H nJIOlI.{a,nblO CeqeHHH A
(pHC. 207). Ero ,nJIIIHa HaMeH.IIeTC.II
Ha tJ.l, Kor,na CHJILI o,nHHaKoBoH Be-
JIHqHHLI F 11 npOTHBOnOJIOJKHOrO
HanpaBJIeHHH rrpHJIOJKeHLI K ero
KOHLlaM nepneH,nHKYJI.IIpHO IIX
nOBepXBOCTHM. MeXaHHqeCKOe Hfirrp.llJKeHHe T Hfi KOHLlax paBHO F / A.
OTHOCHTeJIbHOe H3MeHeHHe ero ,nJIHHbI, T. e. tJ.l/l, Ha3bIBaeTC.II,necpop-
M8n;HeH S CTepJKHH. B TepMHHax HanpHJKeHHH 11 ,necpopM8LlHH 3aKOH
fYKa MOJKeT 6bITb BbIpaJKeH KaK

x
z
PRC. 207
F f..l
T = y. S HJIII A = Y T '
(1)
r,ne Y HaaLIBaeTCH Mo,nYJIeM IOHra CTepJKHH. 3aMeTIIM, ITO HanpHJKe-
HHe CJKaTH.II T cooTBeTcTByeT F < 0 H YMeHblIIeH1I1O ,nJIHHbI (T. e.
tJ.l < 0). TaKoe Hanp.llJKeHlIe HBJIHeTCH OTpHn;aTeJIbHLIM no BeJIHIlIHe 11
CB.II3aHO C ,naB."IeHlIeM p: T = -po
AJIH CTepJKHH O,nHOpO,nHOH nJIOTHOCTH p CKOpOCTb pficnpocTpfiHe-
HHH npo,no."IbHbIX BO."IH (T. e. CKOpOCTb 3BYKa) B,nOJIb CTepJKH.II eCTb
u = .jy /p .
(2)
qacTb 1. OHOpO,nHbIH rrOJIy6ecKOHeqHbIH CTepJKeHb nJIOTHOCTII p
pacnOJIOJKeH MeJK,ny x = 0 11 X = 00 (pHC. 208). CHaqaJIa OH Haxo,nIlTCH
B nOKoe H HeHarrpHJKeH. 3aTeM npH rrOMOU.l1l rrOplIIH.II K KOHLlY CTepJK.
293

H.II (x = 0) npHJIaraeTCH MaJIOe ,naB-
JIeHHe p B TeqeHHe KopOTKoro npo-
MeJKyrKa BpeMeHH t:.t; npli aTOM re-
HepHpyeTC.II BOJIHa ,naBJIeHH.II, pac-
npOCTpaH.IIIOlI.{a.llC.II CO CKOpOCTblO U
BnpaBo.
1. ECJIH noplIIeHb BLI3bIBaeT ,nBH-
JKeHHe JIeBOrO KOHn;a CTepJKHH C nOCTO.llHHOH CKOpOCTblO v (pIlC. 208),
KaKOBa MeXaHHqeCKaH ,necpopMan;H.II S H ,naBJIeHHe p Ha JIeBOM KOHLle B
TeqeHHe rrpOMeJKyrKa BpeMeHH t:.t? OTBeT ,nOJIJKeH 6bITb BLIpaJKeH TOJIb-
KO qepe3 BeJIHqllHbI p, U, v.
2. PaccMoTpHTe rrpo,nOJIbHylO BOJIHY B CTepJKHe, pacnpOCTpaH.IIIO-
lI.{YIOC.II B,nOJIb HanpaBJ"IeHH.II x. ,n:JIH HeKOToporo CeqeHH.II, Koop,nHHaTa
KOToporo B HeHanp.llJKeHHOM CTepJKHe paBHa x, BBe,neM ero CMelI.{eHHe
1;(x, t) aa BpeM.II t (pliC. 209) 11 npe,nnOJIOJKHM, ITO
p -fLr.- p
I CJKaTHe HeAecpoPMH;> .
x=O 00
PHC. 208
1;(x, t) = 1;0 sin h(x - ut),
(3)
r,ne /;oH h - nOCTO.llHHLIe. HaH,nHTe COOTBeTCTBYIOlI.{YIO CKOpOCTb vex, t)
aToro CeqeHH.II, ,necpopMaLlHIO S(x, t) H ,naBJIeHHe p(x, t) KaK cpyHKn;HH
XII t.
x
.1
.i "
Be3 AecpopMaIJ;HH
ABH.iKeHHe BOJIH!'I
PHc.209
LJacTb 2. PaccMoTpHM nJIaCTliHY KpHCTaJIJIa KBapn;a ,nJIHHOH b, TOJI-
lI.{IIHOH h 11 mnpliHoH W (pHC. 210). Ee ,nJIHHa H TOJIlI.{HHa onpe,neJIeHbI
B,nOJIb ocell x H Z COOTBeTCTBeHHO. TOHKHe CJIOIl MeTaJIJIHIeCKoro no-
KPLITHH Ha BepXHeH H HHJKHeH rrOBepXHOCT.IIX CJIYJKaT aJIeKTpo,naMII.
Vet)
x
PHC. 210
294

I bj2 I bj2 I
-------------
. . ,
I I
I
: KBapq
I
:[
ZL
x
PHc.211
.
:3JIeKTpHleCKHe KOHTaKTbI, CJIYJKalI.{He TaKJKe OnOpaMH (pHC. 211),
rrpHrraHHbI K I.leHTpaM 3JIeKTpo,nOB. lJ,eHTpbI ::meKTpo,noB MorYT CqH-
TaTbCH 3aKperrJIeHHbIMH rrpH rrpo,nOJIbHbIX KOJIe6aHHHX B,nOJIb OCH X.
PaCCMaTpHBaeMbIH KpHCTaJIJI KBapI.la HMeeT rrJIOTHOCTb p = 2,65 x
x 10 3 Kr/M 3 H MO,nYJIb IOHra Y = 7,87 X 10 10 H/M 2 . ,n:JIHHa nJIaCTHHbI
b = 1,00 CM, a lIIHpHHa W H TOJIlI.{HHa h TaKOBbI, qTO h « W H W « b.
Kor,na KJIlOq K pa30MKHyT, MbI rrpe,nrrOJIaraeM, qTO B KBapI.leBOH rrJIac-
THHe B036YJK,naroTcH TOJIbKO rrpo,nOJIbHbIe KOJIe6aHHH B BH,ne CTOHqHX
BOJIH B HarrpaBJIeHHII x.
,n:JIH CTOHqHX BOJIH qaCTOTOH f = w/21t CMelI.{eHlIe Sex, t) CeqeHIIH
rrJIaCTHHbI C paBHOBeCHbIM rrOJIOJKeHHeM x B 3aBHCHMOCTH OT BpeMeHH t
MOJKeT 6bITb aarrHcaHO B cpopMe
s(x, t) = 2S o g(x) cos wt , (0 .:;; x .:;; b),
(4)
r,ne So - rrOJIOJKHTeJIbHaH rrOCTOHHHaH, a CPYHKI.lHH Koop,nHHaTbI g(x)
HMeeT BH,n:
g(x) =  sin k( x - %) + B 2 COS k( x - % )- (5)
g(x) HMeeT MaKCHMaJIbHOe 3HaqeHHe, paBHoe 1, a BeJIHqHHa k paBHa
wju.
HanOMHHM, qTO I.leHTpbI ::meKTpo,nOB OCTalOTCH B rrOKoe H 'tITO JIe-
BbIH H npaBbIH TOPI.lbI nJIaCTHHbI CB060,nHbI H ,nOJIJKHbI HMeTb HYJIeBOe
MexaHlIqeCKOe HanpHJKeHHe (H ,naBJIeHHe).
1. Orrpe,neJIHTe BeJIHqHHbI Bl H B 2 B ypaBHeHHH (5) ,nJIH rrpo,nOJIb-
HOM CTOHqeM BOJIHbI B KBapI.leBOH rrJIaCTHHe
2. Orrpe,neJIHTe ,nBe HaHHli31IIHe qaCTOTbI, Ha KOTOpbIX MorYT 6bITb
B036YJK,neHbI rrpo,nOJIbHbIe (B,nOJII, Koop,nHHaTbI x) CTOHqHe BOJIHbI B
KBapI.leBoH rrJIaCTHHe.
llbe303JIeKTpHqeCKHH acl>cpeKT - 3TO oc060e CBOHCTBO KpHCTaJIJIOB
KBapI.la. CJKaTHe HJIH paCTHJKeHHe KpHCTaJIJIa C03,na1OT 3JIeKTpHqeCKOe
295

HarrpHJKeH1:Ie B rrOrrepeIHOM HarrpaBJIeHHH Kp1:lCTaJIJIa, 1:1 Ha060pOT,
BHelliHee aJIeKTpHtIecKOe HanpHJKeHHe, rrp1:lJIOJKeHHOe K Kp1:lCTaJIJIY,
BhI3bIBaeT ero CJKaT1:Ie HJ"IH paCTHJKeHHe B rrOrrepeqHOM HarrpaBJIeHH1:I B
aaBHCHMOCTH OT rrOJIHpHOCT1:I HanpHJKeHHH. CJIe,nOBaTeJIbHO, MexaHH-
tIeCKHe 1:I3JIeKTpHqeCKHe KOJIe6aHHH B KpHCTaJIJIe KBapI.la MOryT Baa1:l-
Mo,neHCTBOBaTb H B3a1:lMOB036YJK,naTbCH.
,n:JIH OrrHcaHHH rrbea03JIeKTpHqeCKOrO 3cpcpeKTa, Kor,na KBapI.leBaH
rrJIaCTHHa Haxo,nHTCH B aJIeKTp1:lqeCKOM nOJIe E, HarrpaBJIeHHOM rro OC1:I
Z, 060aHaq1:lM rrJIOTHOCTH rrOBepXHOCTHbIX 3apH,nOB Ha BepXHeM H HHJK-
HeM 3JIeKTpo,ne -a 1:1 +a. 060aHaq1:lM ,necpopMaI.l1:lIO 11 MeXaH1:IqeCKOe
HarrpHJKeH1:Ie B rrJIaCTliHe B HarrpaBJIeH1:I1:I x COOTBeTCTBeHHO qepe3 S 1:1
T. Tor,na rrbe30aJIeKTpHqeCK1:IH acpcpeKT B Kp1:lCTaJIJIe KBapI.la OnHChIBa-
eTCH C.iIe,nYIOlI.{eH C1:ICTeMOH ypaBHeHHH:
S = (I/Y)T + dpE (6)
a = dpT + ErE (7)
r,ne I/Y = 1,27.10- 11 M 2 /H - KoaCPCP1:lI.l1:leHT yrrpyroH ,necpopMaI.lHH
(T. e. BeJI1:Iq1:lHa, 06paTHaH MO,nYJIIO IOHra) B rrOCTOHHHOM aJIeKTp1:lqe-
CKOM nOJIe H E T = 4,06 . 10- 11 <I>/M - BoCrrpHliMqHBOCTb rrp1:l rrocToH-
HOM MeXaH1:IqeCKOM HarrpHJKeH1:IH, Tor,na KaK d p = 2,25 . 10- 12 M/B -
3TO rrbea09JIeKTpHqeCK1:IH KoaCPCPHI.lHeHT.
PaCCMOTpHM CJIyqaH, Kor,na KJIlOq K (pHC. 210) aaMKHyT. llepeMeH-
Hoe HarrpHJKeHHe U(t) = U m cos wt B 3TOM CJIyqae rrp1:lJIOJKeHO MeJK,ny
3JIeKTpO,naMH, 1:1 B KBapI.leBOH rrJIaCTHHe B HarrpaBJIeH1:I1:I Z rrOHBJIHeTCH
O,nHOpO,nHOe aJIeKTpHqeCKOe rrOJIe E( t) = U( t)/ h. Kor,na ,nOCT1:IraeTCH CTa-
I.l1:l0HapHOe COCTOHH1:Ie, B KBapI.leBoH rrJIaCT1:IHe B HarrpaBJIeHH1:I x Boa-
HHKaeT rrpOAOJIbHaH CTOHqaH BOJIHa C yrJIOBOH qaCTOTOH w.
,n:JIH o,nHopo,nHoro aJIeKTp1:lqeCKOrO rrOJIH E ,nJI1:IHa BOJIHhI A H qac-
TOTa f rrpo,nOJIbHOH CTOHqeH BOJIHhI B rrJIaCTHHe CBH3aHbI COOTHOlIIeH1:IeM
A = u/f. r,ne CKOpOCTb u orrpe,neJIHeTCH ypaBHeHHeM (2). Ho, KaK rrOKa-
3bIBaeT ypaBHeH1:Ie (6), CpopMYJIa T := YS YJKe He BhIrrOJIHHeTCH, XOTH
orrpe,neJIeHHH ,necpopMaI.l1:l1:l 1:1 HarrpHJKeH1:IH OCTalOTCSI He1:l3MeHHhIMH.
TOPD;bI rrJIaCT1:IHbI OCTalOTCH CB060,nHbIM1:I C HYJIeBbIM MeXaHHqeCKHM
HarrpHJKeH1:IeM.
3. lloBepxHocTHaH rrJIOTHOCTb 3apH,na aHa HHJKHeM 9JIeKTpo,ne KaK
CPYHKI.l1:lH Koop,nHHaTbI x 1:1 BpeMeH1:I t MOJKeT 6hITh rrpe,nCTaBJIeHa B B1:I,ne
[ ( b ) ] U(t)
a(x,t) = DJ cosh x -"2 + D 2 T'
r,ne k = w/u. llpHH1:IMaH BO BHHMaH1:Ie COOTHOlIIeHHH (6) 1:1 (7), HaH,nliTe
BhIpaJKeHlIe ,nJIH D 1 H D 2 .
296

4. llOJIHLlH rroBepxHocTHbIH 3apH,n Q( t) Ha HI:IJKHeM 3JIeKTpo,ne CBH-
3aH C Vet) COOTHOlIIeHHeM
Q(t) = [1 + a 2 ( :b tg k; - 1)] CoV(t).
(8)
HaH,nI:lTe BbIpaJKeHlIe ,nJIH CO. a TaKJKe BbIpaJKeHHe H qllCJIeHHOe
aHaqeHl:le ,njJH a 2 .
3a}];aqa 3
qacTb 1. llOKOHlI.{I:IHCH B JIaOOpaTopHoH CHCTeMe OTCqeTa cB060,n-
HbIH HeHTpoH MaCCbI m n pacrra,naeTCH Ha Tpl:l HeBaaI:lMo,neHcTByroU.ll:le
qaCTIIIJ;bI: npOTOH, 3J"IeKTPOH H aHTHHeiiTpHHo. Macca rrOKOH npOTOHa -
m p ' Tor,na KfiK Macca rrOKOH aHTHHeHTpHHo mv rrpe,nrrOJIaraeTCH HeHY-
JIeBOH H HaMHoro MeHblIleH MaCChI rrOKOH 3JIeKTpOHa me. 0603HaqllM
CKOpOCTb CBeTa B BaKYYMe KaK C. M3MepeHHbIe 3HaqeHHH Macc paBHbI:
m n = 939,56563 M3B/c 2 ,
m p = 938,27231 M3B/c 2 , me = 0,5109907 M3B/c 2
B ,naJIbHeHlIIeM Bce 3Heprl:ll:l H CKOpOCTH rrOJIaralOTCH l:IaMepeHHbI-
MH B JIa60paTopHoH CHCTeMe OTCqeTa. llYCTb E - rrOJIHaH 3Heprl:lH aJIeK-
TpOHa, rrOHBJIHIOlI.{erOCH B peaYJIbTaTe pacrra,na.
HaH,D;HTe MaKCIIMaJIbHO BOaMOJKHOe aHaqeHl:le E m8x aHeprl:l1:l Ell co-
OTBeTCTBYIOU.lYIO eMY CKOpOCTb vtJ1 aHTHHeHTpHHo (rrpH E = EmaX>. 06a
OTBeTa ,nOJIJKHbI 6bITb BbIpaJKeHbI Iepea MaCCbI rrOKOH qaCTI:IIJ; 1:1 CKO-
pOCTb CBeTa. lloJ"IaraH, tlTO mv < 7,3 3B/c 2 , paCCqHTaHTe ql:lCJIeHHbIe
aHaqeHI:IH BeJIHqHH v,,/c H E m8x C TOqHOCTblO,nO Tpex aHaIaU.lHX LlHCPP.
qacTL 2. llpoapaIHaH CTeK.1JHHHaH rroJIyccpepa (rroJIyrnapl:le) pa,nH-
YCOM R II MaccoH m I:IMeeT K03q>cpHLlHeHT rrpeJIOMJIeHI:IH n. B cpe,ne,
oKPYJKaIOU.leH rroJIyccpepy, KoacpcpHLlHeHT rrpeJIOMJIeHHH paBeH e,nHHIILle.
IIapaJmeJIbHbIH I.l}WHH,npWlecKHH rrIOK MOHoxpOMaTHqeCKOrO JIaaepHO-
ro 1:13JIIeHHH, O,nHOpo,nHbIH rro CeqeHI:lIO, HOpMaJIbHO rra,naeT Ha LleH1'paJIb-
HYIO qaCTb rrJIOCKOH rrOBepXHOCTI:I rroJIycc:}>epbI (pHc.212). YCKopeHHe
cB060,nHoro na,neHHH g HarrpaBJIeHO BepTI:IKaJIbHO BHH3. Pa,nHYc 8 ce-
tJ:eHHH JIa3epHOro J"IY'ia HaMHoro MeHblIIe R. CTeKJIHHHaH rroJIyccl>epa H
JIyq JIa3epa IJ;I:IJ"IIIH,npHqeCKH CHMMeTpHqHbI OTHOCI:ITeJIbHO OCII z.
CTeKJIHHHaH rroJIyccl>epa He rror J"IOU.laeT JIa3epHOro H3J"IyqeHHH. ,n:JIH
Toro qT06bI c,neJIaTb OTpaJKeHHe JIaaepHOro JIyqa rrpH Bxo,ne 1:1 BbIxo,ne
He3HaII:ITeJIbHbIM, rrOBepXlIOCTb nOJIyccpepbI nOKpbITa TOHKHM CJIOeM
rrp03paqHOrO MaTepHUJIa. OrrTHqeCKHH rrYTb J"IaaepHOro JIyqa rrpH rrpo-
XOJK.n;eHHH aToro CJIOH TaKJKe HeaHaqHTe.iIeH.
297

2. llpeHe6peraH qJIeHaMH 1I0PMKa (8/R)3 H BbIlIIe, HaH,nHTe MOll{-
HOCTb JI8aepa, Heo6xo,nHMyJO ,nJIH Toro, qT06bI CKOMIIeHCHpOB8Tb Bec
cTeKJ"IHHHOH 1I0JIyccpepbI.
1 - 8 2 /2.
YKa.1aHue: llpH 8, HaMHoro MeHblIIHX e,nHHHLlbI, cos 8
z
CmeKJlR.HHa.R. nOJlycljJepa
!g -
JI a3epHbLu JlY'L
--t t-
2B
PeweHMH
s..........
.....
.....
....
....
....
-...
r' '...._
---.
p ........
....
........ ,
Sl'>fJ---
t e,
--e,
PIIC. 213
298
PHC. 212
3a)J;aqa 1
Q qacTb 1. 1. llOCKOJIbKY ,nJIHH8
HHTH L = s + R8 IIOCT05IHHa, TO rrpo-
,nHc]:>c]:>epeHD;HpOBaB ee IIO BpeMeHH,
1I0JIyqHM
ds + R dO = o.
dt dt
2. OTHOCHTeJIbHO TOqKIi 0 TOqKa
Q ,nBHJKeTCH IIO OKPYJKHOCTH pa,nHY-
R  dO
ca . C yrJIOBOH CKOpOCTblO dt ' CJIe-
,nOB8TeJIbHO,
- R de- ds_
v Q = "die, = - dt e,.
3. HaH,neM lIepeMell{eHHe TOqKH
P 3a BpeMH l'>t OTHOCHTeJIbHO TOqKH
Q (pHC. 213).

111' = (sI18)(-e r ) + (l1s)e, = (( s  )(-er) + : e, )M.
Cne,nOBaTeJIbHO,
-, de- ds-
v = -s---e + ---e.
dt r dt'
4. CKOpOCTb qaCTHD;bI OTHOCHTeJIbHO TOqKH 0 eCTb cYMMa ,nBYX CKO-
pocTeH, HaH,neHHblx B IIYHKTax 2 1:1 3. ll03TOMY
-. -, - ( de - ds - ) R de - de -
v = v + v Q = -s-e + -e, + -e, = -s-e .
dt r dt dt dt r
(1)
5. PaccMoTpHM PHCYHOK 214. IIpHpaU.leHl:le (-e,) -KOMIIOHeHTbI CKO-
pOCTH l1ii paBHO
(-e, )l1v = vI18
ll03TOMY e, -KOMrrOHeHTa YCKO-
peHI:IH a = I1v / M paBHa ae, =
de
= -v --. llOCKOJIbKY CKOpOCTb qa-
dt
de
CTHI.lbI paBHa s dt ' TO (e,)-KOMIIO-
HeHTa YCKopeHHH
v de M.
dt
Pnc. 214
-s( = J .
ae,
-v de _ ( s de ) de
dt dt .dt
3aMeTI:IM, qTO pa.n;HaJIbHaH KOMrrOHeHTa YCKopeHI:IH paBHa
(2)
ae = _ 8v = _ 8( s  J .
, 8t 8t
6. llOTeHI.lHaJIbHaH 3Heprl:lH qaCTI:ID;bI B rpaBHTaI.lI:IOHHOM nOJIe iff
paBHa U = -mgh (pIlC. 215). Ee MOJKHO Bblpaal:lTb qepea s H 8:
U(8) = -mg(R(1 - coS 8) + s sin 8). (3)
7. B HI:IJKHeH TOqKe TpaeKTOpl:lH IIOTeHI.ll:laJIbHaH aHeprHH qaCTH-
LlbI IIpHHHMaeT CBoe MI:IHI:IMaJIbHOe aHaqeHl:le U m. LJaCTI:ILla Haxo,nI:lTCH
B COCTOHHI:II:I paBHOBeCHH (pHC. 216). ll03TOMY MHHI:IMYM IIOTeHD;l:IaJIb-
HOH 3Heprl:lll Ha6JIIO,naeTcH npH 8 = 1[/2 I:IJIH s = L -1[R/2.
H3 paBeHCTBa (3) CJIe,nyeT, qTO MHHHMaJIbHaH IIoTeHD;HaJIbHaH3Hep-
rHH qaCTHI.lbI
U m = u( %) = -m g ( R + L _ n: )-
(4)
299

x+....
e,
I
I
hi 'i l
I I
II
__L
p
Rcos 8
Pnc. 215
s
p
Pnc. 216
BHa'laJIe rrOJIHaH MeXaHl:l'leCKaH 3Heprl:lH E = o. llo aaKOHY coxpa-
HeHI:IH 3HeprHH B rrOJIOJKeHHH paBHOBeCI:IH:
1 2
o = -mUm + U m -
2
(5)
HaBbIpaJKeHHH (4) 1:1(5) rrOJIyqaeM
U", = .j -2U n .lm = .j 2g(R + (L - 1tR/2».
'!aCTh 2. 1. COrJIaCHO ypaBHeHl:l1O (3), rrOJIHYIO MeXaHH'leCKYro 3Hep-
rllro 'laCTI:ILlbI MOJKHO rrpe,nCTaBHTb B BI:I,ne:
E = 0 = .!mu 2 + U(8) = .!mv 2 - mg(R(1 - cos 8) + s sin 8).
2 2
300

xt
a
mg sin 8f
f
-I.. 8
PHC. 217
dO
COrJIaCHO ypaBHeHHIO (1), CKOpOCTb v = S-. ll03TOMY
dt
( de ) 2
v 2 :::: S dt = 2g(R(1 - cos 8) + S sin 8).
(6)
IIycTb T - HaTHJKeHl:le HI:ITH, Tor,na e,-KoMIIOHeHTa CliJIbI, ,neHcTBY-
lOU.leH Ha 'laCTI:II.lY, paBHa -T + mg sin 8 (Pl:lc. 217). H3 ypaBHeHI:IJI (2)
( de ) 2
TaHreHI.lHaJIbHOe YCKopeHHe 'laCTHI.lbI a, = -s dt . lIo BTOPOMY aa-
KOHY HbIDToHa:
m( -s( : J) = -T + mg sin 8.
(7)
CorJIaCHO IIOCJIe,nHHM ,nBYM paBeHcTBaM, HaTJlJKeHlie HHTH MOJKeT
6bITb aarrHcaHO B BI:I,ne:
T = m( s( : J + g sin 8 ) = g (2R(1 - cos 8) + 3s sin 8) =
301

2m g R ( 8 3 ( L ) 1 .
= --;- t g "2 - "2 0 - R t 1ll8 =
2mgR .
= -(YI - Y2) Slll 8.
S
(8)
fpa<}>IIKII CPYHKLlI:IH y1II Y2 Ha06paJKeHbI Hfl pHcYHKe 218. llpl:l no-
MOlI.{1:I 3Toro pl:lCYHKa 11 paBeHCTBa (8) cocTaBIIM Ta6JII:ID;Y. YroJI, npH
KOTOpOM Yl = Y2' 060aHa'lHM 8.(n < 0.. < 2n). :3ToT yrOJI aa.u;aeTCH BbIpa-
JKeHHeM:
 ( 8 - !:.. ) = tg 0, ,
2 . R 2
IIJIH
L 2 0
R = O. - "3 tg 2 .
llo YCJIOBIIIO:
L 9n 2 1t ( 1t ) 2 1 ( 1t )
Ii = 8" + '3 ctg 16 = n + '8 - '3 tg '2 11: + '8 .
CpaBHIIBaH rrOCJIe,nHHe ,nBa paBeHCTBa, BH,nIlM, 'lTO 8. = 9n/8.
y
30
20
----------1t
IYI ::::tg- I I I IYI =tg-
--r- 2 r--r--r--r- 2
8,
y =e_t-- 
-p1_-Rt-----
8
10
o
-10
-20
----
-30
1t 1t 3n 21t 5n 3;[
2 2 2
Pnc. 218
302

Y. - Y 2 sin 8 T
0<8<11: + + +
8=11: +00 0 +
11: < 8 < 8, - - +
8=8, 0 - 0
8, < 8 < 211: + - -
H3 Ta6JIlILlbI CJIe,nyeT, 'lTO HaTHJKeHHe HHTH rrOJIOJK1:ITeJIbHO (H1:ITb
rrpHMaH 11 HaTHHYTlIH) npH 0 < 8 < 8.. HIITb rrpOBHCHeT (T  0), Kor,na
8  8.. Ha1:lMeHblIIaH ,nJI1:IHa rrpHMoro y'laCTKa QP ,nOCT1:IraeTCH rrpH
8 = 8. H COCTaBJIHeT:
( 91t 2 n 9n ) 2R n
S - = L - Re = R - + - ctg- - - = - ctg- = 3,352 R.
nun '8 3 16 8 3 16
HaTHJKeHlie T = 0 npH 8 = 8.. PaBeHcTBo (7) ,naeT v 2 = -gs sin 8. QT-
cro,na
V. = .J -gSmlli sin 8.
2gR n. n  4gR n '> r::::r;
-ctg-sm- = - cos- = 113<>vgR.
3 16 8 3 16 '
2. Kor,na 8 > 8., qaCTHD;a ,nBIIJKeTCH KaK cB060,nHoe TeJIO B rrOJIe
THJKeCTH. LJaCT1:IIJ;a Ha'l1:lHaeT ,nBIIJKeHlIe C Ha'laJIbHOH CKOpOCTblO v.
H3 nOJIOJKeHIIH P = (x.' Y.) B HarrpaBJIeHlIlI, COCTaBJIHIOlI.{eM yrOJI
q> = (311:/2 - 8.) C oCbro Oy (pIlC. 219). CKOpOCTb v H 'laCTIIIJ;bI B BepxHeH
TO'lKe rrapa60JIH'leCKOH TpaeKTOp1:l1:l paBHa Y-COCTaBJIHIOlI.{eH Ha'laJIb-
HOH CKOpOCTIi, IIJIH
V n = v. sin (8, - 11:) = 4gR cos 2:.. sin!: = O,4334.Jiii.
3 16 8
PaCCTOHHHe B,nOJIb OC1:I Oy, Ha KOTopoe CMeCTIITCH 'laCTHIJ;a, ,nOCTHr-
HYB MaKCHMaJIbHOH BbICOTbI:
H - v: sin 2(8, - n) _ v: . 9n - 0 4535R
- 2g - 2g sm 4 -, .
Koop,nIlHaTbI 'laCT1:ILlbI rrpll 0 = 0. aa,nalOTCH <}>opMYJIaMH:
x. = R cos 8. - sm," sin 8. = -R cos  + Smlli sin  :::: 0,358R.
Y. :::: R sin 8. - Smlll COS 8. = -R sin  - Snull cos  = -3, 478R.
CJIe,nOBaTeJIbHO, I y.1 > (R + H). lloaTOMY 'laCTHIJ;a ,nOCTlIrHeT MaK-
CHMaJIbHOH BbICOTbI, He y,napllBmllCb 0 LlIlJIIIH,np.
303

y
....-
I
I
I
sl
I
I
I
I

v.
m L
.---------------
yJ
Pnc.219
qacTb 3. IIpHMeHI:IM aaKOH coxpaHeHHH aHeprHH K CHCTeMe rpY30B
(pHC. 220):
-Mgh = E' - Mg(h + D),
r,ne E' - IIOJIHaH MeXaHH'leCKaH3Heprl:lH qaCTI:I:D;bI B TOT MOMeHT, KO-
r,na rpya M OCTaHOBI:ITCH. OTCIO,na E' = M gD. llYCTb A - nOJIHaH ,nJIIIHa
HHTII. :3Ta BeJIH'll:lHa coxpaHHeTcH, noaToMY ,nJIH JII060ro 8
A = L + %R + h = s + R( 8 + %) + (h + D).
3aMeTHM, 'lTO D = aL. llYCTb 1 = L - D, Tor,na 1 = (1 - a.)L.
113 rrOCJIe,nHI:IX ,nBYX CPOPMYJI rrOJIyqaeM:
s = L - D - R8 = 1 - R8. (9)
L
Ax
m
M
Pnc. 220
304

IIoCJle OCTaHOBKH MaCCHBHOrO rpyaa aHeprHH 'laCTHqbI COXpaHHeTCH:
E' = MgD = !.mv 2 - mg(R(l- cos8) + 8 sin e).
2
OTclO,na KBa.n;paT CKOpOCTIi qaCTHbI:
v 2 = (8 : J = 2D + 2 g R( (1 - cos e) +  sin e )-
COrJIaCHO (7), HaTHJKeHHe HHTH paBHo:
T = m(gSine + ( : J) =
= g ( 2: D + 2R(1 - cos 8) + 38 sin e) =
2m g R ( MD 3 ( l ) . )
= - - + (1 - COS 8) + - - - e sm e .
s mR 2R
B nOCJIe,nHeM pRBeHCTBe HCnOJIhaOBaJIOCh ypaBHeHHe (9). BBe,neM
CPYHKD;HIO
(10)
f(8) = 1 - cos 8 + (  - e )sin e.
IIOCKOJIhKY 1 = (L - D) » R, TO MbI MOJKeM HanHcaTh:
fee) ::::: 1 + l.sin 8 - cos e = 1 + A sinCe - <1», (11)
2R
r,ne
A = Jl + ( : r '
3l
q> = arctg 2R .
1 + (: J
H3 paBeHCTBR (11) HRXO,nHM MHHHMYM CPYHKHH:
f min (e) = 1 - A = 1 - Jl + (: r .
IIOCKOJIhKY CHJIa HaTHJKeHHH T HeOTpHaTeJIhHa, TO Ha (10) nOJIY-
1laeM:
MD + 1.. = M(L - l) + 1 _ Jl + ( 3l ) 2  0,
mR It"n mR 2R
11 Cl>H3HJCa 10-11 KIIaccW
305

H."IH
.\IL + 1 > .\Ii + 1 + ( 3i ) 2 "" ( /'.Ii ) + (  ) .
mR mR 2R mR 2R
I1crrOJ"Ib3YH (10), n03IyqaeM
: + 1 # ((  ) + ( : ))(1 - a).
llpeHe6peraH 'lJ"IeHaMH rropH,nKa (R/L) H BbIme, nOJ"IyqaeM:
( ML ) + 1 ( 3L ) - 1
# 1 _ mR = 2R =
a ( : ) + (: ) (  ) + ( : )
1 _ 2R
3L ::::
2!o1 + 1
3m
1
1 + 2M .
3m
llpe,neJIbHOe OTHOlIIeHHe D/ L paBHo:
1
a c = 2M .
1 + -
3m
3a,lJ;a'la 2
LJacTL 1. 1. 3a Mfl."IOe BpeMH t:.t JIeBbIH KpaH CTepJKHH CMelI.{aeTCH Ha
paccToHHHe vt:.t, B TO BpeMH KaK cppOHT BOJIHbI CJKaTHH CMelI.{aeTCH Ha
uM, r,ne u ::::  (pHc.201).
s = t.i = -vt.t = .::.£.
l ut.t u
ut:.t

t=O
p
M/2
I
.E-
JIf
l:1t
8
PIIC. 221
306

113 3aKOHa fYKa CJIe,nyeT, 'lTO
p = - YS = Y.!:. = puv.
u
2. CKOpOCTb v paBHa npOII3Bo,nHOH CMeU.leHHH 1; no BpeMeHIl. llo-
CKOJIbKY (x, t) = o sin k(x - ut) , rrOJIy'lHM
v(x, t) = -kuC;,o cos k(x - ut).
BOCnOJIb3yeMcH pelIIeHlIeM nYHKTa 1:
v(x, t)
Sex, t) = - - = kl;o cos k(x - ut),
u
p(x, t) = puv(x, t) = -kpu 2 C;,o COS k(x - ut) =
= - YS(x, t) = -kYC;,o cos k(x - uf).
LJacTb 2. 1. TaK KaK no YCJIOBHIO LleHTp n3IaCTHHbI Heno,nBIIJKeH, TO
g = !!.. = o. 113 onpe,neJIeHHH g(x) CJIe,nyeT:
2
g(%) = BlsinO+B 2 cosO=B 2
3Ha'lHT, B 2 = 0, Bl =:!:1 (TaK KaK MaKcHMYM g(x) paBeH 1). Tor,n8
- co ( b )
g(x) = :tsm  x - '2 '
 ( t ) 2 - 0 ell ( b )
<; x, =:t <;'0 sm x - '2 coscot.
2. 0603Ha'lllM l;m = :!:l;o. Tor,na
l;(x,t) :::: :!:2l;m sin k( x - % )cos cot
= l;m ( sink( x - % - ut) + sink( x - % + ut)).
(x, t)  ( ( b ) ( b ))
v(x,t) = dt = -kuc,m cosk x - '2 - ut - cosk x - '2 + ut .
llo."lyqeHHoe pelIIeHHe cooTBeTcTByeT JIHTepcpepeHLlHH ,nBYX BOJ"lH.
BocnoJIb3yeMcH pe3YJIbTaTOM nYHKTa 1 'laCTH 1. ,n:JIH BOJIHbI, ,nBHJKY-
U.leHCH B oTpHLlaTeJIbHOM HanpaBJIeHHH OCH x, B Ka'leCTBe CKOpOCTIl CJIe-
,nyeT 6paTL -v, T. e.
S =.!:., r,nep=-puv,
u
II"
307

,n:'-IH BO."lHbI, ,nBII:-KYU.leHCH B rrO."lOJKIITeJIbHOM HanpaBjleHHH OCII X,
S = -V , r,nep = puv.
u
Tor,na rrOJIYIlIM:
S(x,t) = -k1;m( -COSk[ x - % - ut J - COSk( x - %.,. ut)) =
= 2km cos k( x - % )COS (ot,
p(x,t) = -puco1;..( COSk( x - % - ut) + COSk( x - % + ut)) =
-2puwm COSk( x - % Jcoswt.
llOCKOJ'bKY KOHUbI rr.:mCTIIHbI Herro,aBHJKHbI, HarrpHmeHlIe CJKaTHH,
HJIH ,naBJIeHHH Ha HIIX, paBHO 0:
p(O,t) = p(b,t) = -2puco1;.. COSk( % )COS(r)t = 0
3Ha'lliT,
COS( k: ) = O.
kh = 2rrf b = nrr.
u
Ec."l1I BBeCTII ,n.IIIHY BOJIHbI '),' TO
A = 2b
n = 1. 3, 5, ...
n = 1,3,5, ...
n
OTCIO,aa 'laCTOTa
u nu n fY
f = I = 2b = 2b Vr-.
n = 1,3,5,..0
,n:Be HaUHIIallilIe 'laCTOTbI paBHbI COOTBeTCTBeHHO
1. = ..!:!.... = 273 KfLl,
2b
3.113 YC."lOBUH CJIe,nyeT, 'lTO
T = yeS - dpE),
3u
f3 = 31. = - = 818 KfLl.
2b
(1)
() = YdpS + ET( 1 - y :; IE.
(2)
308

II b u 
OCKOJIbKY X = - - yae""I CTOH'leH BOJIHbI. CMeU.leHlle C; II ,necpopMa-
2
LlJlH S onHCLlBalOTCH aHaJIOrH'lHO nYHKTY (2):
(x.t) = ;m sinkr x - !!.. ) cos (cot + cp), (3)
\ 2
S(x,t) = kf"., cos k( x - % )cos (wt + cp), (4)
r,ne q> cooTBeTcTByeT c,nBHry cpaa MeJK,ay Ko."Ie6aHHHMJI cr 11 .
.HcHO, 'lTO nOJIe E MeJK,ny 3."leKTpo,naMII aaBIICHT TOJIbKO OT BpeMe-
HH:
E( ) _ U(t) _ U m coscM
x,t - - .
h h
llO,nCTaBIIM EllS lIa (5) H (4) B (1):
T = yl kf,m COSk( x - % )c0S(wt + cp) - d; U m COSUJt )-
MeXaHlI'leCKOe HanpHJKeHHe T Ha KOHLlax nJIaCTHHbI (x = 0 H X = b)
paBHo O. :3To BOaMOJKHO, TOJIbKO eC."IH q> = 0 H
k kb d U m
<;m cos"'2 = P h O
TaK KaK q> = 0, TO Ha BbIpa>KeHlIii ,n."IH E, S II cr c."Ie,nyeT, 'lTO
(5)
cr (x, t) = cr (x) cos wt,
r,ne
cr(x) = Ydpkf,m COSk(X - %) + ET( 1 _ y :; ) U; =
[ d2 ( h ] l ' d2 ) U
= Y  cos Iz x - - + [T 1 - y...!.. .
cos kb 2 [T h
2
OTCIO,na
d 2
D 1 y
kb'
cos-
2
t ' d2 )
D 2 = ET . 1 - y [; .
4. llOJIHbIll aapH,n Q(t) IIlDKHerO 3JIeKTpo,nfl MOJKlIO lIaiiTII, rrpOJlH-
TerpJlpOBaB cr(x, t) rro nOBepxHocTII 3.'IeKTpo,na:
Q(t) 1 J b 1 J b
- = - (T(x, t)u'dx - - (T(x)wdx =
U(t) U(t) 0 U., 0
309

W b [ d; ( b ] ( di, ))
= - i Y - cos k x - - + E T 1 - Y - dx::::
h 0 cos kb 2 f T
2
( bw )( d ( 2 kb ) [ d ))
= E T - Y-:- -tg- + l-Y- .
h [T kb 2 lOT
113 (POpMy."lbI (8) YCJIOBIIH
Q( t) . ( 2 kb )
- = C (a 2 -tg - + (1- a 2 »
V(t) 0 kb 2 '
oTKy,na
bw
Co = E T h'
d 2
a 2 = y...!:.. = 9,82.10- 3 .
E T
3aJJ:a'la 3
qacTb 1. llYCTb P P p - HMnYJIbCbI aJIeKTpOHa, rrpOTOHa Ii aH-
r' p' v
THHeHTpHHo, a Ee' Ep 11 Ev - HX 3HeprllH. BBe,neM npllBe,neHHbIe 3Ha-
tleHIIH ,nJIH aTUX BeJIII'lHH:
q - - P,
,
c
= P p
qp
c
q = P"
" c
E,
E,. = 2'
e
Ep
Ep = 2'
e
E"
E =
" e 2
Tor,na pe.'lHTHBHCTcKoe COOTHOlIIeHHe MeJK,ny aHeprueH 11 HMrrYJIb-
COM rrpHMeT BU,n:
(c 2 Ep)2 = c 2 (CQp)2 + (m p c 2 )2,
(C 2 E,)2 = c 2 (cq,)2 + (m,c 2 )2,
(C 2 E,_)2 = c 2 (cij,)2 + (mv c2 )2,
HJIH
[ = m + q, E; = m; + q;, E = m; + q;.
113 3aKOHa coxpaHeHliH 3HeprHH II liMrrY;IbCfl CJIe,nyeT:
Ep + f,. = nl n - E.,
qp + ij,. = -q,.
B03Be,nH aTH paBeHCTBa B KBa,npaT, rrOJIY'lIlM
E; + E + 2EpEv = (m" - E,)2,
q; + q; + 2ij p . q" = q; = E; - m;.
310

BbI'lTeM O,nHO ypaBHeHHe 113 ,npyroro:
m: +  + 2(E p F, - iiij,.) = m + m! - 2m n E..
nJIH
2mnE, = m + m; - m: - m - 2(E p E. - qpq,,),
llYCTb e - yro."I MeJK,ny qp II ij,., Tor,na ijpcj" = qpq,. cos e ,.;;; qpq",
a 3Ha'lIlT,
2m n E, ,.;;; m,; + m; - m - m - 2(E p E" - qpq,,).
:3TO HepaBeHCTBO o6pamaeTCH B paBeHCTBO TOJIbKO eCJIH e = 0, a 3Hfl-
'lIlT, B CJIyqae, Kor,na Ee MaKCHMaJIbHa, rrpOTOH 11 aHTIIHeHTpllHO JIeTHT
B O,nHY CTOPOHY.
BBe,neM rrpHBe,neHHbIe 0603Ha'leHIIH ,nJIH CKopOCTeH rrpOTOHa 11 aH-
THHeHTpHHO B JIa60paTopHOH CIiCTeMe OTC'leTa:
I3 p = Up /c, 13. = u" /c.
BBe,neM yrOJI q>v = arcsin 13,. (pIlC. 222), Tor,na
qv = Tn" tg q>" ,
Ev = ,J  + q = Tnv
cos cp"
PIIC. 222
AHaJIOrll'lHO BBe,i:\H q>p = arcsin p' no.rIYIIIM :
qp = mp tg q>p, Ep = .J m: + q: = cO:p
Tor,na HepaneHCTBO ,a.:IH aHeprHII aJIeKTpOHa MOJKHO rrpe,ncTaBHTb B
BH,ne:
2 ' 2 1 - sin cp p sin 'fI,.
. m.E, ,.;;; m + m; - m -  - . mpTn"
cos cp p cos 'fIv
llpeo6pa3yeM 3ftBIICHU.lHH OT CKopocTeH MHOJKHTe;Ib:
1 - sil1l/1p sin 1/1"
cos 1/1 p cos 1/1,.
1 - sin CPp sin CPv - cos CPp cos cp, + 1 =
cos ((Ip cos ((I"
1 - cos(cpp - cp,,)
+11.
cos cp p cos ((I,
311

:3HeprllH MaKCHMftJIbHa, Kor,nfl 3TO HepnBeHCTBO rrpeBpnU.laeTCH B
paBeHCTBO. T. e. npu Cpp = CPv' ITO aKBIIBa."leHTHO paBeHCTBY CKopOCTeH
rrpOTOHa 11 aHTIIHeJlTpIlHO:
I3 p = I\..
Tor,na MflKCIIMfu"lbHOe 3HUtleHHe
1 ( 2 2 2 2 2 )
E,. max = - m + In, - nip - n1;, - mpm"
2m"
1
-(m + nI; - (nip - m,,)2).
2m
MlIKCIIMlIJIbHlIH 3HepniH 3J'IeKTpOHa
2
E -2 _e ( 2 2_ (
- C Fe. "'ax - - m" + me mp
2m"
+ m,,)2) :::: 1,29 MaB.
llOCKOJIbKY rrpOTOH II lIHTIIHeHTpHHo ,nBIIJKYTCH C o,nuHaKoBoH CKO-
pOCTblO,
qp + q,. = q,
Ep + Ev m. - E,
 [2 _ m2
, ,
A = A = qp = q,.
1-',. I-'p f E
p ,
m n - E,.
llo,nCTaBUM E, = E,.mnx :
V m = 13 = E;.m.. - m!
c V.I,....... mrJ - f r . max
.J(m + m; - Ml)2 - 4m;m;
2m - (m + m; - .\1')
.J (m. + Tn. + M)(m. + Tn. - M)(m. - Tn. + 1\,f)(m. - Tn. - M)
:::: 1,27 . 10 3 ,
m - m! + .\1 2
r,ne 1\1 = mp + mv.
Z F
c
m
Pnc. 22:3
312
LJacTb 2. PnccMoTplIM PIlCYHOK 223. llo 3n-
KOHY CHe.'IJIa
n sin 8. = sin 8,.
llpeHe6pernH tlJIeHnMII rrOpH.I{Kn (&/ R)3 H
BbIlIIe, rrOJIyqllM
nO, = 0,.
Tor,nn yroJl OTK.l0HeHIIH :Iyqn 13 paBeH
13 = S, - 8, :::: (n - 1)8,.
:3Hepn:lH cpoToHn Ep CBH3aHft C ero UMrrYJIb-
COM P p COOTHOlIIE'HHeM
Ep = CPr
PaCCMOTpHM yqaCTOK rrOBepXHOCTII .111H3bl
B cl>op:l.le Ko."lbLln pn,nIIYCOM rlllIIllpllHOJI dr. lIo-

CKOJIbKY JIfl3epHbIH nYIOK O,nHopo,neH no Ce'leHlilO, TO Ha aTOT Y'laCTOl-(
rrpllxo,nliTCH MOU.lHOCTb li3J"lY'leHliH
elP = P 2nrdr .
n8 2
3a MaJIOe BpeMH elt 'lepe3 3TOT Y'IflCTOK nepeHOCliTCH aHeprliH
dE = elPelt,
'lTO COOTBeTcTByeT IIMrry.'IbCY
dp = dE = dPdt .
c c
llOCKOJIbKY JIyq OTK."lOHHeTCH HlI yrOJIj3, TO paCCMfiTPliB8eMOMY yqa-
CTKY JIliH3bI rrepe,naeTCH liMnYJIbC
dp = dp(1 - cos 13).
:3TO COOTBeTcTByeT rro,n'beMHOH CliJIe
<>_"J [321
dF = dp, = P 2rdr(1 - cos [3) :::: P = p en - 1)2e:rdr
dt d)2 d)2 d)2.
llOCKOJIbKY e, :::: sin 8, = r / R, liMeeM
dF = p en - 1)2r 3 dr .
c8 2 R 2
llOJIHaH rro,n'beMHfiH cII."la paBHa
fi 6 ( 1) 2 3 d P( 1) 2 1 6 _ Pen _ 1) 2"2
F = J dF = J P n - r r = n - r4 u
o 0 d)! R l 4c8 2 R 2 0 - 4cR 2
:3Ta CliJI8 ypaBHOBelIIHBaeT CHJIY THJKeCTH mg JIHH3bI:
Pen - 1 ) 28 2
= mg,
4cR 2
oTKy,na
p = 4mgcR 2
(n - 1)28 2 .

nporpOMMO
Me)l(AYHOpOAHOM cl»M3MecKOM
OnMMnMOAbl
1. MexoHMKo
KHHeMaTHKa MaTepHaJIbHOH TO'lKH. 3aKOHbI HbIOTOHa. HHepD;H-
aJIbHbIe CHCTeMbI OTC'leTa.
3aMKHYTbIe H HeaaMKHYTLle CHCTeMLl TeJI. HMnYJIbC, 3HeprHH, pa-
60Ta, M0U.lHOCTb. 3aKOH coxpaHeHHH 3HeprHH H HMnYJIbca. CHJILl ynpy-
rOCTH H TpeHHH. 3aKOH BceMHpHoro THrOTeHHH. lloTeHLlHa."lbHaH 3Hep-
rHH H pa60Ta B rpaBHTaLlHOHHOM nOJIe. 3aKOHbI KenJIepa.
CTaTHKa. lJ,eHTp Macc. MOMeHT CH."I. KHHeMflTHKa BpaU.lllTeJIbHOrO
,nBHJKeHHH TBep,noro Te."la. MOMeHT HHepLlHH. OCHOBHoe ypaBHeHHe
,nHHaMHKH BpaU.laTeJlbHoro ,nBHJKeHHH. TeopeMa IIlTeHHepa. 3aKOH
coxpaHeHHH MOMeHTa HMnYJIbca. KHHeTH'leCKaH3HeprHH BpaU.lalOlI.{e-
rOCH TeJIa. HeHHepUHaJIbHbIe CHCTeMbI OTC'leTa. CHJILl HHepLlHH.
2. TepMOAMHOMMKO M MOneK}'nllpHollcj>M3MKO
BHYTpeHHHH 3HeprHH, pa60Ta H TenJIOTa. llepBLlH H BTOpOH 3aKO-
HLl TepMo,nHHaMHKH. H,nefu"lbHLlH raa. ,n:aBJIeHHe H,nefu"lbHOrO raa8. llo-
CTOHHHaH ABorll,npo. YpaBHeHHe COCTOHHHH u,neaJIbHOrO raaa. A6co-
JIIOTHaH TeMnepaTypa. Pa60Ta rrpH H30TepMH'leCKOM H 1I,nHa6aTHOM npo-
Lleccax. lJ,HK.iI KapHo. Kll,n: LlHK.la. 06paTHMbIe II He06paTHMbIe
rrpOLleCCLl. :3HTporrHH.
3. Kone6oHMII M BonHbl
fapMOHHIeCKHe KOJIe6aHIIH. llPO,nOJIbHbIe H rrOrrepe'lHbIe BOJ"lHbI.
JIHHeHHlIH nOJ1HpH3aLlHH. 3BYKOBbIe BOJIHbI. :3cl>cl>eKT ,n:orr.iIepa. HHTep-
cpepeHLlHH BO."lH. CTOHme BOJIHbI.
4.3neKTpoAMHOMMKO
311KOH coxpaHeHHH3JIeKTpH'leCKOrO aapH,na. 311KOH Ky.'IOHlI. :3."leK-
TpH'leCKOenOJIe. llOTeHLlHfi."I. TeopeMa faycca. :3JIeKTpOeMKOCTb. llJIOT-
HOCTb 3HeprHH 3JIeKTpli'leCKOrO rrOJIH.
314

CliJIa TOKU. 3aKOHbI OMU Ii KupxrO(pu. Pu60Ta U MOlI.{HOCTb nOCTO-
HHHOrO Ii nepeMeHHOrO TOKU. 3UKOH .D:JKOYJIH - JIeHl.lfi.
MarHliTHOe nOJIe npHMoro npOBo,na, BliTKU n ,nJIJlHHOH KaTYlliKIi C
TOKOM. CliJIa AMnepa. CHJIfi JIopeHLlU.
3UKOH aJIeKTpOMurHHTHOH liH,nYKLlHIi. MarHHTHLIH nOTOK. 3aKoH
JIeHD;a. CaMOliH,nYKD;HH. HH,nYKTJlBHOCTb. llJIOTHOCTb aHeprHn MarHliT-
Horo nOJIH.
PealiCTOp, KOH,neHCfiTOp H KaTYlIIKlI HH,nYKTHBHOCTH B LlenH nepe-
MeHHoro TOKa. PeaOHfiHC B nOCJIe,nOBaTeJIbHOH Ii napaJ1JIeJIbHOH Llenu.
KOJIe6uTeJIbHbIH KOHTyp. LJacToTfi KO:Ie6aHliH.
CBeTOBbIe BOJIHbI. .D:mppaKLlHH OT O,nHOH Ii ,nBYX U.le.leH. .D:mppaKLlH-
OHHaH pemeTKa. PaapellialOU.laH cnoco6HOCTb pelIIeTKH. HHTep<}JepeH-
D;liH B TOHKHX nJIeHKax. BparrOBCKoe OTpaJKeHHe. llpuHD;lin <l>epMa.
.D:HcnepcliOHHbIe Ii ,nmpPuKLlUOHHbIe cneKTpbI. JIuHeHtlaTbIe cneK-
TpbI.
llOJIHpliaaD;HH aJIeKTpOMarHliTHbIX BOJIH. PuapellialOlI.{aH cnoco6-
HOCTb OnTlitleCKHX CHCTeM. HaJIyqeHlie a6COJ"IIOTHO tlepHOrO TeJIfi. 3a-
KOH CTecpaHa - BOJIbD;MUHU.
5. KBOHTOBOII M penllTMBMCTCKOII cj>M3MKO
<l>oToacpcpeKT. :3HeprHH H liMnYJIbC CpOTOHU. BOJIHbI,ne BPOHJIH. Co-
OTHomeHlie Heonpe,neJIeHHOcTeH.
llpHHLllin OTHOCliTeJIbHOCTII. C."IOJKeHHe CKopocTeH. PeJIHTHBIICT-
CKliH a<PcpeKT .D:onJIepa. PeJIHTUBHCTCKHe ypaBHeHliH ,nBliJKeHUH, UM-
nYJIbCa, aHeprnli. 3aKOH BaallMOCBHaH MaCCbI U aHeprlili. 3aKOHbI co-
xpUHeHliH aHeprnli H IIMnYJIbca.
YpOBHH aHeprliH aTOMOB H MOJIeKYJI. HaJIyqeHHe Ii nOrJIOU.leHHe
aHeprliH. CneKTpbI Bo,nopo.1;ono,no6HbIX aTOMOB.
:3HepreTHtleCKlie ypOBHU aTOMHbIX H,nep. A.lb<l>a-, 6eTU- U raMMU-
lia.iIyqeHliH. BaaHMo,neii:cTBue HaJIYtleHIIH C BeU.leCTBOM. 3aKOH pa,nuo-
aKTliBHoro pacna,na. COCTaB flTOMHbIX H,nep. .D:ecpeKT MaCCbI. H,nepHbIe
peaKD;IIH.

nporpaMMa
MeJKAYHapOAHOM cl»M3MlfecKOM
OnMMnMaAbl


1. MexoHMKo


KIIHeMaTIIKa MaTeplIaJIhHOH TO-qKII. 3aKOHhI HhIOTOHa. IIHep:U;lI-
aJIhHhIe CIICTeMhI OTC1:1eTa.
3aMRHYThIe 11 He3aMKHYThIe CllCTeMhI TeJI.llMIIYJIhC, 3HeprHH, pa-
60Ta, MO:W;HOCTh. 3aROH coxpaHeHllH 3HeprlIll H HMIIYJIhca. CllJIhI yn-
PyrocTH II TpeHHH. 3aROH BceMHpHoro THrOTeHHH. IIoTeHJJ;Ha.nhHaH
;:meprllH H pa60Ta B rpaBHTaIJ;IlOHHOM lIo.ne. 3aROHhI Ken.nepa.
CTaTHKa. IJ;eHTp Macc. MOMeHT cnJI. KHHeMaTHKa Bpam;aTeJIhHOrO
.I1BH.>KeHHH TBep.n;oro Te.na. MOMeHT HHepIJ,HH. OCHOBHoe ypaBHeHHe

I1HaMHRH Bpa:w;aTeJIbHOrO .I1BH.meHHH. TeopeMa IIITeHHepa. 3aROH
coxpaHeHHH MOMeHTa HMlIYJIbca. KHHeTH-qeCRaH 3Hepn1H Bpa:w;alO:w;e-
rOCH Te.na. HeIIHepIJ;Ha.nhHhIe CIICTeMbI OTCqeTa. CIIJIbI HHepIJ;IIH.


2. TepMoAMHOMMKO M MoneKynRpHoll «I»M3MKa


BHYTpeHHHH 3HeprllH, pa60Ta II TenJIOT8. llepBhIH II BTOpOH 38KO-
RbI TepMOJJ;HHaMIIKII. ll.n;eaJIhHbIH ra3. ,D;aBJIeHHe HJJ;eaJIbHOrO ra3a. llo-
CTOHHHaH ABOra.n;pO. 'YpaBHeHHe COCTOHHHH H,[(eaJIbHOrO ra3a. A6co-
}IJOTHaH TeMnepaTypa. Pa60Ta IIPH II30TepMllTIeCROM II a.n;lIa6aTHOM npo-
u;eccax. IJ;HKJI KapHO. Kll.n: IJ;IIKJIa. 06paTllMhle II Heo6paTIIMble
npO:u;eCCbI. SHTpOlIlIH.


3. Kone6oHMil M BonHbl


rapMOHII'tIeCRHe KOJle6aHIiH. llPO.n;OJIbHhle II nOlIepeTIHhle BOJIHhl.
JIllHeHHaH lIOJIHpH3aIJ,HH. 3BYKOBhIe BO
'IHbl. ScPcPeKT JJ;OlIJIepa. llHTep-
4JepeHIJ;IIH BOJIH. CTOHqHe BOJIHbl.


4.3neKTpOAMHOMMKa


3aROH COXpaHeHHH 3J1eKTpll't1eCKOrO 3apHJI;a. 3aKOH KYJlOHa. 8JleR-
TpnqeCKOe nOJIe. llOTeHIJ;na..-r. TeopeMa raycca. 8JIeKTpOCMKOCTh. llJIOT-
HOCTb ::meprIIH 3J1eKTpnQeCKOrO nOJ.UI.


241




CUJla TOKR. 3aKOHbI OMa H KHpxrocl>a. Pa60Ta H MO:W;HOCTL nOCTO-
HHHOrO H nepeMeHHOrO TOKft. 3ftROH A:mOYJIH - JIeH:Q;a.
MarHHTHOe nOJle npHMoro npOBo.n;a, BHTKft 11 ,tJ;JIHHHOii KRTYWKH C
TOKOM. CHJIft AMIIepa. CHJIft JIopeH:Q;a.
3aKOH 3JIeKTpOMarHHTHOH HH.n;YKI.1;HH. MftrHHTHhIH: IIOTOK. 3aKOH
JIeH:Q;a. CSMOHHJJ;YK:Q;IUI. IIH,lJ;YKTHBHOCTL. llJIOTHOCTL 3HeprHH MarHHT-
Boro nOJI.R.
Pe3IICTOp, KOH,lJ;eHCaTOp H KaTymKs HHJJ;YKTHBHOCTII B IJ,enH nepe-
MeUHoro TOKa. Pe30H8HC B nOCJIe.n;OBSTenLHOH II napftJIJIeJIbHOH :Q;eIIH.
Kone6ftTeJILHhIH KOHTYp. "IlacToTa KOJIe6sHHH.
CBeTOBbIe BOJIHbI. .nH<}>paJ<::Q;HH OT O.n;HOH II ,u,BYX
eJIeH. .D;H<}>paR:qH-
OHHaH pemeTKa. PaapelIIalO:W;SSI cnoco6HOCTL pemeTKII. IIHTep«l>epeH-
:Q;HH B TOHKHX IIJIeHKax. BparrOBCKoe OTpameHlIe. IIplIHIJ.Hn <TJepMS.
.z:r;HcnepcHOHHLIe II JJ,Hcl>paK:Q;HOHHLIe cneKTpbI. JIHHeHqaTLIe cneK-
TpLI.
IIoJIHpuaal.\HH 3JIeKTpOMarHIITHLIX BOJlB. Pft3pelIlftJo:w;aH CIIoco6-
HOCTL OnTUqeCKHX CJ:ICTeM. 1I3JIyq:eHHe a6COJIIOTHO qepHOrO Tena. 3a-
ROH CTecl>aHft-BoJILIJ,MftHft.


5. KBOHTOBOII M penllTMBMCTCKOIi cl>M3MKO



oTo3«l>«l>eKT. 3HeprHH 11 HMnYJILC CPOTOHB.. BonuhI l1.e BPOHJIH. Co-
OTHomeHHe Heonpe,u,eJI€HHOCTeii.
IIPHH:Q;HII OTHOCJ:IT€JILHOCTH. CJIOmeHHe CKopOCTeH. PeJIHTHBIICT-
CKHH aCPcI>eKT ,l];onJIepa. PeJIHTHBHCTCKII€ ypaBHeHHH ,u,BH:meHIIH
 HM-
IIYJILCa
 3HeprHII. 3aKOH B38HMOCBH3H MSCCLI H 3HeprHH. 3aKOHhI co-
xpaHeHHH 3HeprHlI H HMnyJILca.
YpOBHH 3HeprHH B.TOMOB H MOn€KYJI. M3JIYQeHHe II norno:w;eHHe
3HeprHH. CrreRTpbI Bo,u,opO,lJ;ono,no6HbIX aTOMOB.
3HepreTHqeCI<lle YPOBHH 8TOMHbIX Hp;ep. AJIL«l>a-
 6eTa- H rftMMa-
H3JIyqeHHSI. BaaHMOl1.eHCTBH€ H3JIyq€HHii C Be:m;eCTBOM. 3aKOH P8.]J;HO-
aKTHBHoro pacna,u,a. COCTaB aTOMHLIX H,lJ;ep. .z:r;ecpeKT M8CChI. Sl,u,epHLI€
pe8KIJ,HH.




TeMOTM..eCKMM
YK030Tenb
30AO..


PU3iJelt
Kypca tjJU3UKU


MeX3Hl1Ka


MOJleKYJI.HpH8R c})1l311Ka
H repMOltllHaMMKa


3JIeKTpo.n;UH3Ml.IKa


KOJIe6aHn.H Ii BOJIHbI


OUTHK8


KBaHTOBaSl <luf3llKa.
ATOMHa.H II .H
epH3.fI
q>113HKa


HOMep OnuMnuaiJw
U nOMep aaiJa1LU


XVI (3), XIX (2), XX (2)
 XXI (3), XXII (1),
XXIII (1)
 XXV (3), XXVI (3), XXVII (1)

XXVIII (1), XXVIII (3), XXIX (1)
 XXX (3)


XVIII(1)
 XX(l), XXIII (3), xxvn(1)
 XXVIII
(1), XXIX (1)


XIV (2), XVIII (2), XX (3), XXI (2). XXII (2)

XXIV (2), XXIV (3)
 XXV (2), XXVII (2)

XXVIII (2), XXX (2)


XVI (I), XVII (2), XVII (3), XVIII (3), XXIII
(2), XXVI (2). XXVII (3)


XVII (1), XIX (1), XXI (1), XXIV (2), XXV
(I), XXVII (1), XXVIII (I), XXIX (3)


XIX (3), XXII (3), XXVI (1), XXVII (l)

XXVIII (2), XXX (1)




nMTEPATYPA


AnSI yrny6neHHorO M3yaieHHII cl»M3MKM
. napWMX KnaCCaX


y.,e6HltKIt It Y'fs6HWe nOC06"A AlIA y"aaqltXCA


1. BYTHKOB E.M., KOH
pRTLeB A.C. cI>H3HKB.: MexaHHRR: ,II;JUi
WRon Ii KJIaCCOB C yrJIy6JIeHHbIM H3YQeHHeM <»H3HRO-MRTeMRTHQeCRHX )J;HC-
I1.HfiJIHH. - M.: BO HSYRa
 1994. - KHHrs 1.
2.KHR0I1H A.K.
KI1ROI1H I1.K.,IIIaMRm C.H.,8BeHQHK 8.E.
cI>li3HK8: ¥qe6HHK ,l];JI.H 10 KJI8CCa mROJl (KJIRCCOB) C yrJIy6JIeHHbIM H3Y'{eHH-
eM <»H3HKH. - M.: TIpocBe
eHHe. 1998.
3. K 0 H Jl. paT L e B A. C. MexaHHKa.: YQe6HHK ,l];JIH cTapmeH mKOJIbI. -
CTI6.: Cnel1H8nbHSH JIHTepRTypR, 1999. - q. I.
4. K 0 H /I, paT b e B A. C. 3JIeKTpOlU-iHRMIIRa: YQe6HlIK /I,JIH cTapmen
lllROJIhl. - CTI6.: CneD;HRJ1bHRH ]IHTepRTypa
 1999. - q. II.
5. K 0 H /I, paT b e B A. C. OCHOBbI cOBpeMeHHoH <»H3HKH: YQe6HlfR ,lI;JUI
cTapmeH lliROJIbI. - CTI6.: Cnen,H8.JILHB..R. JIHTepSTypa
 1999. - q. III.
6. MHKH meB r. H.
 CH HH KOB A. 3. cI>1f3HKR: MOJIeKYJIHpHSH <»H-
3HRS. TepMO,l];HnaMHRR: 10 KJISce: Yqe6HHK AJIB yrJIy6JIeHHOrO H3Y'tleHHH
<»H3HKH. - M.: ,II;po<»s, 1996.
7. M H K H me B r. H .. C H H $I K 0 B A. 3 .. C JI 060 A e K 0 B B. A. cI>ll-
3HKa: 8JIeKTpo.o;HHRMllKS: 10-11 KJIRCCbI: YQe6HHK ,l];JIB yrny6JIeHHOrO Jmy-
qeHH.H <»H3HKll. - M.: ,D;po<»a. 1996.
8. H y pM H H C K Ii H M. If. <I>H3HKa: 11 KJIRCC. ¥qe6HHK )J;JIH lllROJI Ii RJIRC-
COB C yrJIy6JIeHHbIM H3yQeHHeM <»H3IiKH. - M.: HT-QeHTp. 1993.
9. cI>H3HRa: MexRHHKa: 9 KJIacc: ¥'Qe6HHK .D;JIH yrny6JIeHHOrO H3y'tleHHH
<pH3HKH / TIo,n; pe,l];. r. H. MsnmweBa. - M.: ,D;po<»a
 1996.
10. cI>H3HKS: YQe6HHK )J;JIH 10 KJIaCCa mKOJI H KJIRCCOB C yrJIy6JIeH4
HbIM H3yqeHHeM <»H3HKH / TIoA peA- A.A. TIHHcKoro. - M.: npOCBeI.qe

HHe
 2000.
11. cI>H3HKS: ¥qe6HHK )J;JIH 11 KJIRCCR mKOJI H KJIRCCOB C yrJIy6JIeHHb1M
113YQeHHeM <»H3HRH I flo)]; peJ.l.. A.A. IbfHClwro. - M.: IIpocBe
eHHe

2000.
12. HBOPCKHH B. M.
 IIHHCRHH. A.A. GCROBL! <»H3HKH. - M.: Ha-
YRR. 1977. - T. 1.
13. HBOpC.KHR B. M.. TI Ii HCR HH A. A. OCHOBbI <pH3HKH. - M.: Ha-
YRa. 1977. - T. 2.


251




C60pHMKM3Ia,qa..


1. BcepOCCl1llCKlfe OJII1MllHB.)J;bI no
Jf3I1Ke: 9-1 J I<JIaCChI / IIOll; pe)];.
C.M. K03eJIB.. - M.: l{eHTpKoM. 1997.
2. r 0 n b JJ. 4> B. P 6 H. If. cI:J 113 11 1<8 : 3B.JJ;aQHHI<: 9- 11 RnaCCbI: Y qe6HOe no-
c06He ,nJISI 06
e06pa30BaTeJIbHbIX
pe.m,neHI1H. - M.: ,D;poqJa, 1999.
3. Ka6ap,nHH O.
.. OpnoB B.A., 3HJIb6epMRH A.P. (})H3H-
Ra: 3a,naQHHK: 9-11 KnaCCbI: YQe6HOe noc06He ,nJIH 06ll1e06pa30BaTeJIbHbIX
yqpe:iK,lJ;eHHH. - M.: ,D;po4>a, 1999.
4. .K a 6 a P IJ. H H O. cI:J . .
 0 P JI 0 B B. A. MemAYHapO]l;Hble 4>H3HqeCkHe
onliMOHa.rr;hI IIIROJIbHI1ROB / IIo.n; peA. B.r. PR3YMoBCJWro. - M.: HaYKR,1985.
5. K03eJI C.M.. KOPOB"" B.A., OPJIOB B.A. PelllemUIOJIHM-
nl1a,l(HblX 38,lJ;aQ no 4>H3HRe. - M.: IIIKoJIa-IIpecc
 1999.
6. Mene.n; II Hr. B. (})H3I1Ka B a8JJ;aQax. - M.: HaYKS
 1990.
7. II 11 H C R H if A. A. 3a..na
m no 4>H3HKe. - M.: HaYKa, 1977.
8. Phi M K e B H Q A. TI. cI:JH3HKS: 3ap,aQHHR: 9-11 KJIaCCbI: YQe6HOe no-
co61Ie p,JUI 06
eo6paaOBaTeJIbHbIX Y"lpe.mp,eHIiH. - M.: ,D;po4>a. 1997.
9. C60pHI1K aa.n;a'-l 00 qJH3HKe: 9-11 KJUlCCbI / COCT. CTen8HoBa r .H. -
M.: TIpoCBell\eHHe, 1998.
10. C60pHHK aaARl) no .p1l3HKe: 10-11 KJI8CCbJ / ilo,lJ. peA. C.M. Koaen8. -
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11. CJIo60,neu: KHH If. ill., Opn OB B. A. BcecolO3HLIe OJIHMnHR,lJ;bI
no tPH3HKe: TIoc06He ,nJIH yq81QHXCH. - M.: IIpocBell1eHHe. 1982.


Ylle6HO-M8TOltMlleCKMe noco6HII Anll YlllfTeneii


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IIpoeBe
eHHe, 1983.
2. B 0]1 K 0 B hI e R M H P. 10 ., T e M R 11 H a ,D;. A. OpraHH3aIJ;HH ,lJ;n<})(pe-
peHIJ;JipOBaHHoII pa60ThI yqa
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ll\eHHe
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4. E cP a III I( H Hr. B.. Po M 8 HOB C K 8 II H. H .. Tap a COB a A. H .
YQIlTeCb pemaTb 3
aqH 00 4>H3HKe / ilOA pe,lJ;. A.H. TapacoBoM. - M.: TIpo-
CBe
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5. K a 6 a P A 11 H O.
 .. K a 6 a p P; 11 II 8 C. If .. 0 p n 0 B B. A. 3a.n;ElHH.JI
.n;JIH HTorOBoro IWIITpOJUI 3RRHUH YQallll1XCH no 4>113HKe B 7 -11 KJI8CCaX cpeA-
HeM IIIROJIbI. - M.: IIpocBe
eHHe. 1995.
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7. K a 6 8. P lI. If H O. cI:J .. Ie a 6 a p P; If HaC. M ., 0 P .'1 0 B B. A. KOHT-
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1999.
8. JI y n n 0 B r.,D;. MOJIeKYJUlpHaH 4>naliKa II
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10. MeTO));HKB. c:)JBKYJIbTB.THBHbIX 38HHTHii no c:)JH3liKe / nO)); pe.o;.
o.(}). Ks6ap.zUiHB, B.A. OpJIOBa. - M.: npOcBe
eHHe, 1988.
11. HaY
HbIe OCHOBbI 1lIIWJIbHOro KypeR c:)JH3HKII / nO]]; pe.n,. C.H. IlIaMa-
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14. 0 p JI 0 B B. A. (})n:mKB B Ts6JJH
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15. IIpoBepKB Ii o
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19.CaYPoB IO.A.
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 1990.
21. (})H3UtleCKHH npBKTHKYM JJ;JIH KJIBCCOB C yrJIy6JIeHHbIM H3YQeHHeM c:)JH-
3HKH: 9-11 KJIBCCLI / no,ll; peA. 10.11. ,D;HK8, G.(}). Ra6BpJJ;HH8. - M.: IIpo-
CBe
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22. c1>poHTBJIbHble JI86opSTopHbIe pa60TbI no 4>H3HKe B 7-11 KJIaCCBX / ilOA
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23. III a x M B e B H. M. H]];p. cI:JH3HQecl(lIH 3RcnepHMeHT B Cpe));Hen IDKO-
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