Ответы (1115944), страница 2

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Ijbih^klZgh\d_bo\mjZ\g_gby^\b`_gbyihemqbfkbkl_fm^\mo ebg_cguo mjZ\g_gbc gZ ω1 b KI = :I/BI. :gZeh]bqgh fh`gh bkdZlv j_r_gby \ \b^_AIIcosω2t, –BIIcosω2t LZdbf h[jZahf ihemqbf mjZ\g_gby dhe_[Zgbc l_e \ kemqZ_ \ha[m`^_n ξ −nξξ − ξII, lhgbyfh^ihhl^_evghklb?kebξI = ξ1 + n1ξ2, a ξII = ξ1 + n2ξ2, lh ξ1 = 2 I 1 II ; ξ2 = In2 − n1n1 − n2_klvCI = – n2, CII = – n1. ω1, ω2 – ghjfZevgu_qZklhluDhe_[Zgbyfhe_dmeDhe_[Zl_evgu_kl_i_gbk\h[h^uLbiufhe_dmeyjguodhe_[Zgbc \Ze_glgu_ b ^_nhjfZpbhggu_ kbff_ljbqgu_ b Zglbkbff_ljbqgu_ GhjfZevgu_fh^ug_dhlhjuoijhkl_crbofhe_dme1) Lbiufhe_dmeyjguodhe_[Zgbc:<Ze_glgu_fhe_dmeyjgu_dhe_[Zgby±dhe_[ZgbyZlhfh\ijhbkoh^ysb_\^hevgZijZ\e_gbcobfbq_kdbo k\ya_c fh]ml [ulv kbff_ljbqgufb Zlhfu dhe_[exlky kbgnZagh ± \ha[m`^_gZ gbadhqZklhlgZy fh^Z b Zglbkbff_ljbqgufb Zlhfu dhe_[exlky \ ijhlb\hnZa_ ±\ha[m`^_gZ\ukhdhqZklhlgZyfh^Z<Ze_glgu_dhe_[Zgbyhibku\Zxlkyfh^_eyfbk\yaZgguohkpbeeylhjh\kf‹>_nhjfZpbhggu_dhe_[Zgby±baf_g_gb_nhjfufhe_dmeu[_abaf_g_gby^ebgk\ya_c2) Dhe_[Zl_evgu_kl_i_gbk\h[h^u:?kebfhe_dmeZkhklhblbaNZlhfh\lh^ey_zhibkZgbylj_[m_lkyN mjZ\g_gbcdZ`^uc Zlhf ij_^klZ\ey_lky dZd fZl_jbZevgZy lhqdZ bf_xsZy ljb ihklmiZl_evgu_ kl_i_gbk\h[h^uFhe_dmeZ\p_ehfbf__lljbihklmiZl_evgu_bljb^\_^eyebg_cghc\jZsZl_evgu_kl_i_gbk\h[h^uLh]^Zdhe_[ZgbyZlhfh\\fhe_dme_hibku\ZxlkyN – 6 (3N±mjZ\g_gbyfblh_klvfhe_dmeZbf__lN – 6 (3N – 5) dhe_[Zl_evguokl_i_g_ck\h[h^u.GZijbf_j fhe_dmeZ \h^u g_ebg_cgZ b khklhbl ba ljzo Zlhfh\ ihwlhfm hgZ bf__l dhe_[Zl_evgu_ kl_i_gb k\h[h^u >\_ ba gbo khhl\_lkl\mxl kbff_ljbqguf b Zglbkbff_ljbqguf\Ze_glgufdhe_[Zgbyf\^hevH±Gk\ya_cZlj_lvy±^_nhjfZpbhggufdhe_[Zgbyfbaf_g_gbx m]eZ G±H±G Fhe_dmeZ CO2 ebg_cgZ b khklhbl ba ljzo Zlhfh\ ihwlhfm hgZbf__ldhe_[Zl_evgu_kl_i_gb k\h[h^u >\_ ba gbo \gh\v khhl\_lkl\mxl \Ze_glguf kbff_ljbqgufbZglbkbff_ljbqgufdhe_[ZgbyfZ^\_^jm]b_±lZdgZau\Z_fuf\ujh`^_ggufbkdjb\e_gb_fhe_dmeu\^\mo\aZbfghi_ji_g^bdmeyjguoiehkdhklyoLZdb_\ujh`^_ggu_dhe_[Zgby y\eyxlky ^_nhjfZpbhggufb gh bf_xl bgh_ gZa\Zgb_ ihkdhevdm iehkdhklb jZ\ghijZ\gu b khhl\_lkl\mxsb_ bfdhe_[Zgbybf_xlh^bgZdh\mxwg_j]bx Keh`_gb_ ]Zjfhgbq_kdbo dhe_[Zgbc ijhbkoh^ysboih h^ghc hkb Ij_^klZ\e_gb_ h f_lh^_ \_dlhjguo ^bZ]jZffKeh`_gb_ \aZbfgh i_ji_g^bdmeyjguo dhe_[Zgbc h^bgZdh\hcqZklhluNb]mjuEbkkZ`m1) Keh`_gb_dhe_[Zgbcijhbkoh^ysbo\^hevh^ghchkb:>ey keh`_gby lZdbo dhe_[Zgbc bkihevamxl f_lh^ \_dlhjguo^bZ]jZffdhe_[Zgb_ij_^klZ\ey_lky\\b^_\_dlhjZfh^mevdhlhjh]h jZ\_g Zfieblm^_ \jZsZxs_]hky \hdjm] iheh`_gby7jZ\gh\_kby<wlhfkemqZ_ m]he ϕ±nZaZdhe_[ZgbcDhhj^bgZlZl_eZx = Acosϕ baf_gy_lkyih]Zjfhgbq_kdhfmaZdhgmbihkmlbkh\_jrZ_ldhe_[ZgbyBf_gghhgZbf__lnbabq_kdbckfuke<wlhfkemqZ_dhe_[Zgbyfh`ghkdeZ^u\ZlvdZd\_dlhjZZfieblm^?kebdhe_[Zgbydh]_j_glgulh_klvbojZaghklvnZaihklhyggZ\h\j_f_gblh^eyj_amevlbjmxs_]hdhe_[Zgbyξ = Acos(ωt + ϕ0) \_jgu ke_^mxsb_ aZ\bkbfhklb A2 = A12 + A22 + 2 A1 A2 cos (ϕ 2 − ϕ1 ) ;A1 sin ϕ1 + A2 sin ϕ 2.

?keb dhe_[Zgby g_dh]_j_glgu lh j_amevlbjmxsZy Zfieblm^ZA1 cos ϕ1 + A2 cos ϕ 2baf_gy_lky \h \j_f_gb ihwlhfm lj_[m_lky ijhba\h^blv hl^_evguc jZkqzl ^ey dZ`^h]h fhf_glZ\j_f_gbtg ϕ0 =2) Keh`_gb_\aZbfghi_ji_g^bdmeyjguodhe_[Zgbc:< wlhf kemqZ_ m^h[gh ijbgylv ijyfu_ \^hev dhlhjuo ijhbkoh^yl dhe_[Zgby aZ hkbdhhj^bgZl Lh]^Z j_amevlbjmxs__ dhe_[Zgb_ ± keh`_gb_ dhe_[Zgbc ^\mo dhhj^bgZl l_eZx = A1 cos (ω t + ϕ 01 ) , y = A2 cos (ω t + ϕ 02 ) ⇒ sin (ω t + ϕ 01 ) = 1 −x2y2,sin1;ωtϕ+=−()02A12A22cos (ω t + ϕ 01 ) cos (ω t + ϕ 02 ) + sin (ω t + ϕ 01 ) sin (ω t + ϕ 02 ) = cos (ϕ 02 − ϕ 01 ) =xyx2y2+ 1− 2 1− 2 ⇒A1 A2A1A2x 2 y 2 2 xycos ∆ϕ = 1 − cos 2 ∆ϕ ±mjZ\g_gb_weebikZIjhbkoh^blweebilbq_kdZyiheyjb+−A12 A22 A1 A2aZpbydhe_[Zgbc±ljZ_dlhjbyl_eZbf__l\b^weebikZIjbA1 = A2 b∆ϕ = (2m+1)π/2 weebik\ujh`^Z_lky\hdjm`ghklvpbjdmeyjgZyiheyjbaZpbyZijb∆ϕ = πm – \ijyfmx?kebqZklhlukdeZ^u\Z_fuodhe_[ZgbcjZaebqZxlky\p_eh_qbkeh jZa lh ljZ_dlhjbbl_eZ gZau\Zxlky nb]mjZfb EbkkZ`m hgb \ibkZgu \ ijyfhm]hevgbd kh klhjhgZfb A1, 2A2;wlbdjb\u_aZfdgmluZbonhjfZaZ\bkblhlkhhlghr_gbyqZklhlbjZaghklbnZakdeZ^u\Z_fuodhe_[Zgbc5.

AZlmoZxsb_dhe_[ZgbyHkpbeeylhjkg_[hevrbfaZlmoZgb_fQZklhlubi_jbh^ dhe_[Zgbc dhwnnbpb_gl aZlmoZgby \j_fy j_eZdkZpbb Zfieblm^u b wg_j]bb ^_dj_f_glbeh]Zjbnfbq_kdbc^_dj_f_gl^h[jhlghklvHkpbeeylhjk[hevrbfaZlmoZgb_fDjblbq_kdbcj_`bf1) Hkpbeeylhjkg_[hevrbfaZlmoZgb_f:AZlmoZxsb_dhe_[Zgby±dhe_[Zgbyijbdhlhjuobf_xlf_klhihl_jbwg_j]bbImklv gZ dhe_[Zl_evgmx kbkl_fm ^_ckl\m_l kbeZ lj_gby ijhihjpbhgZevgZy kdhjhklb<kemqZc\yadh]hlj_gby Flj = − r ξ . Lh]^Z\lhjhcaZdhgGvxlhgZfh`_l[ulvaZibkZg\\b^_<<<<<<m ξ = −kξ − r ξ ⇒ ξ + 2 β ξ + ω 02ξ = 0, ]^_β = r ω 02 = k oZjZdl_jbklbq_kdh_ mjZ\g_gb_mm22bf__l\b^ λ + 2βλ + ω 0 = 0. <kemqZ_fZeh]haZlmoZgby β < ω0, ihwlhfmj_r_gb_^bnn_j_gpbZevgh]hmjZ\g_gbyijbf_l\b^ ξ = A0 e− βt cos (ω s t + ϕ 0 ) , ]^_ω s = ω 02 − β 2 A^_kv ωs – kh[kl\_ggZyqZklhlZaZlmoZxsbodhe_[Zgbc; ω0±kh[kl\_ggZyqZklhlZg_aZlmoZxsbodhe_[Zgbc2πϕ0 – gZqZevgZynZaZ.

A = A0 e− βt – Zfieblm^Z aZlmoZxsbo dhe_[Zgbc. T =– i_jbh^ aZlmωsoZxsbodhe_[Zgbc.>ey we_dljbq_kdh]h dhe_[Zl_evgh]h dhglmjZ aZlmoZgb_ h[mkeh\e_gh gZebqb_f khijh8lb\e_gby gZ dhlhjhf \u^_ey_lky l_ieh l_jy_lky wg_j]by MjZ\g_gb_ aZibr_lky \ \b^_1qRR− L I = IR + ⇒ q + q +q = 0, lh_klv\wlhfkemqZ_β = CLCLL<<<<2) IZjZf_ljuaZlmoZgbydhe_[Zgbc:β – dhwnnbpb_glaZlmoZgby.<j_fyj_eZdkZpbbZfieblm^u (τ:±\j_fyaZdhlhjh_Zfieblm^Zmf_gvrZ_lky\ejZaA0e − βt1= e ⇒ eβτ A = e ⇒ τ A = .− β (t + τ A )βA0 e<j_fy j_eZdkZpbb wg_j]bb (τW) – \j_fy aZ dhlhjh_ wg_j]by kbkl_fu mf_gvrZ_lky \ _<2m ξ max1jZa W =~ A02e −2 β tω S2 ~ e −2 β t ⇒ τ W =.22βτ A ωS1==.T 2πβ β T>_dj_f_glaZlmoZgby (D)± qbkehjZa\dhlhjh_ mf_gvrZ_lkyZfieblm^Zdhe_[ZgbcaZA (t )i_jbh^ D == eβT .A (t + T )1Eh]Zjbnfbq_kdbc^_dj_f_glaZlmoZgby (γ): γ = ln D = βT =.Ne>h[jhlghklv (Q±iZjZf_ljoZjZdl_jbamxsbcmklhcqb\hklvdhe_[Zl_evghckbkl_fudaZlmoZgbx ijhihjpbhgZe_g kf hlghr_gbx wg_j]bb d m[ueb wg_j]bb aZ i_jbh^ωπ π ωSQ = π Ne = =; ijbfZehfaZlmoZgbb Q ≈ 0 .=γ βT 2β2βQbkehdhe_[ZgbcaZdhlhjh_Zfieblm^Zmf_gvrZ_lky\_jZaNe): N e =3) Hkpbeeylhjk[hevrbfaZlmoZgb_f:Ijb β > ω0 j_r_gb_ ^bnn_j_gpbZevgh]h mjZ\g_gbykhhl\_lkl\mxs_]h\lhjhfmaZdhgmGvxlhgZkfbf__l(bghc \b^ ξ = e − βt Ae]^_ τ1 =1β 2 −ω 02 tτ2 =+ Be− β 2 −ω02 t)= Ae− tτ1+ Be− tτ2,1.

IZjZf_lju:b<β − β 2 − ω 02β + β 2 + ω 02hij_^_eyxlky ba gZqZevguo mkeh\bc <j_fy \ha\jZs_gbykbkl_fu \ iheh`_gb_ jZ\gh\_kby hij_^_ey_lky wdkihg_glhckf_gvrbfihdZaZl_e_flh_klv τ1Ijb[hevrhfaZlmoZgbb τ1 fh`_l [ulv ^hklZlhqgh [hevrbf lh _klv \j_fyj_eZdkZpbbhdZ`_lky[hevrbfqlhg_`_eZl_evghijbjZ[hl_klj_ehqguoijb[hjh\,4) Djblbq_kdbcj_`bf:Ijb β = ω0 j_Zebam_lky lZd gZau\Z_fuc ©djblbq_kdbcª j_`bf dhe_[Zgbc hkpbeeylhjZ \ wlhf kemqZ_ j_r_gb_^bnn_j_gpbZevgh]hmjZ\g_gbybf__l\b^<ξ = e − βt ( A + Bt ). A = ξ(0); ijb: = 0 B = ξ(0); lh]^ZfZdkbfZevgh_ hldehg_gb_ ^hklb]Z_lky \ fhf_gl \j_f_gb /β bBkhklZ\ey_l ξ max =~ ξ (0).

Wlhk\hckl\hbkihevam_lky^eyβehij_^_e_gbybfimevkZihfZdkbfZevghfmhldehg_gbxklj_edb\[Zeebklbq_kdboijb[hjZo<95) Hkh[_gghklbaZlmoZxsbodhe_[Zgbc\kbkl_f_k\yaZgguohkpbeeylhjh\:JZagu_fh^uaZlmoZxlihjZaghfmIjbfZeuoaZlmoZgbyoghjfZevgu_fh^uhklZxlkyg_aZ\bkbfufbihwlhfmboqbkehjZ\ghqbkemkl_i_g_ck\h[h^udZdb^eykbkl_f[_aaZlmoZgbykf‹Ijb[hevrboaZlmoZgbyofh^ui_j_klZxl[ulvg_aZ\bkbfufb<ugm`^_ggu_]Zjfhgbq_kdb_dhe_[ZgbyAZ\bkbfhklbZfieblm^ubnZaumklZgh\b\rbokydhe_[ZgbcZlZd`_Zfieblm^ih]ehs_gbyb^bki_jkbbhlqZklhlu\ugm`^Zxs_c kbeu J_ahgZgku kf_s_gby b kdhjhklb Fhsghklv aZljZqb\Z_fZy gZ ih^^_j`Zgb_dhe_[Zgbc1) MklZgh\b\rb_ky\ugm`^_ggu_dhe_[Zgby:<ugm`^_ggu_ dhe_[Zgby ± dhe_[Zgby ijhbkoh^ysb_ ih^ ^_ckl\b_f i_j_f_gghc \g_rg_c kbeu ;m^_fjZkkfZljb\Zlv lhevdh kemqZc \g_rg_c kbeu baf_gyxs_ckyih]Zjfhgbq_kdhfmaZdhgmF = F0cosωt.Ih^^_ckl\b_f\g_rg_ckbeukbkl_fZ[m^_l\u\_^_gZbaiheh`_gbyjZ\gh\_kbyihwlhfm\g_c\ha[m^ylky kh[kl\_ggu_ k\h[h^gu_ dhe_[Zgby k qZklhlhc ωS;djhf_ wlh]h kbkl_fZ [m^_l dhe_[Zlvky k qZklhlhc ω\g_rg_ckbeuIhkl_i_gghk\h[h^gu_dhe_[ZgbyaZlmogml qZklhlZ dhe_[Zgbc klZg_l jZ\gZ ω ± lZdhc j_`bfdhe_[Zgbc gZau\Z_lky mklZgh\b\rbfky <lhjhc aZdhg<<<<<<GvxlhgZaZibr_lky\\b^_ m ξ = − kξ − r ξ + F0 cos ωt ⇒ ξ + 2β ξ + ω 02ξ = f 0 cos ωt , ]^_f0 = = F0/m.J_r_gb_ ^Zggh]h ^bnn_j_gpbZevgh]h mjZ\g_gby [m^_l khklhylv ba j_r_gby h^ghjh^gh]hmjZ\g_gby khhl\_lkl\mxs_]h k\h[h^guf aZlmoZxsbf dhe_[Zgbyf b qZklh]h j_r_gby dhlhjh_fh`_l[ulvgZc^_gh\ \b^_ ξ = A cos (ωt − α ).

Lh-π]^Z ξ = Aω cos  ω t − α +  , ξ = = Aω 2 cos (ω t − α + π ) .2<hkihevam_fky f_lh^hf \_dlhjguo ^bZ]jZff ih l_hj_f_IbnZ]hjZ2f0A2 (ω 02 − ω 2 ) + 4β 2ω 2 = f 02 ⇒ A =,22 22 2ωω4βω−+( 0 )<<<)(2βω.ω 02 − ω 2KhklZ\eyxsb_Zfieblm^uAP = Asinα bAD = AcosαgZau\Zxlky Zfieblm^Zfb ih]ehs_gby b ^bki_jkbb khf2 βωhl\_lkl\_ggh Ijb ω ω 0 A ≈ 02 , α ≈ 2 ; kbkl_fZω0ω0\_^zlk_[ydZdk\h[h^guchkpbeeylhjgZdhlhjuc^_ckl\m_l ihklhyggZy ih \_ebqbg_ kbeZ F0.

Ijbf2β ff2βω ω 0 A ≈ 02 → 0, ω → ∞; tgα ≈ − ; α → π ; AP ≈ 3 0 , AD ≈ − 02 .ωωωωf0AD ≈ − 2 ≈ − A.ωα = arctg2) J_ahgZgkuZfieblm^ubkdhjhklb:10:fieblm^Z \ugm`^_gguo dhe_[Zgbc fZdkbfZevgZ _keb agZq_gb_ (ω 02 − ω 2 ) + 4β 2ω 22fbgbfZevgh −4ω (ω 02 − ω 2 ) + 8β 2ω = 0 ⇒ ω p = ω 02 − 2β 2 = ω S2 − β 2 ± j_ahgZgkgZy qZklhlZAp ≈Ijbj_ahgZgk_tg α p ≈ω0π→ ∞ ⇒ αp ≈ .2βf0,2 βω Stg α p =ωpβ;ijbfZehf:fieblm^Zkdhjhklbhij_^_ey_lky\ujZ`_gb_f AV =aZlmoZgbbf0(ω02 − ω 2 )Ap ≈f0,2βω 0; fZdkbfmf2ω2+ 4β 2f^hklb]Z_lkyijbω = ω0. AVp = 0 .2β3) FhsghklvaZljZqb\Z_fZygZih^^_j`Zgb_dhe_[Zgbc:<F]gh\_ggZy fhsghklv fh`_l [ulv \uqbke_gZ dZd P(t ) = F (t ) ⋅ ξ(t ), lh _klv P(t ) =π 1ππ = F0 cos ωt ⋅ Aω cos  ωt − α +  = F0 Aω  cos  2ωt − α +  + cos  α −   .

Ihwlhfm kj_^g__2 222 T1agZq_gb_ fhsghklb aZljZqb\Z_fhc gZ ih^^_j`Zgb_ dhe_[Zgbc P(t ) = ∫ P(t )dt =T 011F0 Aω sin α = F0 APω ijhihjpbhgZevghZfieblm^_ih]ehs_gby227. Ehj_gp_\kdZy nhjfZ ebgbb ih]ehs_gby RbjbgZ j_ahgZgkghc djb\hc ih]ehs_gbyIylvhij_^_e_gbc^h[jhlghklbHkh[_gghklb\ugm`^_gguodhe_[Zgbc\kbkl_f_k\yaZgguohkpbeeylhjh\1) RbjbgZj_ahgZgkghcdjb\hcih]ehs_gby:2ω212 ω22 2 ω 2 ≈ 4ω 2 ω − ω 20ωωωωP(t ) = F0 APω ~,1−=−+()( 0 )()0022 ω (ω 2 − ω 2 ) + 4β 2ω 20\[ebab j_ahgZgkZ ihwlhfm P(t ) ~ R(ω ) =β2(ω 0 − ω )2+ β2. Nmgdpby R(ω) gZau\Z_lky ehj_g-p_\kdhc nhjfhc ebgbb b hibku\Z_l j_ahgZgkguc ibd R(ω) = 1 ijb ω = ω0; R(ω) = ½ ijbω − ω 0 = β. Ihwlhfm ihemrbjbgZ j_ahgZgkghc djb\hc lh _klv rbjbgZ ibdZ ijb agZq_gbbfhsghklb jZ\ghf iheh\bg_ fZdkbfZevgh]h ∆ω p ≈ 2β = 1/ τW , ]^_ τW – \j_fy j_eZdkZpbbwg_j]bbkf‹2) Iylvhij_^_e_gbc^h[jhlghklb:I_j\h_hij_^_e_gb_^h[jhlghklb±kf‹ τ W =^_e_gb_.

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