ࡱ> Lq`bbjbjqPqP::8dddd@@@ ...8L./ld2/~0" 1 11a?a?a?ccccccc$ehRhcu@a?}=a?a?a?cdd 11 rdKKKa?rdV 1@1cKa?cKK_@a1t/ Mڋ.@7`cd0dM`6iE:i,ai@a@a?a?Ka?a?a?a?a?ccHa?a?a?da?a?a?a?d..Td@dddddd Soustavy rovnic Clem tto kapitoly je procvi it prci svce rovnicemi vyu~vanmi vtechnick praxi azdoraznit vznam matematiky pYi Yeaen technickch loh. V rovnicch se vyskytuj konkrtn symboly a na rozdl od b~nch matematickch rovnic se neznm nenazv x. V nsledujcch lohch pojde o soustavy dvou nebo tY rovnic, pYi jejich Yeaen budeme postupn dosazovat zjedn rovnice do dala rovnice a postupnmi pravami vyjadYovat hledanou neznmou. Xeaen pYklady  soustava dvou rovnic PYklad 1 Vypo tejte promr ty e kruhovho proYezu d, mte-li kdispozici dv nsledujc rovnice. Pro vpo et promru d pYedpokldejte, ~e hodnoty vaech ostatnch veli in znte. Rovnice [1]  EMBED Equation.3  Rovnice [2]:  EMBED Equation.3 . Xeaen: PYi vpo tu budeme vychzet zrovnice [1], hledme vaak promr d, kter se vyskytuje vrovnici [2]. Proto za hodnotu Svrovnici [1] dosadme Szrovnice [2]. Obdr~me:  Odstranme slo~en zlomek, zskme:  Celou rovnici vynsobme d2 a vydlme Dt, zskme:  Odmocnnm zskme vsledn vztah:  Zvr: Jestli~e dosadme do vslednho vztahu F v [N] a Dt v [MPa], bude vsledn promr d v [mm]. PYklad 2 Vypo tejte promr hYdele kruhovho proYezu d podle obrzku, mte-li kdispozici dv nsledujc rovnice. Pro vpo et promru d pYedpokldejte, ~e hodnoty vaech ostatnch veli in znte. Rovnice [1]:  EMBED Equation.3  Rovnice [2]:  EMBED Equation.3 .  Xeaen: PYi vpo tu budeme vychzet zrovnice [1], hledme vaak promr d, kter se vyskytuje vrovnici [2]. Proto za hodnotu Wkvrovnici [1] dosadme Wk zrovnice [2]. Obdr~me:  Odstranme slo~en zlomek, zskme:  Celou rovnici vynsobme d3 a vydlme Dk, zskme:  Odmocnnm zskme vsledn vztah:  Zvr: Jestli~e dosadme do vslednho vztahu Mk v [Nmm] a Dk v [MPa], bude vsledn promr d v [mm]. NeYeaen pYklady PYklad 1 Vypo tejte promr hYdele kruhovho proYezu d podle obrzku, mte-li kdispozici dv nsledujc rovnice. Pro vpo et promru d pYedpokldejte, ~e hodnoty vaech ostatnch veli in znte. Rovnice [1]:  EMBED Equation.3  Rovnice [2]:  EMBED Equation.3   Vsledek: PYklad 2 Vypo tejte vnitYn promr dutho hYdele mezikruhovho proYezu d, mte-li kdispozici dv nsledujc rovnice. Pro vpo et promru d pYedpokldejte, ~e hodnoty vaech ostatnch veli in znte. Rovnice [1]:  EMBED Equation.3  Rovnice [2]:  EMBED Equation.3 .   Vsledek: PYklad 3 Vypo tejte velikost strany ty e tvercovho proYezu a, mte-li kdispozici dv nsledujc rovnice. Pro vpo et velikosti strany a pYedpokldejte, ~e hodnoty vaech ostatnch veli in znte. Rovnice[1]  EMBED Equation.3  Rovnice [2]:  EMBED Equation.3 .   Vsledek: Xeaen pYklady  soustava tY rovnic PYklad 1 Vypo tejte velikost sly F vyYeaenm soustavy tY rovnic. Pro vpo et sly F pYedpokldejte, ~e hodnoty m, g, f a  znte. Rovnice [1] F.cos   FT = 0; Rovnice [2] F.sin  + FN - m.g = 0; Rovnice [3] FT = FN.f. Xeaen: PYi vpo tu budeme vychzet zrovnice [1], jsou vn vaak 2 neznm  F a FT, proto za hodnotu FTvrovnici [1] dosadme FT zrovnice [3]. Obdr~me: F.cos   FN .f = 0 Vzskan rovnici jsou vaak opt dv neznm  F a FN, proto za hodnotu FN dosadme FN vyjdYenou zrovnice [2]: FN = m.g - F.sin . Obdr~me: F.cos   [(m.g - F.sin ).f] = 0 Vrovnici je ji~ jen jedna neznm  hledan sla F, kterou postupn vyjdYme. 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"""&"("*","^"`"b"d"f""""""""""""""T####$J&J(JFJPJJJ⵱}{}}Uhc hc 6hc h`}6hh`}6 h6hV#hh6H*hh6hh`}hm ehx6 hx6H*ht/hx6hxhx6H*hxhx6hxhht/6H*hht/6ht/h/"0.f + F.f.sin  = 0 PYevedeme len m.g.f na pravou stranu rovnice: F.cos  + F.f.sin  = m.g.f Na lev stran rovnice vytkneme neznmou F: F.(cos  + f.sin ) = m.g.f Vydlenm cel rovnice lenem vzvorce obdr~me vsledn vztah:  EMBED Equation.3  Zvr: Jestli~e dosadme do vslednho vztahu vzkladnch jednotkch, bude vsledn sla F v [N]. PYklad 2 Vypo tejte velikost sly F vyYeaenm soustavy tY rovnic. Pro vpo et sly F pYedpokldejte, ~e hodnoty m, g, f a  znte. Rovnice [1] F.cos   m.g sin  - FT = 0; Rovnice [2] FN - F.sin  - m.g cos  = 0; Rovnice [3] FT = FN.f. Xeaen: PYi vpo tu budeme vychzet zrovnice [1], jsou vn vaak 2 neznm  F a FT, proto za hodnotu FTvrovnici [1] dosadme FT zrovnice [3]. Obdr~me: F.cos   m.g.sin  - FN.f = 0 Vzskan rovnici jsou vaak opt dv neznm  F a FN, proto za hodnotu FN dosadme FN vyjdYenou zrovnice [2]: FN = m.g.cos  + F.sin . Obdr~me: F.cos   m.g.sin  - [(m.g.cos  + F.sin ).f] = 0 Vrovnici je ji~ jen jedna neznm  hledan sla F, kterou postupn vyjdYme. NejdYve odstranme zvorky a zskme: F.cos   m.g.sin  - m.g.f.cos  - F.f.sin  = 0 PYevedeme leny sm.g na pravou stranu rovnice: F.cos  - F.f.sin  = m.g.sin  + m.g.f.cos  Na lev stran rovnice vytkneme F, na prav stran m.g, obdr~me: F.(cos  - f.sin ) = m.g.(sin  + f.cos ) Vydlenm cel rovnice lenem vzvorce na lev stran rovnice obdr~me vsledn vztah:  EMBED Equation.3  Zvr: Jestli~e dosadme do vslednho vztahu vzkladnch jednotkch, bude vsledn sla F v [N]. NeYeaen pYklady PYklad 1 Vypo tejte velikost sly F vyYeaenm soustavy tY rovnic. Pro vpo et sly F pYedpokldejte, ~e hodnoty m, g, f a  znte. Rovnice [1] F.cos   FT = 0; Rovnice [2] FN - F.sin  - m.g = 0; Rovnice [3] FT = FN.f. Vsledek:  EMBED Equation.3  PYklad 2 Vypo tejte velikost sly F vyYeaenm soustavy tY rovnic. Pro vpo et sly F pYedpokldejte, ~e hodnoty m, g, f a  znte. Rovnice [1] F.cos   m.g sin  + FT = 0; Rovnice [2] FN - F.sin  - m.g cos  = 0; Rovnice [3] FT = FN.f. Vsledek:  EMBED Equation.3  PYklad 3 Vypo tejte velikost ramene sly l vyYeaenm soustavy tY rovnic. Pro vpo et pYedpokldejte, ~e hodnoty b, h a Do znte. Rovnice [1]  EMBED Equation.3 ; Rovnice [2] Mo = F.l ; Rovnice [3]  EMBED Equation.3 ; Vsledek:  EMBED Equation.3  PYklad 4 Vypo tejte velikost strany proYezu h vyYeaenm soustavy tY rovnic. Pro vpo et pYedpokldejte, ~e hodnoty Mo a Do znte. Rovnice [1]  EMBED Equation.3 ; Rovnice [2]  EMBED Equation.3 ; Rovnice [3]  EMBED Equation.3 . 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