Oscillations - 7 7 fl W "W 73 OSCILLRTIONS...

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Unformatted text preview: 7 7 fl W "W 73 OSCILLRTIONS Sig?“ hagnic 0} gillgtg; Consider a; Particle. free to move in one JimenSiOn 03¢ a. Condition, of Stable. 02¢...“bl‘im {.Q. 4 Moves Pom pantie». (assume at lie ’ Y. *9) ha 4:!" a} Subject [’9 a tester-£013 force “.6 4.9m: an it: Jayne...“ [3(3) 3 —-kx “Ila M mum.“ Whoa I} “Caitlin? . ‘7 03219 7“. Mint! example (3‘ a. .' - X:0 ex‘) " - - - -‘ ‘ It. '3 E, -— k3! — mo. where k «3 skim .'. 41x :0")? . ‘M 71"} , a; course ’ ,(S o. Jifferent'o'a( ezval‘cbh, or x(t') = C cos[w.t’—fl) (or guy-hurl, C the! I) .7 we neeJ 2 guanth be specified al’ a. time , can“, lie. 03:17.01 Condition: , Ylo) cud Er maple , IT Yfl“) = Ace; NJ“ + 133.40.? Q04 = #4:». 3:». «At 43qu 71*” = A ’ S ’3”. E12333 {a Simple Harmonic flgh‘on If: we Specify X(o)= a tie” x0) = 20.") :: 3w, €0.3th 85c}! w,t' The. Potential 5‘99: lle Spat-.3 I} U = J.sz = ékB‘m’uJ : 1.0”)?" = 1.016 U.‘co:‘o,t 1 1. M to} = B. M - K = Jihs‘m‘ax 75' a t 75 ’7 fl; Phase. Slag; 713 mm... a; a. one —c(¢hven:¢‘am[ ascédcl'or t} Compldfil, Specified a: a. function of if [in (core! “1051;: are. , x “at AR (7700 cooreléde: are neealecl because it :3 a Qd-orcler al-‘Fferenh'a! Wish.) Me com (rounder Me rut-77¢: X("J am! )Plt) to be [Le Conn/Jolt: of a. Point- :3- 2" thumb-ml If.“ all“ plum M, known of an of time [ F" a. general ascetArw win: n degrees of freedom , “rev. pluue, Jpace z} a. 1n - dimensional Space .1 AS hm: Vow-£2: ’ Ila pom} P(x,)?) we'll Move along a anQCI‘on‘ «3.. PM: Spence which describe: “.9. Motion c; “a “and?” . 5:059:12!” t3 CQ‘IQJ a ghue Jib. mm. ye For our Simple Inwmonic oscillator we. had X(l') = ’3 SanJ' x“ Yr" = 3’1 + 61-22.? I . _I 2_ I 2 ‘ (remembenha Unl’ 5-3M _ image) (Mad. 1.} five gaffer: of an :34»: Correspoan (‘0 a. [77131 energg E. 0M» energies camped b one? claim: Noll [kt for “re €003?”th $33120» “lat NO chose , (la l’m‘eclbrg “:3 alum”: clockuée Lame. Far [9033759 A’ , X «(News 49.00.41.003. T? No 1’ we plvue [‘mjecl‘avies can on: . If “an: (:0le (at a [True [3 ) dud" 000 «H 3 that ’1. e Cahek’i'on: x03) , )?[a) CoulJ pruch «(0093 [Zoo <16]:me p.11: . 13...? flu} c3: imposst'UQ “cues! are “(Jinn a; [la ism... tail yd?“ é: (0.94:2 . 80 MPQJ OSCKIIQIT Simple humane}: Marion OVQV‘ *Sc'mpln'rhi: . 712 Wald‘bé (P‘y3‘éul) Case weak! :3. dude fwél'c’ong/ or «mafia force: “I‘ch woulcl e mutually Mace Me emu“ anal flue motion S'o “mt “dilation: Nil! no longer occur , 27m:th an em Am #5 Wu? rm? + b)? + lax = o we write Mn} 0.: §+2fli +woaX :0 when: fl = .3; is Me {wrigme Ile Mmih'c 04 (do = E unhaulu M’% (J. M absence of Jumping : [REMEnJeri W? C on Solving Jiffevoql‘iql QIUOh'onij 1'73 «1 Solution 09 Me. form xw = Ce" {(0') = Cre M )?(t') = Cr’e't Substctulb Lita Me Jiffenenkol EIuafion 3) Crier, + ilfl Crew + w.‘ce"' =' a ir‘+M¢+a.‘ :0 , 1 1k general sciatica is do. t JET? " “4" t m) =- e’” (4e *Az¢ The“ line Soluh'oa is “fit. . at. ". I xlt) : e (/hem + A,e w t) M a.“ = COS «ht - (Sin 0),? M, “muse Y0) In: to be rec(,we on» MR “'9 Solution 0.: x0) = (“’74 Car qt 4- Bakqt’) (3' at “'1 Maximum QXQD-n'on My. x(a-) = e'm'fl co: m“ i i if if ‘ 7 i i i x 33 Wummd Motion («)3 < /3 ) New Um S’um roof mod N: chcne an)a s 103‘— w:- ahel [1.2 Solul‘éon L} .. 4.1.6- Xa’) = e flt[A¢e%b+ 413 J Not! because wz<fl So 52g. [Ems we "flue Specéfd: beLaer depencl: a» 11¢ dih‘al Conclih'ans (Me relative Sign A. Mel A1). C-ril’icdlu Dmfecl Notion (of =/3‘) So solutim is xa') =A «.V/3 I5 “is tin cal, Saint-a... 3’ anal-1 I muslin? , 14 , but 2 ¢;.;(;;,( Conditio»: X (0) and A: (0) ~ [‘2’ “is M a den water “(when xm = Bte‘flw aéo works [-395 check...” 3:“) 6' O '3 (e'm- fibe‘flb) Substituting => t. J 66% +/3‘t)e""+ m3(e"’,ate‘”’)+fl18te :o “ding B and 8-”: Common "0 0b“ (3'53 3) 4.43:: +2/x-2/3‘t +fi‘t- :0 0K / gm __.2 F_:D_9_MP_9~‘_9 _’Z§_Ipon:e s oVQ'C‘cmPQC‘ Criticdlj Jmptd uncle'- clam/mi ...
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Oscillations - 7 7 fl W &amp;quot;W 73 OSCILLRTIONS...

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