Introductory Physics I
Introductory Physics I PHY 231
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This 43 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 231 at Michigan State University taught by Remco Zegers in Fall. Since its upload, it has received 22 views. For similar materials see /class/207625/phy-231-michigan-state-university in Physics 2 at Michigan State University.
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Date Created: 09/19/15
PHYSICS 231 Chapter 13 Oscillations Remco Zeger39s PHY 231 Hooke39s law I F5 3 a W I Fskx Hooke39s law I X baa pl 0 F 0 If There is no fricTion The mass b WWW x nues To oscnIIaTe back and 0 F x S gt C 39 WW 39 39 x x 2003ThomsonBmukslcole x 0 If a force is proporTionaI To The displacemenT x buT opposiTe in direcTion The resuITing moTion of The objecT is called simple harmonic osciIIaTion PHY 231 Simple harmonic motion displacemen r x a WWWquot Ampli rude A maximum dis rance from uh equilibrium uni139 m Per39iod T Time To comple re one full FFO b Mac oscilla rion Uhi l39i 5 xio Frequency 1 Number39 of comple red C I oscilla rions per39 second 39 x uni r 1s 1 Her39z Hz mmmmm x0 f1T PH 231 3 Simple harmonic moTion displacemenT x a whaT is The ampIiTude of The harmonic osciIIaTion b whaT is The period of The harmonic osciIIaTion c whaT is The frequency of The harmonic oscillaTion PHY 231 The spring consTanT R Q 2003 Thomson BrooksCole When The objecT hanging from The spring is noT moving F5 39 39 Fspring 39FgraviTy d kd mg k mgd k is a consTanT so if we mg hang Twice The amounT a b c of mass from The spring d becomes Twice larger k2mg2dmgd PHY 231 displacement vs acceleration displacement x New ron39s second law Fma gt kxma gt akx m accele a riona PHY 231 quesTion A mass is oscillaTion horizonTally while aTTached To a spring wiTh spring consTanT k Which of The following is True a When The magniTude of The displacemenT is largesT The magniTude of The acceleraTion is also largesT b When The displacemenT is posiTive The acceleraTion is also posiTive c When The displacemenT is zero The acceleraTion is nonzero PHY 231 example A mass of 1 kg is hung from a spring The spring s rre rches by 05 m Nex r The spring is placed horizon rally and fixed on one side To The wall The same mass is a r rached and The spring s rre rched by 02 m and Then released Wha r is The accelera rion upon release PHY 231 energy and velocity Ekinmvz EpoTspringkx2 Sum O kA2 kA2 b mv2 O mv2 C O kA2 kA2 2003 Thomson BrooksCo x 0 conservation of ME mvx02lkA2 so vx01Akm 9 PHY 231 velocity more general Total ME at any displacement x Total ME at max displacement A Conservation of ME kA2mv2kx2 So v r AZx2k m mv2kx2 kA2 position X velocity V acceleration a A O kAm o AVkm o A O kAm PHY 231 Generally also add gravitational PE MEKEPE spring PEgr aviTy mv2 kx2 mgh 12 PHY 231 An example A 04 kg objecT connecTed To a lighT spring wiTh a spring consTanT of 196 Nm oscillaTes on a fricTionless horizonTal surface If The spring is compressed by 004 and Then released deTermine a The maximum speed of The objecT b The speed of The objecT when The spring is compressed by 0015 m c when iT is sTreTched by 0015m d for whaT value of x does The speed equal one half of The maximum speed 13 PHY 231 PHY 231 2 circular motion amp simple harmonic motion A par ricle moves in a circular orbi r wi rh angular veloci ry 0 corresponding To a linear veloci ry v003r03A gt 39 The horizon ral posi rion as a func rion c of Time X 39ACOSGACOS03 I39 903 l xl a x i I The horizon ral veloci ry as a func rlon of Time sin9vXv0 vx rvosin903Asinoa r Time To comple re one circle Ie one period T X T27cAvo27cA03A27c03 to 032nT275f 1 frequency 0 an ular fre uenc 1 PHY 231 9 q y 5 Circular motion and simple harmonic motion The simple harmonic mo rion can be described by The projec rion of circular mo rion on The horizon ral axis xharmonicTACOS0T VharmonicTO AS39 03 where A is The ampli rude of The oscilla rion and 0327cT227cf where T is The period of The harmonic mo rion and f1T The frequency 16 PHY 231 For The case of a spring posiTion X velociTy V acceler a rion a A O kAm 0 AVkm 0 A O kAm 1 veloci ry is maximum if viAkm 2 circular moTion v ptmg r03Asinoa r maximal if Vsprmg103A combine 1 Si 2 cojkm Acceler a rion a rkAmcosoa roa2Acosoa r PHY 231 17 xhar39moniccr Acos 39 co2nfznT lkm quotquotquotquot vhar39moniccr 39 DASin Dr gt ahar39moniccr quotDZACOS03139 39PFW7339139 39 39 39 Example A mass of 02 kg is aTTached To a spring wiTh k100 Nm The spring is sTreTched over 01 m and released a WhaT is The angular frequency 0 of The corresponding circular moTion b WhaT is The period T of The harmonic moTion c WhaT is The frequency f d WhaT are The funcTions for xv and T of The mass as a funcTion of Time Make a skeTch of These 19 PHY 231 quesTion An objecT is aTTached on The lhs and rhs by a spring wiTh The same spring consTanTs and oscillaTing harmonically Which of The following is NOT True a In The cenTral posiTion The velociTy is maximal b In The mosT lhs or rhs posiTion The magniTude of The acceleraTion is largesT c The acceleraTion is always direcTed so Th aT iT counTeracTs The velociTy d in The absence of fricTional forces The objecT will oscillaTe forever e The velociTy is zero aT The mosT lhs and rhs posiTions of The objecT 20 PHY 231 Period T 6 3 2 1 Frequency 1 16 1 3 n mk 627c 3275 2275 3 0 2706 2703 2702 f a k 21 PHY 231 PHY 231 2 AnoTher39 simple harmonic oscillaTion The pendulum ResTor39ing for39ce Fmgsin9 The force pushes The mass m back To The cenTr aI posiTion L sin99 if 9 is small lt150 radians Fmge also 9sL 9Ton9sL so FmgLs mg cos 0 2003 Thomson BrooksCole 23 PHY 231 2003 Thomso A A A A A A A A A A A A A AA pendulum vs spring Vlw ww A A n A A A A m A A A n A n A A A A parame rer spring pendulum resToring Fkx FzOngLJs force F z period T T 27cmk TzZmLg frequency f fkm27c fgL27c angular ookm oogL frequency l BrooksCole A PHY 231 T27r m 27 mgL g 24 example a pendulum clock The machinery in a pendulum clock is kepT in moTion by The swinging pendulum Does The clock run fasTer aT The same speed or slower if a The mass is hung higher b The mass is replaced by a heavier mass c The clock is broughT To The moon d The clock is puT in an upward acceleraTing elevaTor Ll ml moon elevaTor fasTer T 272Z same 8 slower PHY 231 example The height of The lecture room demo T27rZ 8 T2 L g 2 025T2 7239 26 PHY 231 27 PHY 231 Jug cas example A pendulum wiTh a lengTh of 4 m and a swinging mass of 1 kg oscillaTes wiTh an maximum angle of 10 WhaT is The graviTaTional force parallel To The sTring perpendicular To The sTring The ToTal graviTaTional force and The cenTripeTal force when The mass passes Through The equilibrium posiTion and when iT reaches iTs maximum ampliTude 28 PHY 231 The wave carries rhe dis rur39bancem hot The wafer posi rion y KR m i l l J l Each poin r makes a simple harmonic ver rical oscilla rion 29 PHY 231 Types of waves WGV a o 39 oo 390 z a neg 39 1 395 a 353m 3 59M quot 0 O at a g 390 39 o 351 iigi39zquot awe3 OSCquot39 39 239L quotEZ39Erz s a39r393 39quotquot339f gyr aw w Tronsversol movemenT is perpendicular To The wave moTion a o 0 C 3 a 39 w39g f 3o o o o 1 I 5 P o 39 4 quot39 quot quot 35133 939 3quot 5 39 oscilloTion O 0 l 393 4quot LongiTudinol movemenT is in The dir39ecTion of The wave moTion 30 PHY 231 A single pulse gtP veloci ryv gtl Time To Time T1 muNhf gt x0 x1 J gt vX139Xo 1391391390 A wwwiq a 2003 Thomson BrooksCole 31 PHY 231 A mm pulsar is rming as illhllsminlciL with unifnrm spasm v alnng a mph Whixsl39fl ml hug graphg 11 4 huluw mmcllly alum395 hm whi mm mm1331 the L isp ahmmm 5 nl39 puinl P and lith I P 139 s S A T A C 139 s Ms f xf B D PHY 231 describing a Traveling wave Pl 7 waveleng rh l 39 F 5 p pk gp ff dls rance be rween l wo mamma Vilnaling 1 blade g A Rg x x LikedTX V39l l 1 if I El While The wave has rr39aveled one y l wavelength each poin rl on The r39ope Ex lm ll has made one period of oscilla fion K M amp ll vAxA1MT hf 1 l atlng blade 33 PHY 231 exmnle w A Traveling wave is seen To have a horizontal dis mnce f 393 3 3 L a L 1 of 12 bpe rwee n a maximum I ix in and The nearfe sf mini and 1 V 39iv39er Tical heigh r of If W I 12 1 1 CI I lt 7 moves wi lm whra r is its a amplitude b period cf frequency gtAA I J W V V 734 PHY 231 80 WGVBS An anchored fishing boaT is going up and down wiTh The waves IT reaches a maximum heighT every 5 seconds and a person on The boaT sees ThaT while reaching a maximum The previous wave has moved abouT 40 m away from The boaT WhaT is The speed of The Traveling waves 35 PHY 231 Interference Two Traveling waves pass Through each o rher wi rhou r affec ring each o rher The resul ring displacemen r is The superposi rion of The Two individual waves example rwo pulses on a s rring rha r mee r PHY 231 36 Interference II gt a gt quotu MFXIM e e harm cons rr39uc rive i n rer39fer39ence desfrucIive inferference PHY 231 demo i n rer39fer39enceJ I I Interferen e III W 42 I I I I I I 1 l a I I l b i i b I I C 2003 Thomson BrooksCole 2003 ThOITISOH BfOOkSCole cons rr39uc rive in rer39fer39ence deSTr39UCTiVB ih fer39fer ehce waves in phase waves 7 DUI of phase PHY 231 38 Interference IV pm AAAD 39 Two i n rer39fer39ing waves can If The TWO infer39fer39ing a r rimes constructively waves alw f ys We The quotWarfare and GT Times same ver39hcal displacement a r any poi n r along The waves bu r are of opposi re sign s randing waves des rr39uc rively i n rer39fer39e 39 PHY 231 Interference holds for any wave Type D The pulses can be sinewaves r ec rangular waves or39 Triangular waves PHY 231 40 Interference in spherical waves maximum of wave r 5 minimum of wave 1 2 R4 7 1 ffg a posi rive cons rr39uc rive i n rer39fer39ence Q nega rive cons rr39uc rive in rer39fer39ence 0 destructive interference if r39Zr391nk Then cons rr39uc rive i n rer39fer39ence occurs if r39Zr391nk The destructive interference occur gi1 PHY 231 Inferference of water waves PHY 231 Example Two speakers separaTed by 07m produce a sound wiTh frequency 690 Hz from 0 7m The same sound sysTem A person sTarTs walking from one of The speakers perpendicular To The line direcTion of connecTing The speakers AfTer whaT walking person disTance does he reach The firsT maximum And The firsT minimum Vsound343 15 43 PHY 231
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