Chapter 10. Simple Harmonic Motion and Elasticity. Goals for Chapter 10


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1 Chper 0 Siple Hronic Moion nd Elsiciy Gols or Chper 0 o ollow periodic oion o sudy o siple hronic oion. o sole equions o siple hronic oion. o use he pendulu s prooypicl syse undergoing siple hronic oion. o sudy how oscillions y be dped or drien. o sudy sress, srin nd elsic deorion. o deine elsiciy nd plsiciy. Periodic Moion he Idel Spring nd Siple Hronic Moion Periodic oion or oscillions: n objec repes he se oion oer nd oer; e.g. rising o he sun, roion o bicycle wheel, ss on spring,... F pplied spring consn Unis: N/ Eple: ire Pressure Guge he spring consn o he spring is 30 N/ nd he br indicor eends.0 c. Wh orce does he ir in he ire pply o he spring? Hooe s Lw: Resoring Force o n Idel Spring he resoring orce o n idel spring is F F pplied ( 30 N )( 0.00 ) 6.4 N
2 Siple Hronic Moion nd Unior Circulr Moion Connecion beween Roionl nd Oscillory Moion bll is ched o he ri o urnble o rdius he ocus is on he shdow h he bll css on he screen When he urnble roes wih consn ngulr speed, he shdow oes in siple hronic oion φ Y oion y R sin φ X oion R cos φ Siple Hronic Moion nd he Reerence Circle Siple Hronic Moion nd he Reerence Circle DISPLCEMEN ( ) cosθ cos θ pliude : he iu displceen period : he ie required o coplee one cycle requency : he nuber o cycles per second (esured in Hz) π π pliude pliude, he pliude is he iu posiion o he objec relie o he equilibriu posiion In he bsence o ricion, n objec in siple hronic oion will oscille beween ± on ech side o he equilibriu posiion Period nd Frequency he period,, is he ie h i es or he objec o coplee one coplee cycle o oion Fro o  nd bc o he requency, ƒ, is he nuber o coplee cycles or ibrions per uni ie
3 Eple Velociy he grph shown in he igure closely pproies he displceen o uning or s uncion o ie s i is plying single noe. Wh re () he pliude, ( 0.4) θ θ V R π sin θ sin { sin (b) period, (c) requency ( s) ( / /(s) 500 Hz), Eple: Miu Speed o Loudspeer Diphrg he requency o oion is.0 KHz nd he pliude is 0.0. Wh is he iu speed o he diphrg? Where in he oion does his iu speed occur? sin { sin θ ( π ) 3 3 ( )( π)(.0 0 Hz).3 s he iu speed occurs idwy beween he ends o is oion. ccelerion θ θ c cosθ c cos R R cos cos, Eple: Miu ccelerion o Loudspeer Diphrg he requency o oion is.0 KHz nd he pliude is 0.0. Wh is he iu ccelerion o he diphrg? cos { cos c θ ( π ) 3 3 ( )( π) (.0 0 Hz) s huge! Siple Hronic Moion (SHM) For SHM: Resoring orce is direcly proporionl o he displceen ro equilibriu Hooe s lw orce he orce lwys cs owrd he equilibriu posiion I is clled he resoring orce he direcion o he resoring orce is such h he objec is being eiher pushed or pulled owrd he equilibriu posiion Spring  ss syse eple or Hooe s lw nd siple hronic oion
4 Frequency o Vibrion cos F cos Oscillion requency o ss spring syse he orce is gien by Hooe s Lw F   / ccelerion or siple hronic oion πƒπ Frequency Unis re cycles/second or Herz, Hz Period his gies he ie required or n objec o ss ched o spring o consn o coplee one cycle o is oion ƒ ngulr requency π π Eple: Body Mss Mesureen Deice Gien nd, ind sro ol ol + ol ( π ) chir sro π π he deice consiss o springouned chir in which he sronu sis. he spring hs spring consn o 606 N/ nd he ss o he chir is.0 g. he esured period is.4 s. Find he ss o he sronu. sro ( π ) chir ( 606 N )(.4s) 4π.0 g 77. g Sury: Siple Hronic Oscillor () Iniil Posiion nd Phse Shi: cos sin he os generl soluion is cos( + φ) where pliude requency φ phse cos
5 Eple ss g on spring oscilles wih pliude 0 c. 0 is speed is iu, nd is + /s. Wh is he ngulr requency o oscillion? Wh is he spring consn? Energy nd Siple Hronic Moion copressed spring cn do wor. W elsic ( F cosθ)s o ( + ) cos0 ( ) o o MX s MX 0s 0c lso: W elsic o So ( g) (0 s  ) 800 g/s 800 N/ Energy nd Siple Hronic Moion DEFINIION OF ELSIC POENIL ENERGY he elsic poenil energy is he energy h spring hs by irue o being sreched or copressed. For n idel spring, he elsic poenil energy is PE elsic SI Uni o Elsic Poenil Energy: joule (J) Eple: Lnding je on n ircr crrier ssue no chnge in griionl poenil energy Ny je o ss 0,000 g lnds on n ircr crrier nd sngs cble o slow i down. he cble is ched o spring wih spring consn 40,000 N/ I he spring sreches 5 o sop he plne, wh ws he lnding speed o he plne? Ki + Ui K + U i, i 50 / s. 40,000 0,000 5 Concepul Eple re Shorer Spring sier Spring? 0coil spring hs spring consn. I he spring is cu in hl, so here re wo 5coil springs, wh is he spring consn o ech o he sller springs? Finl coers up o pge 96.
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