Phys 331: Ch 9,.6-.7 Noninertial Frames: Centrifugal and Corriolis forces 1

Transcription

1 Phs : Ch Nnneral Frames: Cenrfugal and Crrls frces Wed /8 Thurs /9 Fr /0 Mn / Tues /4 Wed /5 Thurs /6 Fr / Free Fall & Crls Fucaul Pendulum 0- Cener f Mass & Ran abu a Fed As 0-4 Ran abu an As Inera Tensr Prncple Aes 05-6 Fndng Prncple Aes Preces 07-8 Euler s Equans Mn /0 Revew fr Fnal Prjec HW9c (95 97) HW0a (06-) HW0b (06 09) Equpmen: Glbe Ball wh crdnae aes Turnable wh paper aped Pendulum spnnng n urn able Smulan fr HW 4 Nn-neral Frames: Rang Las me we learned ha when ne frame s rang relave he her sa he Earh relave he fed sars hen velc and acceleran measuremens made n he w frames are relaed b r r V (r) f where ( r) r e V r ) r ˆ V f f ( as And r r Af ( r) ACrr ( r ) where Af Acen r r as rˆ as and A r crr S S O

2 Of curse Newn s nd Law apples nl n an neral frame Fne r m S r F ne m A ( r ) A ( r ) f Crr Fcus Ineral / Frame Frce mr F ne F frame Fframe m Af ( r) ACrr ( r ) Fcenrpeal Fcrls where F ma m r mr rˆ and F cen cen as as crr ma crr m r Tda and mrrw we ll lk a sme effecs f he cenrfugal frce: F cf m r and he Crls frce: F mr cr Hw can we bserve an effec f he Crls frce n he mn f an bjec near he earh? (We alread cnsdered hurrcanes bu he are cmplcaed ssems f parcles) Our calculans wll be dne fr he nrhern hemsphere Bh Frces: T ge a lle pracce wh a relavel smple suan le s cnsder he mn f a frcnless puck n a hrnal rang urnable Cmpared he spnnng f he urnable n s wn as he spnnng f he rm (sng n he face f a spnnng Earh) s neglgble s we ll rea he rm as an neral frame Of curse n he neral frame he puck wll smpl mve n a sragh lne because here s n ne frce A (nnneral) rang bserver ma bserve mre cmplcaed mns whch wll be eplaned b he cenrfugal and he Crls frces

3 Eample #: Prb 90 (backgrund fr 94) Suppse a frcnless puck mves n a hrnal urnable rang cunerclckwse (vewed frm abve) a an angular speed Wre dwn he equans f mn fr he puck n he rang ssem f he puck sars a an nal psn r 0 wh an nal velc v v v as measured n he rang frame Ignre Earh s ran! In an neral frame here s n ne frce n he puck s r 0 and we d see he puck mvng wh cnsan velc / n a sragh lne m F ne Hw wuld lk a lle bug rddng n he urnable? In ha nnneral frame ha raes wh he urnable Newn s secnd law s: mr F F m r mr cf cr Takng he angular velc f he urnable be 00 he psn s r 0 Calculae he crss prducs: ˆ ˆ I d: r de 0 0 The d: r de ˆ 0 ˆ ˆ ˆ ˆ ˆ ˆ The d: r de S he equan f mn gves (dvdng u he mass): r he equans fr he and cmpnens are: and A rck fr slvng bh f hese cupled dfferenal equans a he same me (see Sec 7) s defne Cmple nan: We dd smehng lke hs back when we were dealng wh damped drven harmnc scllars In hs case he nerprean s even smpler: hs s essenall hand nan fr he -D vecr where he plas he rle f -ha A he end f he

4 da we ll be able break ur slun back apar n he and cmpnens b ug hs fac If we add mes he -equan he -equan we ge: Ths lks an awful l lke he damped harmnc scllar (asde frm ha facr f and he lack f a negave sgn n he lnear erm) S we can guess he basc frm f he slun ha we d guessed n ha case Snce hs s a lnear dfferenal equan guess he slun aular equan: The d: 0 e whch gves he Ths mples ha There s nl ne slun fr s we need a secnd slun (he dfferenal equan s secnd rder) Ths sunds a l lke he prblem wh he crcall damped smple harmnc scllar S we ve g a gd chance ha a smlar slun wll wrk Jus as wh crcal dampng u can check ha n addn e e s a slun s he general slun s: where he ceffcens ma be cmple e C C Impse Inal Cndns r 00 and r v v 0 r and v v The frs cndn mples ha and he dervave s: S he secnd cndn mples: C e C Ce 0 C v v C v v Ths gves: e v v cs v v The real par f s and he magnar par s whch gves (Eq 97): 4

5 The d: v cs v v v Yu wll eplre (cmpuanall) he behavr f he mn fr dfferen nal velces n he hmewrk (Prb 94) Free Fall: We wll use he crdnae aes and wh he rgn n he surface f he earh a he claude (belw n he lef) Thse crdnaes pn n he same drecns as rang crdnae aes and wh he rgn a he cener f he earh (belw n he rgh) (nrh) r (eas) (up) ' r ' R ' R The psn f he parcle can be wren as R r where R s a vecr frm he cener f he earh a pn n he surface a claude and r s he psn relave he pn n he surface We ll assume ha he epermen akes place near he surface f he earh s r R and R r R The cenrfugal frce s apprmael: F cf m R A plumb lne wll pn alng he bserved g whch s g g R (as dscussed las me) The drecn f g defnes he drecn f he as We wll use he claude as he angle beween he as and he angular mmenum vecr gnrng he slgh crrecn dscussed las me Newn s secnd law n he rang frame gves: mr mg mr m r g r R The angular velc n he rang crdnae ssem s: 0 cs 5

6 s he crss prduc s: r ˆ de 0 ˆ ˆ cs cs cs The cmpnens f he equan f mn are: cs cs g Suppse ha an bjec s drpped frm res a 0 and h The bk ges abu makng erave apprmans n he same ven as when we were lkng fr he range f a prjecle subjec drag Alernavel hs prblem (and he analgus ne fr a charge mvng n bh an elecrc and a magnec feld) can be slved eacl Here are nes n bh appraches: he erave apprman and hen he eac Ierave Apprman slun As a ereh rder apprman we can drp all erms cnanng s negrang wce gves: The bjec wll land ( 0 and g 0 g and h g 0) a abu he me: h g Ths gves: T ge a frs rder apprman pu he ereh rder apprman fr he cmpnen f he velc n he rgnal equans f mn ge: 0 g g Qualavle: remember ha pns Eas s frm he neral perspecve a ball ha s drpped lks lke s hrwn wh an Easward nal velc As he ball fall s ha Easward nal s cmparavel larger and larger han he easward velc f buldngs rees ec a s smaller and smaller radus s lks lke s accelerang Eas The negrang he equan wce gves: g g 6

7 When he bjec lands: A a claude (& laude) f be: g h g rad/s h g 45 and hegh f 00 meers he easward deflecn wuld 00 m 98 m/s m 55 cm T ge a secnd rder apprman fr he cmpnen f he acceleran ( s er n he frs rder) subsue he frs rder apprman fr he cmpnen f he velc n he rgnal equan fr : g cs T keep erms f rder fr we ll gnre he small crrecn he cmpnen Inegrae he equan wce ge: When he bjec lands: 6 A a claude (& laude) f be: g h g Ý 6 cs g cs g 4 cs h cs g 45 and hegh f 00 meers he suhward deflecn wuld rad/s 00 m 98 m/s 45 cs m cm Gradens n he gravanal feld f he earh can als cnrbue he suhward deflecn f a fallng bjec bu hs calculan gves he rgh rder f magnude Eac Slun Reurnng he ssem f equans ha we wan slve: g cs cs wh he nal cndns ha If we negraed he secnd and hrd equan we have g cs (0) (0) (0) (0) (0) Then subsung hese epress back n he frs equan we have h (0) 0 0 7

8 4 cs g cs 0 g S ur ask s jus slve hs dfferenal equan; nce we have an epres fr we can easl ake s dervave and hus fnd he epres fr he accelerans n and as well A gd guess wuld have he frm ( ) A( B Pluggng ha n we fnd ha wrks fr all values f f g B and Impg he nal cndn ha (0)=0 ells us ha 0 Impg he nal cndn ha (0) 0 hen ells us (0) 0 Pung hs all geher we have A B A g ( ) 4 B g 4 Befre prceedng ne ha f he argumen f e s que small u can replace wh he frs few erms n s Talr seres hs gves g g 6 g g 4 cs Ths s a ad decepve ce ur g self s dependen n he c-laude In parcular g g g g R g R rad an based n equans 945 and 947 Wh hs n hand we hen knw ha g cs g g g cs g S g cs cs h g cs g 4 cs 8

9 Easward deflecns were measured b several epermens beween abu 800 and 900 The fllwng s a summar f epermens frm MS Tersen and H Sdak Am J Phs 68 () 9-4 (000) In her nan s easward and s suhward The suhward deflecn was small measure n he epermens An alernave apprach he calculan abve s hnk f he pah f he bjec as an rb n an neral reference frame The ran f he Earh has be aken n accun afer deermnng he rb fnd he pah seen b a rang bserver (The rbal mn f he Earh abu he Sun wll be nsgnfcan durng hs pe f epermen) The Fucaul Pendulum: Wha s specal abu? Fucaul s wnderful dscver was he realan ha he small effecs f he Crls frce culd be greal mulpled b ug a pendulum Wha a wnderful da mus have been fr Fucaul when he nced ha he rghward deflecn f ne swng wuld n be undne n he reurn swng; he effecs wuld accumulae! RH Rmer Am J Phs 5 (8) 68 (98) Thus he pendulum has he advanage ha he effecs [f he Crls frce] accumulae and hus he effec mves frm he dman f her ha f bservan Lén Fucaul A lng sphercal pendulum (n cnsraned mve n a plane) ha has a ver small amplude f scllan mves apprmael n a hrnal plane We ll use he same crdnaes ha we dd fr free fall We can gnre he dsplacemen ( 0) and velc ( Ý 0) n he vercal drecn Befre we ge sared slvng hs le s hnk abu wha we d epec lkng a he suan frm he neral / nn-rang frame T make reall smple we ll cnsder a pendulum a he Nrh ple The pendulum swngs back and frh raher blvus he fac ha he Earh beneah s rang Fcug n jus he mn f he bb n he plane we d see ) Acs ˆ ( 9

10 Bu he rang frame s aes are rang relave he neral nes; relave hse aes we d sa ) Acs cs ˆ ˆ ( Nw we re gng prve hs guess rgh Fr small scllans T T mg because he acceleran f he bb s ver small and L The and cmpnens f he en are prprnal he dsplacemens: T T L and s T T mg L and T T mg L L L The equans f mn (wh an m facred u) are he same as fr free fall ecep fr he addn f he en: T T m m cs cs g L T T g L L cs cs The hrnal cmpnens f he Crls frce when a pendulum s swngng n he Nrhern hemsphere are shwn belw (use he cmpnen equans abve and recall he dscus f hurrcanes eserda) 0

11 v F cr Defne he naural frequenc f he pendulum g L and he cmpnen f he earh s angular mmenum cs ge: The slun can be fund b defnng he cmple funcn Add he frs equan and mes he secnd ne ge: Guess ha he slun wll be f he frm e whch gves he aular equan: 0 The slun he quadrac equan s: because The general slun s: C e C e e C e C e s: Ce Ce If we chse he nal cndns A 0 and v v 0 hen he nal cndns fr are: These are apprmael sasfed ( ) f: 0 A and 0 0 C C A and C C 0

12 The ceffcens are C C A and he slun s: S e ( ) Acs Acs Acs Acs cs e cs cs ˆ Acs cs Acs ˆ Acs Ths las epres ma be he eases nerpre: he pendulum swngs back and frh a s naural frequenc whle appearng rae clckwse relave he reference frame whch s self rang cunerclckwse The amplude f he scllan s A he frequenc f he scllan s and cs s he frequenc f he ran f he drecn he pendulum s swng The angle beween he drecn f swng and he as s The angular speed f he earh s ran s 60 da s a he Nrh Ple ( 0) he pendulum raes nce a da A a laude f 4 (claude 48 ): cs48 60 da 40 da 0 hur Dem: Pendulum n a rang plafrm The pendulum cnnues spn n he same as n an neral frame bu changes drecn n he rang frame The analg wh he Fucaul pendulum s n perfec because he rae f change n he drecn f he pendulum s swng (n he rang frame) des n depend n s lcan n he urnable Als he frce n he pendulum s alwas n he same drecn n hs case bu changes as a Fucaul pendulum ges arund he earh