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

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1 Phs 331: Ch Nnnetal Fames: Centfual and Cls fces 1 Mn 1/5 Wed 1/7 Thus F Mn 1/ Fctnal Fces: Centfual and Cls 98-9 Fee-Fall Cls Fucault 101- Cente f Mass & Rtatn abut a Fed As Rtatn abut an As Ineta Tens Pncple Aes Fnal Just Ch 9 & bennn f 10 F 1/17 6 pm Fnal Eam Equpment: Glbe Ball wth cdnate aes Tuntable wth pape taped t t Ch 30-mn test HW Cente f Mass & Rtatn abut a Fed As Nn-netal Fames: Rtatn Last tme we leaned that when ne fame s tatn elatve t the the sa the Eath elatve t the fed stas then velct and acceleatn measuements made n the tw fames ae elated b V () f whee ( ) e V ) ˆ V f f ( as And Af ( ) AC ( ) whee Af Acent as ˆ as and A c S S O Of cuse Newtn s nd Law apples nl n an netal fame Fnet m S F net m A ( ) A ( ) f C

2 Fcttus Inetal / Fame Fce m F net F fame Ffame m Af ( ) AC ( ) Fcentpetal Fcls whee F ma m m ˆ and F cent cent as as c ma c m Tda and tmw we ll lk at sme effects f the centfual fce: and the Cls fce: 96 Centfual Fce: F cf m F m c Suppse a mass has a fed pstn n a tatn cdnate sstem wth anula velct What s the centfual fce n ths case? Label the anle between and as The daam belw wll be helpful as = sn F cf m The se f the fst css pduct s: sn It s tanent t the path f the mass (ccle) as t tates The esult s als pependcula t the anula velct: That means that the se f the centfual fce s: The centfual fce F cf s pependcula t F cf m m sn and pnts awa fm the as f tatn Suppse we ae descbn an bject nea the suface f the eath The se f the centfual fce s lae nea the equat As descbed b an bseve n the eath the dectn f the centfual fce (hw much N-S and hw much In-Out) wll depend n the lcatn We wll

3 use the anle fm the anula mmentum (the Nth Ple) Ths s knwn as the clattude and s 90 mnus the lattude See the daam belw adal (upwad) F cf m tanental (hntal) In the tatn fame the effectve fce n an bject that s nt mvn (elatve t the tatn fame) s: F eff F av F cf Defne as the avtatnal acceleatn that wuld be felt f thee was n tatn We wll nw use " " f the effectve avtatnal acceleatn n the tatn fame The ndvdual and effectve fces ae shwn belw adal m m" " m " " F cf F cf ad sn tanental The effectve avtatnal acceleatn n the tatn fame s ven b: Fcf " " m We can splt ths nt adal / n-ut and tanental cmpnents / nth-suth The se f the adal cmpnent s: ad ad F sn R sn (sn flp when dppn the abslute value sns snce wll eneall be lae than the centfual cntbutn) cf 3

4 At the ples 0 ad and at the equat t s less b: R s m 0034 m/s The tanental cmpnent s: tan tan F cf cs Rsn cs The anle between and the adal dectn ( ) s alwas small s (n adans): tan 1 tan ad tan ad Rsn cs A plumb lne (a bb n a stn) wll han at ths anle t n equlbum n the tatn fame The anle s laest at 45 whee: 97 Cls Fce: R 0034 m/s 98 m/s ad 01 Let s cnsde an bject mvn clse t the suface f the eath The Cls fce depends n what dectn an bject s n (velct) elatve t the anula velct Nte that what t means t nth depends n the lcatn n eath! F c v F c v F c v v F c 4

5 Cmbnn all f the pctues we can eplan wh hucanes tend t tate cunteclckwse n the Nthen Hemsphee If a s mvn nwad twad an aea f lw pessue t s deflected b the Cls fce n the wa shwn belw (vewed fm abve) lw pessue The ppste tatn esults n the Suthen Hemsphee Ths effect s t small t detemne the wa wate tates as t flws dwn the dan (e lke when the Smpsn s vst Austala) Bth Fces: The mtn f a fctnless puck n a hntal tatn tuntable s an nteestn eample F an netal bseve (IGNORE EARTH S ROTATION!) the puck wll smpl mve n a staht lne because thee s n net fce A (nnnetal) tatn bseve ma bseve me cmplcated mtns whch wll be eplaned b the centfual and the Cls fces Eample #: Pb 90 (backund f 94) Suppse a fctnless puck mves n a hntal tuntable tatn cunteclckwse (vewed fm abve) at an anula speed Wte dwn the equatns f mtn f the puck n the tatn sstem f the puck stats at an ntal pstn 0 wth an ntal velct v v v Ine Eath s tatn! In an netal fame thee s n net fce n the puck s 0 and we d see the puck mvn wth cnstant velct / n a staht lne m F net Hw wuld t lk t a lttle bu ddn n the tuntable? In that nnnetal fame that tates wth the tuntable Newtn s secnd law s: m F F m m cf c Takn the anula velct f the tuntable t be 00 the pstn s 0 Calculate the css pducts: ˆ ˆ det 0 0 det ˆ 0 ˆ ˆ ˆ

6 ˆ det 0 ˆ 0 ˆ 0 0 S the equatn f mtn ves (dvdn ut the mass): the equatns f the and cmpnents ae: and A tck f slvn bth f these cupled dffeental equatns at the same tme (see Sect 7) s t defne If we add tmes the -equatn t the -equatn we et: Ths lks an awful lt lke the damped hamnc scllat (asde fm that fact f and the lack f a neatve sn n the lnea tem) S we can uess the basc fm f the slutn that we d uessed n that case Snce ths s a lnea dffeental equatn uess the slutn aula equatn: 0 e t whch ves the Ths mples that Thee s nl ne slutn f s we need a secnd slutn (the dffeental equatn s secnd de) Ths sunds a lt lke the pblem wth the ctcall damped smple hamnc scllat S we ve t a d chance that a smla slutn wll wk Yu can check that n addtn t e t te t s a slutn s the eneal slutn s: t e t C 1 C t whee the ceffcents ma be cmple The ntal cndtns ae 0 0 v v 0 and v v The fst cndtn mples that C 1 and the devatve s: s: C t t t e C1 Ct Ce 0 C C1 v v v v C1 v v and Ths ves: t e t v t v t cs t sn t v t v t 6

7 The eal pat f s t and the mana pat s t whch ves (Eq 97): t t v t cs t v t sn t v t sn t v t sn t Yu wll eple (cmputatnall) the behav f the mtn f dffeent ntal velctes n the hmewk (Pb 94) Hw can we bseve an effect f the Cls fce n the mtn f an bject nea the eath? (We alead cnsdeed hucanes but the ae cmplcated sstems f patcles) Ou calculatns wll be dne f the nthen hemsphee Fee Fall: We wll use the cdnate aes and wth the n n the suface f the eath at the clattude (belw n the left) Thse cdnates pnt n the same dectns as tatn cdnate aes and wth the n at the cente f the eath (belw n the ht) (nth) (east) (up) ' ' R ' R The pstn f the patcle can be wtten as R whee R s a vect fm the cente f the eath t a pnt n the suface at clattude and s the pstn elatve the pnt n the suface We ll assume that the epement takes place nea the suface f the eath s R and R R The centfual fce s appmatel: F cf m R A plumb lne wll pnt aln the bseved whch s R (as dscussed last tme) The dectn f defnes the dectn f the as We wll use the clattude as the anle between the as and the anula mmentum vect nn the slht cectn dscussed last tme Newtn s secnd law n the tatn fame ves: 7

8 m m m m R The anula velct n the tatn cdnate sstem s: 0 sn cs s the css pduct s: ˆ det 0 ˆ sn ˆ cs cs sn cs sn The cmpnents f the equatn f mtn ae: cs cs sn sn Suppse that an bject s dpped fm est at 0 and h As a eeth de appmatn we can dp all tems cntann s nteatn twce ves: The bject wll land ( 0 and 1 0 t and h t 0) at abut the tme: t h Ths ves: T et a fst de appmatn put the eeth de appmatn f the cmpnent f the velct n the nal equatns f mtn t et: The nteatn the equatn twce ves: 0 t sn t sn When the bject lands: 1 3 t sn h 3 sn 3 h 3 sn 8

9 At a clattude (& lattude) f be: 45 and heht f 100 metes the eastwad deflectn wuld ad/s 100 m 3 98 m/s sn m 155 cm 9

10 T et a secnd de appmatn f the cmpnent f the acceleatn (t s e n the fst de) substtute the fst de appmatn f the cmpnent f the velct nt the nal equatn f : t sn cs T keep tems f de f we ll ne the small cectn t the cmpnent Inteate the equatn twce t et: When the bject lands: 1 6 At a clattude (& lattude) f be: h Ý sn cs t 3 sn cs t 4 sn cs h sn cs 3 45 and heht f 100 metes the suthwad deflectn wuld ad/s 100 m 3 98 m/s sn45 cs m cm Gadents n the avtatnal feld f the eath can als cntbute t the suthwad deflectn f a falln bject but ths calculatn ves the ht de f mantude Eastwad deflectns wee measued b seveal epements between abut 1800 and 1900 The fllwn s a summa f epements fm MS Testen and H Sdak Am J Phs 68 () (000) In the ntatn s eastwad and s suthwad The suthwad deflectn was t small t measue n the epements An altenatve appach t the calculatn abve s t thnk f the path f the bject as an bt n an netal efeence fame The tatn f the Eath has t be taken nt accunt afte detemnn the bt t fnd the path seen b a tatn bseve (The btal mtn f the Eath abut the Sun wll be nsnfcant dun ths tpe f epement) 10

11 The Fucault Pendulum: What s specal abut t? Fucault s wndeful dscve was the ealatn that the small effects f the Cls fce culd be eatl multpled b usn a pendulum What a wndeful da t must have been f Fucault when he ntced that the htwad deflectn f ne swn wuld nt be undne n the etun swn; the effects wuld accumulate! RH Rme Am J Phs 51 (8) 683 (1983) Thus the pendulum has the advantae that the effects [f the Cls fce] accumulate and thus the effect mves fm the dman f the t that f bsevatn Lén Fucault A ln sphecal pendulum (nt cnstaned t mve n a plane) that has a ve small ampltude f scllatn mves appmatel n a hntal plane We ll use the same cdnates that we dd f fee fall We can ne the dsplacement ( 0) and velct ( Ý 0) n the vetcal dectn F small scllatns T T m because the acceleatn f the bb s ve small and L The and cmpnents f the tensn ae pptnal t the dsplacements: T L T m L and T L T m L The equatns f mtn (wth an m facted ut) ae the same as f fee fall ecept f the addtn f the tensn: T T m m cs cs sn L L cs cs The hntal cmpnents f the Cls fce when a pendulum s swnn n the Nthen hemsphee ae shwn belw (use the cmpnent equatns abve and ecall the dscussn f hucanes esteda) 11

12 v F c Defne the natual fequenc f the pendulum L and the cmpnent f the eath s anula mmentum cs t et: The slutn can be fund b defnn the cmple functn Add the fst equatn and tmes the secnd ne t et: Guess that the slutn wll be f the fm e t whch ves the aula equatn: 0 The slutn t the quadatc equatn s: because The eneal slutn s: t C 1 e t C e t e t C 1 e t C e t s: t t t C1e Ce If we chse the ntal cndtns A 0 and v v 0 then the ntal cndtns f ae: These ae appmatel satsfed ( ) f: 0 A and 0 0 C 1 C A and C 1 C 0 1

13 The ceffcents ae C 1 C A and the slutn s: e t Acs t cs t sn t Acs t t t The ampltude f the scllatn s A the fequenc f the scllatn s and cs s the fequenc f the tatn f the dectn the pendulum s swn The anle between the dectn f swn and the as s t The anula speed f the eath s tatn s 360 da s at the Nth Ple ( 0) the pendulum tates nce a da At a lattude f 4 (clattude 48 ): cs da 40 da 10 hu Dem: Pendulum n a tatn platfm The pendulum cntnues t spn n the same as n an netal fame but chanes dectn n the tatn fame The anal wth the Fucault pendulum s nt pefect because the ate f chane n the dectn f the pendulum s swn (n the tatn fame) des nt depend n ts lcatn n the tuntable Als the fce n the pendulum s alwas n the same dectn n ths case but t chanes as a Fucault pendulum es aund the eath 13