Phys 331: Ch 9,.1-.3 Noninertial Frames: Acceleration, Ties
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1 Py 33: C 9,.-.3 Nninetial ae: cceleatin, Tie i., / Nninetial ae: cceleatin, Tide, nula Velcity n., / Nninetial ae: Tie deivative, Newtn nd. HW9a (9., 9.8) Nn-inetial ae Say yu dwn t te e Paade and ee a Pl-all lat lidin by. On tat lat, cue, tey ave a pl table and tw peple playin pl, let ay I ne te. t e ent, elative t yu, te nt te lat a pitin, velcity, and acceleatin, V,. I yu beve te cue ball, elative t yu, yu d eaue it pitin, velcity, and acceleatin t be,, and. O cue, I, idin at te nt te lat, wuld eaue te ball pitin, velcity, and acceleatin elative t yel a V nd yu S e S entially wat yu eaue, inu i wn pitin, velcity, and acceleatin. (nte: ti i te claical appiatin, a i apppiate ti cla; takin int cnideatin tat lk vin at dieent peed eaue tie and ditance dieently i te ubject pecial and eneal elativity) Nne tee ae paticulaly eakable kineatic bevatin; weve, kineatic eet dynaic (inteactin, pue and pull) in Newtn nd aw wee ce i elated t acceleatin, but nly a eaued in a nn-acceleatin ae. O net net S, eaple, i yu culd eaue all te ce n te cue ball, ten yu culd ue ti t pedict w it wuld acceleate elative t yu n te cubide.
2 Tat ean, it wuld acceleate away te pen idin te lat a net Tada! It tat iple. I, ten wy n eat ave we ied away analyzin tin in nn-inetial ae ln? Ove te cue ti capte we ll ee tat e ituatin ae ate eay, wile te ae ate ticky. ictitiu Inetial / ae ce I we ultiply u acceleatin elatin by a, cue we et net In te ae piit wit wic we d deined a centiual ptential and an eective ptential in te lat capte, we culd deine a ae ce and an eective net ce ee (indeed, centiual ce i a ae ce a tatin yte.) ae net. e net ae net. e In wat ene i te ae (, a te bk call it inetial, wic ee a tane cice ince it an atiact bein in a nn-inetial ae) ce like and nt like a eal ce? Nt a ce. In pyic, we deined ce in de t quantiy inteactin pull and pue. ind yu, ti in t a eal ce in te ene tat tw patie aen t pullin puin n eac te. eaple, wen yu it te a and yu pne e lyin te da wee yu d et it, it nt becaue tee an inviible and puin it back, ate, it becaue te da acceleated ut unde te pne. Tat quite bviu t ene tandin n te cub watcin yu zip by. ike a ce. On te te and, a eaued by yu w ae acceleatin aln wit te ca, i yu didn t knw any bette (ay yu e in te back a windwle van), te pne lew back a i ene tuned n a tact bea tat pulled it. net e. ae S it ve elative t te acceleatin ae a i tee wee ti ae ce. Nte: cue, yu t ve in te ca a i epeiencin a ae ce eveytin eaued aaint te acceleatin ae de (tu callin it a ae ce). In yu cae, yu and te eat pe int eac te a i te tact bea wee pullin yu back.
3 ae ce and avitatin Bette yet, it a i a avitatinal ield wee pullin yu back. ike te eal avitatinal ce, te appaent ae ce i pptinal t a (tat, cue, i te nly way tat yu culd eplain wy all tin, in pite avin dieent ae, wuld acceleate te ae in epne t te ce) ae a te ae a te ce due t a uni avitatinal ield. I yu wee in a plane in a clud, te nly in yu d ave tat it wa bankin wa tat yu elt eavie. eat. e eneal elativity aide: it intein ad tackled te pecial cae elatin eaueent in tw cdinate yte tat wee vin elative t eac te but nt acceleatin; tat wa pecial elativity. Ten e wanted t develp te e eneal elatin between eaueent ade in tw eeence ae tat wee acceleatin elative t eac te; tat wa eneal elativity. Wy ten d we aciate eneal elativity wit te avitatinal ce? Becaue intein ade ti bevatin te eect avitatin i equivalent t tat an acceleatin ae ae ce. 3
4 Steady State / Static. Jut like in Int Pyic, te eaiet ituatin t wic we can apply te ce acceleatin elatin i wen tee i n acceleatin eaued in te nn-inetial ae. aple: bca pendulu Te bk de ti, but it t a i enu pay-/et ati tat it wt epeatin. ailad ca wit a pendulu anin it ceilin (lent, a ) acceleate wad at. et ay we knw te a te ball and te acceleatin te ca, ten we want t lve te ce te tin and te anle at wic te ball will an. I d Inetial pepective tin at a I elp daw, tey d Nn-Inetial pepective tin ae at a net a ˆ : in yˆ : c c net. e ˆ : in a yˆ : c ae in c nle: S, takin te ati te tw equatin tell u tat tan in c actan anitude: Squain and addin te ive 4
5 in c in c ective avitatinal ce: ti i te ae eult we d et i tee wee a avitatinal ce av. e e and av. e e tin av. e e Peid: Wile it in t eally tat uc ade t ind te pendulu equency cillatin abut ti anle tan it i t ind it peid in an inetial (nn-accleatin) ae, Quetin: I te pendulu ween t in an acceleatin b ca, w wuld te equency it cillatin be elated t and? : Nw, wat u new eective e? nd wat te equency we d epect in te acceleatin b ca? We can kip t te eult in ti cae, e 5
6 aple: balln. In pble 8., te bk entin a un cpanin t ti pble a latin balln. t it blu, yu it epect it t an like But tink a inute abut te ce tat ld it in place. Tee te avitatinal ce dwn, te ae ce back, te buyant ce wee de te buyant ce pint? ecall tat in a nn-acceleatin ituatin, buyancy eult te act tat denity and peue te ai inceae a yu deceae elevatin (ince pat tat peue i te eult all te ai abve beain dwn n yu tank t te avitatinal pull dwn n it), jut belw a balln te peue i litly eate tan jut abve te dieence between tee tw peue ive a net ce up / te ppite diectin avitatin. Nw, in an acceleatin b ca tee al a peue dieence let t it ince te tailin wall i puin int te ai in nt it wile te leadin wall i pullin away te ai beind it. In t, te ai i al vin aln wit te acceleatin eeence ae, it al epeience an eective avitatinal pull wit and te buyant eect i tat peue deceae a yu ve up and wad / te ppite diectin avitatin. e av. e e Buy tin Te balln will lean wad! One way t tink ti i tat te balln i like a bubble, a lack a. all te a in te ai et pued dwn and let, tat ppaate te bubble up and it. 6
7 aple #: nkey & Hunte (te eay way!) unte t a bullet at a nkey in a tee ldin a ccnut. Te initial velcity te bullet i twad te nkey and ccnut. t te intant te t i ied, te nkey eleae te ccnut. Sw tat te bullet will it te ccnut. In te unte (inetial) ae, b b te bullet and c c te ccnut. We culd w tat i te initial velcity te bullet i twad te ccnut, tey will cllide a te ccnut all. Hweve, ti equie a bit et. Inin ai eitance, te acceleatin te ccnut i. In te ccnut (nninetial) ae, we ut include te inetial ce b b in Newtn ecnd law te bullet: b b b cnt Teee, i te bullet i initially in twad te iin te ccnut ae, it will cntinue t twad it. b b 7
8 9. Tide. Nw, tee ay be e caue analyzin te tin tin in acceleatin b ca, but by a te t cnly epeienced acceleatin eeence ae i te at. It pinnin n it ai, it bitin te un. It eve--litly wbblin in epne t te n pull. We ll ate ti lat eect it. Puely in te ce, te n avitatinal pull deceae a yu et ute it. Tat ean tat te nea ide te at eel a eate avitatinal pull tan de te iddle wic eel a eate pull tan de te a ide. Since wate i uc ee-e t de tan i lid ck, te eult i tat te wate nea te n ve twad it a little, te lid eat beneat ve twad it a little le, and te wate n te a ide ve twad it till le tu, elative t te lid eat, te wate bule up n bt te nea and a ide. Dp wate n d d at Wee ˆ d ˆ d d an ipatial, inetial beve pepective, Newtn nd law applied t te wate dplet i net d a a d buy Nw yu, idin n te at a yu ae, ae nt an ipatial inetial beve. te ake ti analyi, we ll jut wy abut w yu and te at tat yu e idin acceleate twad te n in epne t it avitatinal ce, in paticula, ˆ ˆ ˆ S, lked at yu nn-inetial pepective, 8
9 net d a d ˆ ˆ a buy ˆ e. tidal buy a buy buy a ˆ a S te tidal eect i due t tat dieence between wat te n avitatinal ce n te wate dplet i and wat it wuld be i te dplet wee at ae lcatin a te cente te at. a Dp wate tide n tide tide at tide S, wate nea te n et pulled ut, wate a te n et pued ut, and wate al way between et pued inwad. ecie: ind an appiate epein te tidal ce n a dp at te pint neaet te n (can ue tat d << d and a binial epanin) t ti pint, all ce ae aln te -ai, and d = -. tide ˆ ˆ tide ˆ ˆ 3 ˆ ˆ ˆ 9
10 anitude Tide In equilibiu, tidal buy tidal buy a Nw, te buyant ce wate n a dplet at te uace te wate i eentially a nal ce peventin te dplet inkin. Tat ean tat te te tw ce ut, in equilibiu pint nally int te uace te wate. Nw, i tidal i nal t te wate uace, ten i we iaine abbin a dplet and dain it aln te uace, tee tw ce ut d n wk / cntitute n cane in tei aciated ptential eney. S, i we cnide te ptential eneie aciated wit ten ce, ten we can ay te ttal ptential at ne lcatin i equal t tat at ante. Tat will allw u t evaluate tei elevatin. We can ewite te ce a U U wee U U d tidal d d tide at tide tide Vaie a dplet ve U tide at, ti ut be cntant at any tw pint n te wate uace, let cpae te i and lw tide pt: buy Cntant, pint in diectin tidal buy O H H H.. w y. H. Hi Wee. H H H O t a petty d appiatin,.. H.
11 Nw, we culd calculate at ti pint, but we culd al ipliy akin ue te act tat >>, and ten d a binial epanin n te tw actin. ivin tat, / / 3 3 Te bk calculate ti t be abut 54c. Pein te ae analyi te un eect wuld eactly te ae; iven tat te un i uc ute away, but al uc e aive tan te n, yu et a cntibutin ante 5c.
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