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l28 - Pbtcwttq Eng/55‘ OE<1 S‘PQ‘we 60nd me¥ic\e...

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Unformatted text preview: Pbtcwttq Eng/55‘ OE <1 S‘PQ‘we 60nd me¥ic\e Covxsidur 0o Poucsh‘de oi: waSS M 0396ch is $cdied of q “CLJK‘MS Y VJ! “om +Re «,me 09- cc gmqmogagfleooS sobi -;.:..th12¥'€ 0% 3 mass M Git/tel ‘(QJUJS R m, mum CQSQ I 1 Y > The force on a particle when it is outside a uniform soiid sphere is given by GMm/r2 and is directed —§o\id spgwave is made 0? 0‘? . toward the center. The force on the my} 382x41 “S Somme; ‘Lo ma K \ m U W‘ particle when it is inside such a ’ sphere is proportional to rand Y‘aé k MS R . qnev goes to zero at the center. "$K“Ced'fi)‘f€€ OQAA Fo‘k‘Ci/A‘Cajfl 06 +0 GKV shit étP-QYU‘: oni on mass me 58¢“ cum! kit‘sirwce +0 Pafiécie m rem cewjth) F=-GmM C ma) \(1 u : «£14 W2— R) \C Case 11: Y S R 'Fxrom 00¢ resoH“ ovx spRcvicak SQQNS I Wm ‘Covce OV\ Ox. ripaxsiicke is 40C QWLW‘C‘ '10 at“ +21 mSS Ee‘i‘toeev» ~r~=D as“! \(‘=\F. No kgamma do: +0 ‘H/{Q mass [octweeh \"T—V and. K. F = Gwflnm LYSR) \(1 5H4» 700% E amovcl' 043 mass 'm co sloeevc a? mes r. M E ‘lb‘lmk mass 0? ouv's‘oee‘rc o?- V‘aJiMS R. '- 7Y2(‘(\ : M Ins @vovovhovm‘ +0 vo'oMe> :1: “6va <~‘ (YSRX QhQY‘ Text Foltcwsc‘waiw ‘lexe ‘reac‘ov‘ \(‘SK i3 «ah/en 1m? OQT usual ite-Cini‘howf \- R Y 110d: ‘f :4}? : 'GmMJY! +3‘ GmM‘r' Jr! co co Yea K R3 = -G~mW\[3R°‘—T1 (ism 2K3 pom] energy a, R (a) Usasafunction ofr Separation r (b) F as a function ofr Separation r Potential energy and gravitational attraction of a particle m by a solid sphere of mass M and radius R Two 5PQWVICCLHLk SIMMQ NC o‘od‘ed’s qfimc} QOLCQ o’chqv mike on \rm .—-—‘7 Fflfi Z G} mAmB T 1 Y :. SQPV‘Q} ‘5“ b €+LU€ €Y\ C QVVLCY$ . chgx e‘emevflLcu‘xé Mass o-e O‘o‘ed‘ B ad‘s OV\ +€\e Cm' O‘e ob‘ec“ pr (see FO'm'k' mass+ 3 in‘CcJ SQQH)- Summin oveY A V ‘3 all e‘emewfs Am 0&1 B ’rc‘e $60chan {‘0ch 0W {s 6* CW1, B?) sukmm¢ \k 0N 01“er ‘ffie ‘che on B is 4130 OJ” ( r :8 CT, Eva New) \AS 55“! Kano {Ru} qve ezoq/ ant, FFOS(? II; Stu} B Aces “0+ che SFQfY‘fCQ/I name/f” *er Segue OV\ 9 st” QC+$ 4% +8? cm :3 IQ. W6 72? 'CCN‘CQ °V\ B Romfuev AQES “0+ (16+ (1+ “fee CM 01? B .”. vaHAiomal l:o~rce5 ovx EW‘TflC‘eJ Claim}; 9.7-5 TRQ abfavi¥uiflomal 'FOVCQ 40C +0 Q6 eqv+€~ qC‘Lt'vua’ §0V\ a) Paw-Md? 0C mass \m’; is gum-g i chcejérdiion clue 4o (EMVJ‘K 40¢ T£\a:LR\ie|&8 SW“ o‘occis i?- a: (an {0? qxsomeé +0 ‘36 Cons‘lmvl- OWY +9 dimensions 09 4R9 obact}; A: hi: —"‘I -; — _';’._ —5 F rlg»: f‘ml-a LZMJ'B— M3 lY\ : +043} mass 0Q 0L69c+ Q Alf mew} o'uvd’ on ‘er o‘o'efli 4093. “Wu”; 51 P €¥Ccc§iifl€ V «e cult? ’3 If Fig. The individual particles m,- comprising an object are acted upon by the gravitational 3 forces mg. The sum oi all these individual forces is equivalent to a single force that acts at C. the _ GM of the object. 1 l CaJco‘aJ‘e +Re 40$ 0:; aLod' “Rec a wsoH’ich‘ {from a“ +92 430va E. 723 Smx‘e che ? mUS+ also fN‘ i: rL vz‘o‘m4oce5 4&2 PPRUES o—P a‘o00+ &Q (m 449. 3 Z ifln-T T Cim A .au {Q inéiuhloal 4:3«95. M 0“: Oded‘ ENE? L OAOC€ no +0 _~, R 20 CM «394' V5 025 quc‘g elqck’oa 0n “1% CW‘ 0(3 fie (Jaded? Teas rmfi is also CaUeri 4% 03; “Re o‘aéecv I"; JFRQ 0‘0 'eci A} ove\( +R€ €146 <avwxo+ Be RAQAI'CMJ exter, ‘ \ owhku i3 30 [core 71qu 0-(3 {32 okagd’, ‘(Ke i: uavfles ew‘cv—OP- + QQT sFQericoJ 0&3 flag lcemécv ~ 0? jam/Ant! WUA? 7.3:ch Fovces «29-8 'Wemw \mvi‘tajfiova fie” is unifiovm all Pat/+5 “Pr 0\ QNQ‘ (foul/{in olo’ecf exfevimre Scume accel. ‘ I; "\Q‘Q 0“ (“[28 {S “city/xiii . QFOVW ‘3‘qu —~\~o—- (QCQ. ‘ R€%O\¥ A tit? e‘ftNLCaA ‘Co\[(e —o\oéed~ Yuma \oe {LY‘CJV(Q€A oy- com‘arcséecl, VW\( aCcefev (on vav‘t'cs (a) (b) Fig. (a) In a uniform gravitational field. all parts of an object experience the same gravitational acceleration. (b) When the field is not uniform, gravity acts differently on the different parts of the object. The result is a differential force that tends to stretch an object along the field and compress it at right angles to the field. 'bi?€€vw\£ Rag erces ——>> Tidal Foxfces ~ \Ocemfl‘LtéQ‘. Consider Jtuoo 0‘08“ij OE mass 7m m ’ m SePaV‘ACA <1 AC$‘(’€W\C€ 1‘" .Q 0 loCcigxé (1* 21than e \C‘ Q vow/'3 The tidal force on an object arises from ka$$ m - the difference in gravitational field strength across the object. \TCXCJ ‘Qsflfll on “it oioxecf 2: Stgfexrcvufi Between {:2 3(th {1&0va co? C-Q ov\ ei‘tevev {“4 .’ E: GMm _ 6mm : GMm meal—weal Q2431 Q”? U431 (“rqv‘ .11 ll 4 €7me (Jtic‘iatl ‘COVTQB Y3 q<<‘( -Falls%oi3£ as xix»3 ' Moon move ‘tmyaovliavd‘ tam massive stun. Moon Earth Fig. Ocean tides result primarily from the differential force of the moon's gravity. The gravitational force is strongest on the moonward ocean, weaker on the solid earth, and weakest on the distant ocean. Thus the water on the moonward side is pulled away from the solid earth, while ate-l4 earth itself is pulled away from the distant water. Two tidal bulges result, so that a given location 1 experiences two high tides each day as earth rotates. (Not shown is the differential force associated with the sun's gravity, which is weaker but also important.) l WQi%Q\’\' (FAA G‘mvfi‘rtahovmj 'FCN'CQ 2‘1-9 jNemr' +sz sov‘gqce 0% “L €CLY+Q\ ‘Hie magi“ 0?- a ‘ooétt 0‘: mass m i3 Us: YWCX "Me com 2 0d": +R§$ 41> (”Pm {Ewe qcitmg OV\ +8? @023} QcAkvx?‘ b643, m%= Gflgm _RE <3 2 6 Me [AeQMilriOV‘ 0Q 31 SOP?03€ Okaeot m is (1+ 4 Refs/OAT R qLoUQ +Q€ «Al‘s sow-gate 0V 0k alx‘swtcmce \r (3mm +9“ (PN‘E‘Y‘P \(‘3 RE +2 1-? 422 \moéxé 3s ‘m £fie fink“. thS acce(~evwlu'bh 3 f8, “val = gmem : Gmgm Y; @593" cg“— C1795 -_ GME Y; 02ng L8" Jecwases 9036 incvmsihg 4%“;th 3‘90 avg-700 ._Covxs{tl£\r PE 0? fqrhcle a} («Jinn \(7 ubd : " GMEY“ Y :PE 0‘? Fq-Y‘kric‘P QJV swear? 09 «Ma, r: RE U(RE\ 3 ’ GMEW) RE ‘Cexqwkekw DE A“: (LU‘A- 710%}: ~6mEm [L— L] V RE -= GMEm <T~ RE) YRS . Lér Y—Re :3 430‘- 2<<Rg .cm‘é ‘(EE ’v RE VCRCUR< 1% PE 0&3 Cu {3:16th 406 +0 Cu (Ox .l‘V\ cech’doM -2 new! eadE‘S sung/1M. «Mfr A‘H ficum'hwé M Q g I‘QIN CaseI '- C(vcu‘qv 0v\ni3r$ -\(3« bOALES 09 mass m is mouh‘g in q cho‘qr ov‘bi¥ o?» MAW To aloofi cu massive o‘oa‘ecjf mike mass M (sovq . a... .. M A body of mass m moving in a circular orbit about a Foxme 0‘4?ch on ‘er PIQHQ+ bodyofmassM. is F = GMM <1} .I(» 'W‘Q g‘gngl OE +Q\Q Y3\qv\e+ is \T +even H‘s (ev{\ Vern al acce‘eraflmh i3 ; QC: 3 Yo I Gmm : ““7 1 To To 0( «3'1 7— 6..."). [RQAHAS o? o~(\oi'\' Exes swag] ‘Co ' szx‘oé ‘CO‘( I- ‘(CVO‘O'LI'OVU ' —v I = Q\‘V‘CUW\Q(‘{(V\te/$PQQA (KT-(‘50 ’\5' 95H 1 ow \=4— 0 . “(€9ka Lave) ‘ ”Home «“8 0"\3;\S cor: heCWMk ckrcO‘av (BAA +963 skm‘ak ommhksis is 3046 30061- * 36mm <1qu sis afflies ‘l‘Q +Re chrculav moILt'oI/x o? Hie mow oxr qr‘kCch‘J 3&9” ht? m-OO‘AJ o: F‘QMJE In Jrexis cqse Jrexe 30mg mass is \j‘firaaceé [Na Wm mass 09 we ’la/qme'f. “MO‘LCOM JomZHJeJ L 41.00 Boil (f‘ §Qn+Plah€D "WLCVCLQJHOM. PcV‘JRNLliVC 994?“; 0& o+€ev (Jamal; is gmaM . €941 1 k ’3 __‘_ U- :- _‘ MW! (2*{ 3m 2 GT0 [K n c EMVXBI u : ~GMm [R‘rsz‘ioJ mefl To TO‘\’CJ E‘nevgxk: E = K+ u : LGMM _ Gfl 5* ‘90 w; E = -_I_ Gmm ToJch E s “88$an 1 V0 . . Ana u\q\( momevdmm‘ q K = E I : mV‘o AI L1: 21K : amvfl 1 6 mm 1 r L;,\ : GMMlVo \‘o : L:Z Case ‘11 -. E\\\“;)ficq\ O(\o{\’$ ‘Kcflevs |s¥ Lam'- wfig \‘Re 722 OTL( +5 0? FLU/“9L5 ave ell'sesF 50“ cu“ one Focd Powd. I So“ (4—9 Nana‘s Ewe 4A Ptsdle“ 1:3 M m m<<M. 11Consic‘er *ee PICUAC+ J owe '0'? ”36 “wins %W\“¥S ‘13 0“ \(‘ql 1 E z iqu“ GM {lo +qt EVR = C0W§LV\‘\V] 3 ‘r W“ : L = \“VT [75124 Anfio‘av MOMPVJUM: (OMS-Q J: —— GIL!“ - E 1 MW; Y ammw-J—j =0 E :ME \‘1 — 61‘»: 6315111 4L2 1 E K5 1m; 1‘. 1 a f (a: dek .1 ( 3 +- 1: fig: vak - K 3 — S A: («*f‘a: -611“! E ,i Lancga 0‘? magoY‘ Q‘JHS is Witltpcnclcnf of L. .Or\aijr5 (JAR same ma‘boY rods Qave +29 same +07LJ mm, (con 520w *Rd} E = 5 1+ QELD‘ mLGMmY‘ A7”! 3 W A“ ok\ec¥ is \p ‘eété urmqvg wk”: can in(‘\(&1 ‘3‘0694 ’UA‘ . HOLD $an 4093 Jrexe (Aegea- o -? I E (Anna/MS assume) 34c} em?) ENVY“ is Cowswve. -. KX+ 714- : Kg‘L’MC An object of mass m projected 2 upward from the earth's J. ‘mfwl- G "Ling : _'_n'\’\J~ — G Mm surfaoewith an initial speed ”uf A " a. Y‘ reaches a maximum altitude ‘(A' 4 h (where Me >m). H5? {If : \CMQK / N¥ :0 _ \C- : RE keqvlfex (Q4Uk3) \ — “of = (16%, (L RE (“at had} is JTQQ mmimum BPQEA SOY (“039* 413 €$CQ6X QCL‘VXQ {BMYA-A’e‘ 7 Tfiifi Cmmieg‘oowAé VeachvchS (ma12oo £03?er ”‘5‘; =0. "J‘ : RENE (DC Rx: W 2 2 xé-(flxlo ' $539 “0&4 \Q; (0-37x10é’ '1 Hflx 104mb 0V anoemk/g O‘OXCCk c?- Y‘nqu W\ }Y\ +Ke qul‘LA/Ll‘onw‘ CH2” f0? 00 ,quéle (00:17) 0‘? mass M 1’04“} ‘3‘“ka '- E:K+u : im'xrl— 6 MM (mmm) 3 ~( :TASSUMC Ohio? {3 Yewmfiei ‘prom surface 0? M 30-!7 a} («Aims “n+3 q V’JOCJ‘A \J' suck +M 14$ KE is sewn 0)“ \(‘=00. I EA' = E5; i‘mvl- GMM _._ O + O a R "\f2- ZGM (esc e niacifi R “e 5 M: Mo} ”50‘ ugh/s, R? R0 M: “E } \g.- (kawds '3‘ Kg genera} p‘anefl’qug Mohm 30,; Two —?cN\*CC\€ SSE§‘\T{W\§ ‘Comiclev’ +wo KNHVQA‘KAB PQV¥|.C\CS On‘a‘ m, A pair of interacting particles with masses m, and m. Two equivalent coordinate representations are shown—that consisting of the individual coordinates f1 and r2 in the inertial frame 0, and that consisting of the relative coordinate r = r, - r1 and the center-of—mass coordinate B. The central forces. F12 and F”. are shown as repulsive forces; they could equally well be attractive forces. _ Two rawstLAes 0‘3- mass m‘ QYWL ml aqvc Position. vet/{bis j: and “at.“ om iwev‘tial “gramme O, A (av/Jim! £~o¢<e CLitA‘S \02+LQQ€Y\ HQ qu‘tt‘c‘es. F is +ctf fins“: ’0. ovx m‘ &\)L +0 W3. "-3.! 2 (9 Fa = m if 7 4t?” Jescwilaes motions 0"? mI ch ma. , :- a ‘34 1 W1 3:3 At?“ w: Loam} +0 [coltL 4* 1R6 mol‘imn using CM coovéimjte> {30¢ tfit qu+_(c\fs- E": huh)“ \Icc’mv 0&9 CM sttdxm vech-tve +0 0. "I? nelakve coovdinagr 091% {we furl‘t'oles. Ul "“«' , “5 ‘9 a a —-? @ K — “M“ + "5“5)‘ MR = WHY" 4— max; mt‘Hma a "$ @ 1? - r‘a -—‘q a *5 “‘7 So‘u W3 Y‘: R-W‘l‘r @ m a» a A Y}: R ‘7 Ym Y (9 M u.) £\f( M: W\\-\—Ws:L a» _ fi Fla. ' -‘—:\\ MQNLOIAIS 3&4 Law 0+® "46:0 =>£=V=mwu A a? dJc CW‘ ' CM “who“ clebcvdmct Qa‘) 4405 [9/466 42‘ consfw’j .VC‘Oc'tha qu is incl-eftm‘enj a? ’ch [W‘fYad’iom befweem ref-\‘\IC\€S. mAxE‘tf9 - m‘xE16) cum!» uni/n3 Egg->41“! (6) mm {3* : Mg - elf“ Le" /,(: me‘a Rufuccnl mass o9 2“ YWLl'c‘rsvavw mm: . F“ 7 CW” _ E7 ' /“J'E“ - a“ @ 30-3 7K: relafi‘fl'w coovéinofe $(é) 0-? ‘I'Qe "‘iLuoo qV'lTCIC SazWLtm R45 HG scum? Began/[Or as +e{ cochlL'rwaf 0 cu sin “1 Hide one mass movén in q port? , 12¢!th 66:10:»), wQJEQx 18/45: mAu+Uajg ov‘ im/{rna/ . RVCC Leldeen +£€ QQ‘IUJ {3qu Op qu+,‘cles. Kmekc Enev mAA Momew'fo 7K2 4&3 ‘h'ne‘Lic Querw .‘n 1‘29 .‘nrNLt'cJ gramme O Lam Le exfrcsseJ in kvms of— {7= (IE/if I '{Kt “velociL-k be {2Q QM qVKL $= dig/CH" fat re/coA'vP Ue‘ou'L} 09‘ Hit 4101) far‘kCIfs. ~ 1 2 K- Jimnfi 4— 191mm“); a «a U.“ ,3 4:1 «53> <13; <45: d’c At (11' at 7341 KE {k gran/V6 O {s “Hie SUM 0F +6! KC: 0‘? 6L R7FOWC‘HCI Hick 09 mass M=Vm+m11mov3m1 (n+2 WIOCH"? U 0‘? CM flu; E; ME «94 Pay-Fake Owe mass fl moving wii'fi We w/JCve ve’omly V. flit“! L‘thY MOmiJUM {h {Eva/1:9 O —-5~ -—3 -—7 _ 1). “+133 — MV (ReLJNe' livxeox momevd‘q 0% HA Pair 0‘? qu+{c/es .3, /“’\—? : m1mn(\E-\N) M : hm‘li— whfi) M I-C 49m s‘és‘km 0'? ‘oqufLCLeS [8 (SO [Ix/{9C1 (no each/{ml lea/(rs) ——'7 P : Con$+€Wk1P 734‘! F, in CM {imgme =0. ...
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