Newton s Law of Gravitation

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1 Physics 106 Lctu 9 Nwton s Law of Gavitation SJ 7 th Ed.: Chap 1.1 to, 1.4 to 5 Histoical ovviw Nwton s invs-squa law of avitation i oc Gavitational acclation Supposition Gavitation na th Eath s sufac Gavitation insid th Eath (concntic shlls) Gavitational potntial ny Rlatd to th foc by intation A consvativ foc ans it is path indpndnt Escap vlocity Gavitation Intoduction Why do thins fall? Why dosn t vythin fall to th cnt of th Eath? What holds th Eath (and th st of th Univs) toth? Why a th stas, plants and alais, not just dilut as? Aistotl Eathly physics is diffnt fo clstial physics Kpl laws of plantay otion, Sun at th cnt. Nuical fit/no thoy Nwton Enlish, 1665 (a ) Physical Laws a th sa vywh in th univs (sa laws fo lnday fallin appl and plants in sola obit, tc). Invntd diffntial and intal calculus (so did Libnitz) Poposd th law of univsal avitation Dducd Kpl s laws of plantay otion Rvolutionizd Enlihtnnt thouht fo 50 yas Rason pdiction and contol, vsus faith and spculation Rvolutionay viw of clockwok, dtinistic univs (now datd) Einstin - Nwton + 50 yas (1915, a 5) Gnal Rlativity ass is a fo of concntatd ny (Ec ), avitation is a distotion of spac-ti that bnds liht and pits black hols (avitational collaps). Planck, Boh, Hisnb, t al Quantu chanics (1900 7) Eny & anula ontu co in fid bundls (quanta): atoic obits, spin, photons, tc. Paticl-wav duality: dtinis baks down. Th should b a aviton (quantu avity paticl). No poss yt. Cunt Issus Dak Matt Luinous ass of alais is too sall to plain stas obits. Dak Eny and inflation Possibl anti-avity at lon an fuls acclatin pansion of th univs, and also aly Bi Ban. 1

Rlatd to th foc by intation A consvativ foc ans it is path indpndnt Escap vlocity Gavitation Intoduction Why do thins fall? Why dosn t vythin fall to th cnt of th Eath?

2 Gavitation Basic Concpts Evy paticl in th Univs attacts vy oth paticl with a foc that is dictly popotional to th poduct of thi asss and invsly popotional to th squa of th distanc btwn th G 1 o a pai of asss: 1 Intial ass: 1 Masus sistanc to acclation,..: a. Masus spons to avitational acclation - a fild. Gavitational ass: Mass is th souc of th avitational acclation fild - always Gavitational ass asus stnth of a avitational fild poducd. Duality/Equivalnc: Evy bit of ass acts as both intial and avitational ass with th sa valu of in ach ol. Gavitational oc No contact ndd: action at a distanc. Cannot b scnd out, unlik lctical focs. Always attactiv unlik lctical focs (cpt fo dak ny, ayb). Vy wak copad to lctical focs. Too sall to notic btwn ost huanscal objcts and sall (.., p + and - ). Gavitation is lon an, has cosoloical ffcts ov lon tis. But it is a wak foc on th huan scal. Nwton s Law of Univsal Gavitation 1 1 d law pai of focs 1 1 oc on du to 1 Gavitational Constant G N /k G 1 1 ˆ ˆ 1 ˆ Displacnt 1 fo 1 to oc is btwn pais of point asss Sytic in 1 & so 1-1 Not scnd o affctd by oth bodis Easy to iss btwn asss na a thid la ass (.. on Eath sufac) 1 Unit vcto alon 1 I IS REMOVED, IS ANYTHING AT POINT DIERENT BECAUSE 1 IS STILL AT POINT 1? IELD tansits th foc (no contact, action at a distanc) G1 Acclation fild ˆ Eath Invs squa law: as sph ows fild (o foc) aa is constant A4π fild at location of du to 1

Gavitational ass: Mass is th souc of th avitational acclation fild - always Gavitational ass asus stnth of a avitational fild poducd.

3 Hny Cavndish fist asud G dictly (1798) Two asss a fid at th nds of a liht hoizontal od (tosion pndulu) Two la asss M w placd na th sall ons Th anl of otation was asud Rsults w fittd into Nwton s Law indin th Valu of G G N. /k G vsus : G is th univsal avitational constant, th sa vywh a is th acclation du to avity. It vais by location /s at th sufac of th Eath Why was th Law of Gavitation not obvious (cpt to Nwton). How bi a avitational focs btwn odinay objcts? G 1 1 Nwton is about th foc ndd to 1 suppot 100 as of ass on th Eath k a lit of soda 1 k sandwich 1 t N. 100 k a pson 100 k anoth pson 1 t N k a ship 10 6 k anoth ship 100 ts 0.67 N. still had to dtct Conclusion: G is vy sall, so nd hu asss to t pcptibl focs Dos avitation play a ol in atoic physics & chisty? k lcton k poton t obit adius N.

80 /s at th sufac of th Eath Why was th Law of Gavitation not obvious (cpt to Nwton). How bi a avitational focs btwn odinay objcts?

4 Supposition: Th nt foc on a point ass whn th a any oths naby is th vcto su of th focs takn on pai at a ti + + on 1 i, 1, 1, 1 4, 1 i 1 All avitational ffcts a btwn pais of asss. No known ffcts dpnd dictly on o o asss. Eapl: 4 ' ' at 0 by syty 1 G i i, 1 ai,1 - on 1 1, i o continuous ass distibutions, intat on 1 d1 ass dist Nuical Eapl: ind nt foc on 1 du to and Us supposition Basic focs at iht anls nt,j i,j i j y a a

Eapl: 4 ' 4 5 1 ' 5 1 1 1 1 1 14 41 4 1 at 0 by syty 1 G i i, 1 ai,1 - on 1 1, i o continuous ass distibutions,

5 Supposition fo a tianl 9.1. In th sktch, qual asss a placd at th vtics of an quilatal tianl, ach of whos sids quals s. In which diction would th top-ost chunk of ass ty to acclat (ino th Eath s avity) with th botto two hld in plac? A) B) C) D) E) a 0 s + s s 9.. Anoth chunk of ass is placd at th act cnt of th tianl in th sktch. In which diction dos it tnd to acclat? nt,j i,j i j Shll Tho: supposition fo asss with sphical syty 1. o a tst ass OUTSIDE of a unifo sphical shll of ass, th shll s avitational foc (fild) is th sa as that of a point ass concntatd at th shll s ass cnt Sa fo a solid sph (.., Eath, Sun) via nstd shlls + +. o a tst ass INSIDE of a unifo sphical shll of ass, th shll s avitational foc (fild) is zo Obvious by syty fo cnt Elswh, nd to intat ov sph. o a solid sph, th foc on a tst ass INSIDE includs only th ass clos to th CM than th tst ass. Eapl: On sufac, asu acclation a distanc fo cnt Eapl: Halfway to cnt, a / 4 V sph π 5

. Anoth chunk of ass is placd at th act cnt of th tianl in th sktch. In which diction dos it tnd to acclat? nt,j i,j i j Shll Tho: supposition fo asss with sphical syty 1.

6 Gavitation na th sufac of th Eath: h Whn is on o na th sufac: What do and wiht an? Eath s ass acts as lik a point ass at th cnt (by th Shll Tho) Radius of Eath Objct with ass is at altitud h abov th sufac, so +h Wiht W a with acclation ivn by Nwtons Law of Gavitation (any altitud) a G ˆ ( + h) at any altitud h << o, in oth wods + h G a wh 9.8 /s Eapl: Us th abov to find th ass of th Eath, ivn: 9.8 /s (asu in lab) G /k.s (lab) G k (ava - asu) k Altitud dpndnc of Wiht dcass with altitud h Th wok ndd to incas Δh dclins, sinc wiht dcass 6

acclation ivn by Nwtons Law of Gavitation (any altitud) a G ˆ ( + h) at any altitud h << o, in oth wods + h G a wh 9.

7 fall acclation 9. What is th anitud of th f-fall acclation at a point that is a distanc abov th sufac of th Eath, wh is th adius of th Eath? a) 4.8 /s b) 1.1 /s c). /s d).5 /s ) 6.5 /s a G ( + h) at any altitud 9.8 /s Gavitational fild tansits th foc A pic of ass 1 placd sowh cats a avitational fild that has valus dscibd by so function 1 () vywh in spac. Anoth pic of ass fls a foc popotional to 1 () and in th sa diction, also popotional to. G 1 Concpts fo -filds: 1 1() ˆ 1 1 No contact ndd: action at a distanc. Acclation ild catd by avitational ass tansits th foc as a distotion of spac that anoth (intial) ass sponds to. Gavitational fild is Consvativ (i.. can hav a potntial ny function). cannot b scnd out, unlik lctical filds. is always attactiv (cpt cosoloically, ayb), unlik lctical filds. Th fild 1 () is th avitational foc p unit ass GM ˆ catd by ass 1, psnt at all points whth o not th is a tst ass locatd th Th avitational fild vctos point in th diction of th acclation a paticl would pinc if placd in th fild at ach point. ild lins hlp to visualiz stnth and diction. clos toth ston fild, diction foc on tst ass 7

Anoth pic of ass fls a foc popotional to 1 () and in th sa diction, also popotional to. G 1 Concpts fo -filds: 1 1() ˆ 1 1 No contact ndd: action at a distanc.

8 Gavitational Potntial Eny ΔU ΔU Δh fails unlss a is constant foc dpnds on Dfinition: dw d Wok don by avity on a tst ass ovd thouh d foc vais alon path du displacnt potntial ny chan Choos: avitational potntial zo at R wh th foc zo i..: U(R) 0 as R. Mass M cats th -fild. Intat alon a adial path fo R to infinity ΔU R dw GM R d d R GM GM d R R U GM R Th avitational potntial ny btwn any two paticls Not: vais as 1/R. Th foc vais as 1/R NOT R Th potntial ny is nativ bcaus th foc is attactiv and w chos th potntial ny to b zo at infinit spaation. Anoth fo of ny (tnal wok o kintic ny) is convtd whn th potntial ny and spaation btwn asss incas. Gavitational Potntial Eny Mutual potntial ny of a syst of any paticls U total U ( ) shad, su ov ij all pais ij all possibl paiins Th total avitational potntial ny of th syst is th su ov all pais of paticls. Gavitational potntial ny obys th supposition pincipl Eapl possibl pais U + U + U Utotal 1 1 G1 1 G1 + Th slop of th potntial ny cuv is latd to th foc. Rcall: dw d du 1 G + foc du to avitation inus du d Slop of potntial ny function (divativ, adint) 8

Intat alon a adial path fo R to infinity ΔU R dw GM R d d R GM GM d R R U GM R Th avitational potntial ny btwn any two paticls Not: vais as 1/R.

9 Gavitational Potntial Eny, cont As a paticl ovs fo A to B, its avitational potntial ny chans by ΔU But th chanical ny ains constant, indpndnt d of path, so lon as no oth foc is actin E ch K + U() Gaph of th avitational potntial ny U vsus fo an objct abov th Eath s sufac Th potntial ny os to zo as appoachs infinity Th chanical ny ay b positiv, nativ, o zo E ch E ch1 Consvation of chanical ny with avitation E ch dtins whth otion is bound, f, o at scap thshold E ch is constant Ech K + U() G U () always nativ o E ch < 0, paticl is bound and cannot scap. It cannot ov byond a tunin point (.., ) o E ch > 0, paticl is f. It can ach infinity and still hav so KE lft U 0 E ch U( 1 ) E REE 1 KE 1 BOUND U G Tunin point KE 0 E ch 0 is th scap condition. i A du paticl at any location would nd at d last KE -U () to ov off to th iht and nv tun. How uch ny dos it cost p kiloa to scap copltly fo th sufac of th Eath? G1 U KWH/k Th tannt to th potntial ny aph asus th avitational foc 9

with avitation E ch dtins whth otion is bound, f, o at scap thshold E ch is constant Ech K + U() G U () always nativ o E ch < 0, paticl is bound and cannot scap. It cannot ov byond a tunin point (.

10 Escap spd foula divation and apl Escap condition fo objct of ass fo th sufac: E ch K + U Th ass cancls: 0 1 v sc G - 1 G v G 0 sc - vsc Eapl: ind th scap spd fo th Eath s sufac 9.8 /s 670 k v sc , / s 7 i/s Eapl: Jupit has 00 tis th Eath s ass and 10 tis th Eath s diat. How dos th scap vlocity fo Jupit copa to that fo th Eath?.G.jup.G..00 v jup 0 v 5. 5 v jup. 10 How uch dos vay acoss objcts na th ath s sufac? d ~ siz of objct Optional Topic dpnds on atio of objct siz to Eaths adius conclusion: can tat as constant 10

7010 11, 100 6 / s 7 i/s Eapl: Jupit has 00 tis th Eath s ass and 10 tis th Eath s diat.

11 Whn is it valid to appoiat U by Δh? Optional Topic Answ: whn Δh << 11

Gravity and the Earth Newtonian Gravity and Earth Rotation Effects

Gravity and the Earth Newtonian Gravity and Earth Rotation Effects Gavity and th Eath Nwtonian Gavity and Eath Rotation Effcts Jams R. Clynch, 003 I. Nwtonian Gavity A. Nwtonian Gavitational Foc B. Nwtonian Gavitational Fild C. Nwtonian Gavitational Potntial D. Gavity