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howe fet! a& i ® Gyovitotion @ e itotional force . se % i yo Q> = — : Fo HIN FXG Qz ~ iN2 =e Fol 4 —_ — (distance)? ! t= Ko Ge i ~4 Fz GHiMa e ye | = ~~ notes 5 | = @= : = * y -m2 l Universal Qrovitakonal Constant CQ) = 6-67 X10 ue = = a} Grovilokont force 1s SAL, point masses e. 3 Olas Duy adtrackive @.. ways kokoro) force blu mass bavtdes ts e@- adependond of Py@semse af third mass - e Wis a _qnevvalve bree = Gis a central foyte e_ i} obey's inerse squmt squave low a 2 eke ibis independent of medium o. Vt obeys piable of Super bosihon e qe Ko B= Glixg lik = @- 6. CFX/6'N <. ML ( \m)* aq 6| <—- L Ha Se an o z tambon ¥ foyce hetween Rod 2 Porh'ole a& %K an element lomoth da a Scanned with CamScanner Of mass dm: MH du AL distomce from partck Cm) => Force blw element Nass @ Partide des Gmde dpf-ém [UH ds) - 2 xe He Fe ferce blu fartcle 2 Qod =Gmmn [de 2 (Corde 2 fe! L J 12 ) _ (-1 | ~f pay pe -(at] re a Lage a) C a(aty J F=eGum_} — Gum L alesth) a (atc) if aspse E= Gum a(a) mass density Uneay mass density z } = dm dm= Adi di suyface mass density © = dm are i GA dA votbme mass density ff = dm dm= fay aw force on O parvticlem cern ph Fe Fe F= Gmt az ae > D 0/0 8 BD 66) O 6 HOO Y @ MH @ @ I SF Scanned with CamScanner uJ —= AE Eo mm Enek = Fe = my* Y Se Ef FES Gm? 3 = mv? G? G/B m mer iP =~ Sp Fret = fo f . C { - E (e+ | = mv? il Gm \ ES erus 2 ES, is Gm? { (e+) =mv* = oz \ *! alla = = : 13. Mou | Qvavitakonal fetd C¥) a= —_% u 5 P i SOE gs 3 3 + gta: field T= © = -Gmu/ 7? 7 m ™ F=-Gmm. ¥ T--GM + m= test mass y2 y2 grav force = mass x gyow. fd eam. Unit of grav Fd: No =m Kg 32 ts a vector quanitity , always towards the sovvce marss id obey's prinuble of Super basiton Obje ck grav. fd. Point macs yoy eee P T= GM, ving (H.R) a 2 T=-GM x olong %X-eus te ( ata? )%* Scanned with CamScanner BN ° at centye (x= ti IN 49 alta’ |X cg At t= ER Tmox = = SGN —= Ja re) 3I3BR ; = a : bisecto Ayc of Ring fix\ \ T conte: ~ 2Gd_sin along SENOE — Az Mass f of ! << length Ws 2kA = R pe Disc (RM) Bn o=Mass - M 17 Toxis = & (4x G) (+ cos) — oreo wa® (ae = (ang) [i-x__\ i= = surface mass Y 2 x FR? = density a. : ~ \ is = » infinite Stroight wire ae e = A=\\near Hass density .> 2. Finite Straight wire * Ins Gd [sin i+ Sin $2) H oy = <> e e = Sin UP 479, Sao GA (med) stabs) a . 5 > 5 . Thor = TE . - 7 » Gem - Infinte wire & > ig 2 KR E ( \n ze rf - hue eR pe = zK i A= N — L Foam I doete semi-circle m= =m [ 2&4 sing) => 2m& ML = ——————__ Sin {80 =X Gro a a 2 e= A (2 “ = . ; eS mass of eorth is Sl mes of mass of moon. if sep- ™ eyation bekoeen theiy centers’ is D_ then Find distance e- From centre of eayth where ge gravitatonl Feld ; intensity is 0. a SN Ey wy ~ (oi 2) = YY” Re neo r eont San Ua, G SIN GN ® "2 (D=x)2 = = m_ ALP Inet = 0 : Te + Im =6 Q- Tet In i GBIN = GY aq -l » Ye (o- % DK S O-x = & ®. X24 p * & 10 be ~ Sta => Prion ax:tp es 10 a Scanned with CamScanner oe TS Fraga 00.) 0-2 , 2,8 (oo); rm 44/10/14 119/21, 351 RE JuCAL, Gravitational Potential work done per unit mass by external agent to bring ® tesi mass from yefference point (generally infinity) ra Qiven point P, without change in Ws kinetic a) 8) >— P) ») > » energy. . } Vp = (WexthorP _ (Wyrfone) we bd M eT a 7 . sures ORE GM 2 ¥ S Ss Note Ss t4 is ascaley quantity it may be tve » -ve SY zero » ond is Joule ] kg » tt depends on reffeyence point Fp) a objeck Potential ® | Point mass H > e veered 7 axis =H Ss @| Ring CuiR) jaws bp a? X u == 5 . Necente ou ri = — S 7 5 Ss @ | solid shheve - a £ volun tis™ 6 doty 2 = [yu \ a \4/xn3] 2 Scanned with CamScanner _grovitotional potential diffevence. difference in gravitokvonal botentiol QYavitotional Potential energy - (Wext jo > P= Us - (Udgy- fd) o> P mV=s ~ Gite se ee Ae SOlid sphere = earth ee oy (oom m U= - Grim 3 | : Lee oa a Us-Gmm_ «3 ag aS x je mm SS gan (et Uz = Gm24 = Gm?2x2 . a —. — mL—,—In ad Qa _bayticle of mass two is divided in fow bayts: => mym,H-me , M-m ahd placed Ot < corners of =s equoxe of side a ,~ find UH so that potential energy of system is mar eat He — =3 Nays= - Gm (M-m) 4- G(N-m)? = Gm2 =§ a alz OZ N-m _ m Ss ws -G [Go Cmm—m?)+(M-mPem2 | o Red J 3——— } dy =- G& [aliM-an +2( =m! (o-t) tam] -& wm oa v ? ; Ze a a fe 20 [ oN-&m + -OM +m 49m| 7) oa ie ] — 4m-3m + Pmaetmstt GM bmM+ GA WZ 2 Scanned with CamScanner tx a 8, ‘e Rowe Fit tg: ne ot = 56l%S2 24,24 —Gi 4m-Sm+ (4m -24) % = & Q am — 9am shim 2M os mala Bf2 tM (O83 2) 20 — Ja v2 %& m (4-3ta = - M442 - 2) & miq-¢)= 4 (2-2) 0 G-sh . Wl 29 =~ e 2-45 ms =~ %& \ncvease ‘in gravitational _P-E as Be 7 . GPE aoe sin — a i _ oe \ VS ) s : ea 2. Gee ot Bed -at height h = U2=-GUm 2 Qh _ 1 \norease_ia_gy PE 7 Yr AV= U2 -v, , = GNm pfs GMm_) ee eee 7 Lis Reh (oe ) [Lorn RI 2 = Raforh) AUe Gum Uh = » R(Rsh) R (R+h) Ts= gy. fa od surface of earth 8 = GN. 2g ® Ree Any \ncvease in gy PE due ta earth. » PV = GNem k . Re Reth) * > Mm GNe —h Avemoh h2CE0 Re be V+ Reke | Vee) Re = Gvdonw, = Scanned with CamScanner Acceleyotion dueto gvowity / grav: Fd due to earth. oviside y>R a = yah, f Tout =9.= GM AS 1 (/ {*) 4 Y=Reth YF a) Qout= GU “ (Re+h)? " if heighk heeeRe z [Re = 6400kmI a go= GN = GM (lun \ >? ce 2 Zz Re! R2 (teh) R i aut=/9/ (-2h \) ~ VY R/7 ve Chonge in Aceh de to gravity e Ag = govt-J x =g{I-2h)-g =» 9[-2h) . —— a 7+ charge in acc due 40 _gyowity. rv Ag x 1007 ~2h vino 7% v 3 R = a) o Surface of eayth “ Q=GM_ = 0-8 mls? ® Rite: ™ ® Inside earth [yer] / eho \ Zin _GMr ee) ‘i R3 wT x = GM 3 ain= 4 ¥ 4 RR = ~~ R s Scanned with CamScanner At depthd rd =2 y= R-d Gin? 9 Dag { Bd ix C_ eT gin =9 [\-4 | ' R/ Change tn occ due to_growity. AQ= Qin —9 = o {+d \-92 9 [4] Chonge_in occ due 9 gyowity Ag xtc % = -d x 100 3 9 R p ot centre > =O > a -| | ¥ eR inside F) e~ > sae YR OBubide > | | a gin XY Qo XL > Wier eels iil ioe d aa & -| acc dveto Growity is max. at Suvface (g= 9-3 mis) y -| when we move towards the centre of earth from surface ¥ __| then ace due te growity decreases incveases L _7 [When we move awoy From contre Of earth »~fom i __|_surface then ace due te gyavity cdlecveases i——] ° i Factors affecting acc due to Gravity * ». Scanned with CamScanner Y Pagono. \: Se ) cqlewlote height above the earth Surface where aS gvowttadional field is G'957. of suvface {mB — > go = GN = 6-25 os pane Alina s_ 13 es f iS GH. a= ts = ey | Te | :. a 2 (s Q2 Wi iss Koo bh Ha: gh botevs 48 ior et i = hev-R = GR-R=3R End distance From centye of earth where GYovityotonal field is 9/4 case | —thside | gin = al 1 A oie = 19 WAN a ij y= 2/4 se T= outside Go = GH - Q_ ye oe GN _ GU at 7 rR 4 LL. toss y= QR y R 2 find gravitational Field at depth of 64 km _gin = q{ L- 64 \ 64000) g (1-L ). of gc) { tood { loo} => dat. of Scanned with CamScanner a y| Find tne gyavitakonal fd _at height of 64 om % gout = g { 1- 2x64 \ ri 64x100/ - & g: | 4g \ =a87 ofg % (Oo _* Escape velocity * minimum Sheed requived to send a body out of ~ gyovilakonal field of planet a a {ZI K Z! [ ®. [AAR = [om 4 mye |= Oro “ \ tis “YN { na? i) pasate . Planet CNR) — Noz= GH -y Vet 2GU 2 ¥ Y 2 Ne= | 9Gu : % ® escoabe Velouty debends only on mass and 3 Radius of blanet Ly Position From where the Partde is projected a escape velocity dosen't depends on » moss of parhde ( object angle of projection x na escape Velo tity Ot sveyface of earth - » » Ve = |2GUe [84 PxR* = _ll2kem J Re J 3 few % * Black hole ve = a > Scanned with CamScanner ) , ® #\ #)\ of a Rodus R then al body consichy a uniform _speric Shave ts Cohenitial* ceeereliie’ wus dud eavudline bregsuve oak @ distance _¥ fan centre of pe Prax @ | R2-7?) NAAURIRIA AR 3x2 2 2. +s veg he in (r= - (xy) pore R Pe Pane (et(R \*) i & e cL Cal l Eel, Pian tase Get) R=) ti & U } &—- Kebley’s law of Planetary motion sl Kepler's law of ovbitt each planet moves avounds the ™| sun ino civeslay bath oy eleibHcot path With the S| sun oO§ ts foous. = fo h — ss P 8 Ht. f 24 Cs) ht \ f Nene” —) =| lows of Aveol velodty:yote of ave swaped by the =! SE vecrg iS Constant oe a = position: of plonet wri. Sun (Gs “ve e a} j a E= Gro. fre et on planet we bo sun —> 5 V.2iy ay of Plane hd | ‘ “ = fe t: ang momentum 2 ly) y : WS | 3 7) Vay aT dy. € = th=0 (0) coam “LTU7 ~~ at ~~ Vitis rei QYE_ swapped - da= pw (Lt) 4 n_ difference dt ul Ss Scanned with CamScanner aa - ecto Rate sf aver Swabped = } pv = da” = Ll eed 2 d+ 2M ee ee ee ee Dem Qm en as \ arafet = Tr ak Livig we <4 Joma ~ V = = Yin +Y¥mox = Qa YmGxK Ven - VO Jnor © Qiein = all-e) Ymac a(\te) max = | 2G4 (+e) Ymin= | 24 _[t-e) N 9a U-2) 24 (te tal energy of a planet: kE+PE beyyte Gm — Gitm |“ Cahways) = sable ei Yoviarie {F vavincle 3 2a — Sas) _— e R -7xV - Ls Kebler's second law is indeed ak 2 Orn low of ConServalion Of Angular momentom | Bkebley'’s low of time bevied. +? <73 Q= Seni —Mojoy Quis 2 Soselight bh And B ay orbiting Gygvh d-tho earth in ovbits of vadius ¥ and Gy then robes of haiy dime powed Scanned with CamScanner 
