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ees a Gvavitodion e itational force ee gyavi onal = ¥ wee beset sear <) (an Qo B= i — | | Fu Q Qe ™~ F a Hy N2 = = = Fol) | = (distance)? | te KO Oe = F= GUN = Ye | 4 | ~~ ; Nokes — | 2 e. \| Universal Gravitakonal constant (@) = 6-67 X10 ae = — a} Grovilolent force 1s Shy point masses e. 3] ow oy odtrackive e. mravitodinod fore blo 2mass bavtides iS e. ‘inde henden! of Pyesemse af thived mass - - ‘is a cwervalive re : cl avis a central forte eo 4 obey's inerse squmt square low a ye ek ibis independent of medium Ae ‘hobs pieble Of Super posihon o 9 ke 1M — {kg F= Glico? Uikg = @-6. CX /I6"'N = Mat tm)? LN __ 4a ral x = > @ os — —o Bhai ~ de am o foyce between Rod 2 Porhele & aS an clement length da 7 Scanned with CamScanner oe &’ Yooh gg Of mass dm- iH dy At Ustance 4 from partck (m) => Force blw element Hass 2 Partide dts Gmde df= én { Hoda): - ee xe Me F= frce blu Particle 2 Rod sGmmu [de 2 (Corde te 3 fe! L J 92 Jj pe -(atu] Raia sew aor [at a) Va (aty J FeGum_) _ _GUm La (oAv ao (ate) if aspxsC F= GQUm % 6 6|8 G @ BO @ @ & |% ala) mass density a Uineay mass density z 4 = dm dm= dal di suytoce mass density & = dim dm= < dA aA volbme mass donstty f = dm dm = faly aw force on oO particlem a 07 —————— Mm —=>--~—-.—CSS > F E F= Gmz az 4S) % 10/90 8 B 0/6 Scanned with CamScanner can a Enek = Fe = my Y. FY3 Gen2 3 = my? OF Gi “ Fat Fret = Fe li = * F (B41) emu % Gm? { (2 €1\ = mv" “2 | “! alte 13. Mau | Oravitatonal feld C¥) 9 ¥ U Souree fest u i = SOVrES, 65 ¥, a gtav: field T= F = -GmM/ 72? ™ ™ F=-Gmm: ¥ T.-GM 7 m= test mass y2 y? ayav foyce = mass x grow. fd eam Unit of grav Fd: No =m Kgs? is avetoy quanitity , always towards the sovvce MOASS id obey's principle of Superbasikon oF — Obje ck a fms ce yay. fd. —@ ; 3 ~ ro Point mass yo P T= Gt =o % ving (YR) a p T=2=-GUX olong %-or a iy Nez ( aR?) rn Scanned with CamScanner =0) al it T=9 at centye (x — | } ain | 5 La ' t/2 > eZ —- h\ b = aes y= At a= ER Imox_= Sa —2= J2 si 3{3R ; —z = ae s} bisecto Ayc of Rng f=) a Tener" 2GA_sin§, along “i a -_ 2 Az Mass ( of f = length NZ A Q2KA Ra R B= (AR) | : Disc (TRIM — co = Noss Ny {\ Toxis = & (4x &) (+ cos®) = area na? (pees (4nG)[1-x__\ et 2 = = svrfoce Moss Y ~ < ( vl xF+R2 } & density , -t- \ Ts 26 Infinite Stoight wire —— e = d= \ineor Hass density 1 2 Tinie straight wite : Tx= G4 [sinort sin be) _ a z ot ~ ai wy Ty: G4 (cos, - Cos 62) = 5 . TInekr = | Ixn2+4 Ty Ly . - = Q. Semi ~ Infinte wire & py Tn = G1 pti: & - %y RQ “RR , fi Thet= G4 J2 R Wy Scanned with CamScanner in foym Semi- a _yod of moss UH, length L, is bent eiycle _ point moss m iskebt at cemtve of semi ciycle then growitational force athng on dne_pavtde ig 2 KE (eNa hue pagan A A=M i Fam I dueto seme circle =m (2GA sin) => 2mG Mw ee | cL oT = U2 mass of eorth is Bl Hmes of mass of moon. if sepb- — eyation between theiv centers’ is D then Find distance e- From contre of eayth where g¢ gravitatonl Feld r intensity is 0. ». ar m7 YBes \ = _ a 2 Teawn dma? y eon th G 3IN GN 2 "2 ( D=x)2 ~ = = _ AL PB net = 0 ‘| Te Im =6 Q- Te # In . G3IN- : Gy aq esl > a (o- % Dm > O-x = % ®. X24 p 7 e- a) aL . = X +H => O=(0K x= p . 9 10 a Scanned with CamScanner =) SING 2 ¢ te" W page no. \\ : 0-2 42,8 ep) Ti a= TM 14,4 710,14 119721391 BS v > Gvavitotionol Potential - work done ber unit mass by external agent to bring v ® tesi mass from yefference point (generally infinity) Do i _HOiven oink P, without change in Ws kinetic PP) energy . ; Vp = (Wert bor P _ (-We7 fore) wo @ Me RN eal 7 . an 8 lots) 1 FS OM > Y S 3S Note ~ it is a scaler quontty it moy be tve » -ve OY zero SS ~ und is Tovle | kg i» it depends on veffeyence point =D. > obje ck Potential Point mass 4——y—— P Ve= —_ = OW axis -&h S Ring (NR) [nt tp JR? % 5S Vv Ncentve = -GN 4 R od = —F re 7 _e he veorive Ss solid ebheve ; 47/7 mc 92 voluiwdmnitae f/f ian Ss a \ if S/} a ovity . // q “Ta se s Wars s Scanned with CamScanner gravitational potential diffeyence. difference in _gravitokyonat ~ botential Qrowitodi onal Potential energy - (Wext jo > P= Us — (Udgy- fd) no —> P mV= - Gite ® | SO0lid sbheve = eayth CCR RAAT a \ m ™ U= - Gam m. 2 Ss ™ hee 2 Fiat V=-Gmm_ «3 [= Ws \o + |= mn = [=e Sout Pee a Us -Gm24 = Gm? x2 Y wz Qa bayticle of mass two is divided in ow bayts: mym,H-Mme , Hom ahd placed a+ 4 corners of squoyve of side a , find U__ go that potential ° ral % a eneygy of system 1S Max. —— Nays= - Gm (N-m) 4 G(N-m)? - Genz Ga alz OYZ N-mL—___Im W= -G Tg Can h—m?)+(M=m)2cm? | ol Ia. 5 NASFROSS PPA TE dy =- & [4lim-a, +2(M-m! (o-t) + 2m] am a tl ) 7 O= = -a [ 6M- 8m + -OM 49m
Ui=- GN m aA R Solids MR) ®: GPE _ ob Be> -ot height h 3 U2=_~GNm 7 Reh ‘ 1 \novease_ io _ay PE 7 Yr AU= U2 -v, = Gm ‘2 [Gum] os GHmpty «h] Lis R+h (Re) [en &J 2 ~ Rafesh) _AUs Gum hh = Dm. R(Qshb) R (R+h) Ts= gy. fF od surface of earth . > GN . fey ® Re A (Increase in gy PE de to easth. » BV = GNem 4 “ Re Reth) % > Mm GNe —h Avemah hedCe Re , Ve ReRe | Vee) Re = GudokKin = Scanned with CamScanner Accelevotion dueto qvowity / gyav-Fd due to earth. —~ tlovtside y>R A Tout =9s= GM Z 2h = (Miia ¥=Reth we 2 Qout= GI _ (Re+h)? 7 if height h<<éRe t [Re = 6400ln] a go> GN ='GmM (te, \ a> ae 2 i R! R2 =) Re ‘ aut =(9/1-2h)) - a % Chonge in ace" dhe to Qrowity ec Ag = govt-g “2 =g[I-2h)-g => 9[-2h) . ea ar 7- charge in acc dve 40 gyowity. t Ag xioo0ve ~2Qh vino % m i) R 2 a) oat Surface of eayth “ Q= GM o - 4-8 mis? ® R%e al vr (Inside earth Free] / 2h ©\ Lin _GMr ( "er 4 RB ee, x = GN x Qin = qv 4 Qa? R Y —R s Scanned with CamScanner At depth d Y+d =R y= R-d in=9 DL =q { Bed ain R al ze gin =9 [1-4 | ' R/ Change in ace due to _gyowity. Ag= gin 92 9(*-d \-gs 3 [~d | S7 Change in acc _dve 4 gyowity, Ag xtao'% = -d xioo 3 9 in Py ot centre 4 =Q > L — . iy ; - ¥ €R_ inside d ate >= oA YSR © ovbide > | Se gn oy geal : a a ——— Xe x2 y yeR Y>R 7 -| acc due to growity is max. at surface (g= 2-8 mis) y -| when we move towards the centre of earth from Surface ay = then acc due to gravity decreases incveases ® -| when we move awoy From contre Of earth » Hom 5 _ susfoce then ace due to gyavity decyveases 5 7 Foclors affecting acc due to gravity 3 » eT Scanned with CamScanner cqlwlote height above the earth Surface where gvowitadional field is G:257. of suvface — go = GN = 6-25 os et Midbdin Y2 (oo f. I + Pari |\ GN sobt GU 2 (4 Q? $i Ve seer he EL = -G ¥= GR h-=y-R => GR-R=3R Lind distance rem _contye of earth where GYovityotional Field is 9/4 case | — Inside | gin = 9(4 2 ie oR = 14 in can sles aT os a y= 2/4 ose U— outeick Go = GU. 4q 7 & GM _ GH ct ia R? 4 lL. toss ye QR gy f& 2 find Gravitational Field at dopth of ¢&km gin = q( L- 64 \ 64000) g {t-t \. of gc) \ Coo ( loo} => 4a. of 9 Scanned with CamScanner y| Find tne gyavitakenal fd_ot eight of 64 om 5 gout = gf 1- 2x64 \ ri 64x100/ & g: [ 4g \ =a87 ofg & { 100} > Escape velocity aS minimum Sheed yequived to send % body out of 5 ayovitakonal field of planet 4 a {7 AN Zz v => aA = = “> Rate of aver webbed => ! 4 7 aa Vmoy Tn 2 Benet Den Qn | ae S. N wtafet A Se Prax a 4 Venn +y = 2a Ym Gx Vin - Von Yn er ES Ymin max = Qirrin = all-2) Yman 7 a(\e) —— max = | 2Gu (te) Nmin= am (t-e) J 9a U-2) 2a (Ite “Total emoergy of a planet: kKE+PE Linyt- Gm = ~ -— GUm (always) ~ voviatle ; F vavierle 3 26a . dn -FW -~ 1.5 Kebler's second law is indeed at 4 Onn tow of ConServation Of Angela, momentum | Bkebley’s tow of time bevisd. +? X73 Q = Seni —Mojoy QWs 2 Soselight b And ¥) or oy biting Qydvh tho earth in_O¥ bis Of vadiuvs ¥ and Gy then Yotres of thaiy Sime pewed. Scanned with CamScanner 
