Get Solution of These Packages & Learn by Video Tutorials on GRAVITATION

Transcription

1 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on INTODUCTION Th motion of clstial bodis such as th sun, th moon, th ath and th plants tc. has bn a subjct of fascination sinc tim immmoial. Indian astonoms of th ancint tims hav don billiant wok in this fild, th most notabl among thm bing Aya Bhatt th fist pson to asst that all plants including th ath volv ound th sun. A millnnium lat th Danish astonom Tycobah (546-6) conductd a dtaild study of plantay motion which was intptd by his pupil Johnaas Kpl (57-6), ionically aft th mast himslf had passd away. Kpl fomulatd his impotant findings in th laws of plantay motion. UNIVESA AW OF GAVITATION Accoding to this law "Each paticl attacts vy oth paticl. Th foc of attaction btwn thm is dictly popotional to th poduct of thi masss and invsly popotional to squa of th distanc btwn thm". mm m F o F m G wh G 6.67 Nm kg is th univsal gavitational constant. This law holds good ispctiv of th natu of two objcts (siz, shap, mass tc.) at all placs and all tims. That is why it is known as univsal law of gavitation. Dimnsional fomula of G : F F m m [MT ] [ ] [M [M ] T ] Nwton's aw of gavitation in vcto fom : Gmm Gmm F ˆ & F ˆ W h F is th foc on mass m xtd by mass m and vic-vsa. Now G m ˆ ˆ, Thus m F ˆ. Compaing abov, w gt F F Impotant chaactistics of gavitational foc (i) Gavitational foc btwn two bodis fom an action and action pai i.. th focs a qual in magnitud but opposit in diction. Ex. GAVITATION (ii) Gavitational foc is a cntal foc i.. it acts along th lin joining th cnts of th two intacting bodis. (iii) Gavitational foc btwn two bodis is indpndnt of th natu of th mdium, in which thy li. (iv) Gavitational foc btwn two bodis dos not dpnd upon th psnc of oth bodis. (v) Gavitational foc is ngligibl in cas of light bodis but bcoms appciabl in cas of massiv bodis lik stas and plants. (vi) Gavitational foc is long ang-foc i.., gavitational foc btwn two bodis is ffctiv vn if thi spaation is vy lag. Fo xampl, gavitational foc btwn th sun and th ath is of th od of 7 N although distanc btwn thm is.5 7 km Th cnts of two idntical sphs a at a distanc. m apat. If th gavitational foc btwn thm is. N, thn find th mass of ach sph. (G 6.67 m kg sc ) Gm.m Sol. Gavitational foc F on substituting F. N,. m and G 6.67 m kg sc w gt m.5 5 kg Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

2 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on Ex- Sol. Ex- Sol. Two paticls of masss m and m, initially at st at infinit distanc fom ach oth, mov und th action of mutual gavitational pull. Show that at any instant thi lativ vlocity of appoach is wh is thi spaation at that instant. Th gavitational foc of attaction on m du to m at a spaation is Gmm F F Gm Thfo, th acclation of m is a m G(m + m )/, Gm Similaly, th acclation of m du to m is a th ngativ sign bing put as a is dictd opposit to a. Th lativ acclation of appoach is G (m + m) a a a dv If v is th lativ vlocity, thn a dt d But v (ngativ sign shows that dcass with incaing t ). dt dv d d dt.... () dv a v.... () d Fom () and (), w hav v dv G (m + m) d v Intgating, w gt At, v (givn), and so C. v G (m + m ) + C G(m + m ) G(m + m ) t v v whn. Thn v Th idntical bodis of mass M a locatd at th vtics of an quilatal tiangl with sid. At what spd must thy mov if thy all volv und th influnc of on anoth's gavity in a cicula obit cicumscibing th tiangl whil still psving th quilatal tiangl? t A, B and C b th th masss and O th cnt of th cicumscibing cicl. Th adius of this cicl is sc. t v b th spd of ach mass M along th cicl. t us consid th motion of th mass at A. Th foc of gavitational attaction on it du to th masss at B and C a along AB and Th sultant foc is thfo along AC cos along AD. This, fo psving th tiangl, must b qual to th ncssay cntiptal foc. That is, Mv Mv [ / ] o v Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

3 Gt Solution of Ths Packags & an by Vido Tutoials on Qus.4 Find out th tim piod of cicula motion in abov xampl FEE Download Study Packag fom wbsit: & Ans. Ex.- 5 Sol. (π / ) A solid sph of lad has mass M and adius.a sphical hollow is dug out fom it (s figu). Its bounday passing though th cnt and also touching th bounday of th solid sph. Dduc th gavitational foc on a mass m placd at P, which is distant fom O along th lin of cnts. t O b th cnt of th sph and O' that of th hollow (figu). Fo an xtnal point th sph bhavs as if its nti mass is concntatd at its cnt. Thfo, th gavitatinal foc on a mass `m` at P du to th oiginal sph (of mass M) is Mm F G, along PO. Th diamt of th small sph (which would b cut off) is, so that its adius OO' is /. Th foc on m at P du to this sph of mass M' (say) would b M m F G ( ) along PO. [ distanc PO ] As th adius of this sph is half of that of th oiginal sph, w hav M 8 M. Mm F G along PO. 8( ) As both F and F point along th sam diction, th foc du to th hollowd sph is m F F m 8 ( ) m 8( ). GAVITATIONA FIED Th spac suounding th body within which its gavitational foc of attaction is xpincd by oth bodis is calld gavitational fild. Gavitational fild is vy simila to lctic fild in lctostatics wh chag 'q' is placd by mass 'm' and lctic constant 'K' is placd by gavitational constant 'G'. Th intnsity of gavitational fild at a points is dfind as th foc xpincd by a unit mass placd at that point. Ex-6 F E m Th unit of th intnsity of gavitational fild is N kg. In vcto fom E ˆ F [MT ] Dimnsional fomula of intnsity of gavitational fild [M T ] m [M] Find th lation btwn th gavitational fild on th sufac of two plants A & B of masss m A, m B & adius A & B spctivly if (i) thy hav qual mass (ii) thy hav qual (unifom) dnsity t E A & E B b th gavitational fild intnsitis on th sufac of plants A & B. thn, E A Similaly, Gm A A 4 G π A ρ A 4Gπ ρ A A A Gm 4G B E B π B E A B (i) fo m A m B E B A ρ B B. Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

4 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on E A A (ii) Fo & ρ A ρ B EB B 4. GAVITATIONA POTENTIA Th gavitational potntial at a point in th gavitational fild of a body is dfind as th amount of wok don by an xtnal agnt in binging a body of unit mass fom infinity to that point, slowly (no chang in kintic ngy). Gavitational potntial is vy simila to lctic potntial in lctostatics. t th unit mass b displacd though a distanc d towads mass M, thn wok don is givn by dw F d d. dw d. Thus gavitational potntial, V. Th unit of gavitational potntial is J kg. Dimnsional Fomula of gavitational potntial Wok [M T ] [M mass [M] T ]. 5. EATION BETWEEN GAVITATIONA FIED AND POTENTIA Th wok don by an xtnal agnt to mov unit mass fom a point to anoth point in th diction of th fild E, slowly though an infinitsimal distanc d Foc by xtnal agnt distanc movd Ed. dv Thus dv Ed E. d Thfo, gavitational fild at any point is qual to th ngativ gadint at that point. Ex.7 Th gavitational fild in a gion is givn by E (N/kg) ( î + ĵ). Find th gavitational potntial at th oigin (, ) (in J/kg) Sol. (A*) zo (B) (C) (D) can not b dfind V. d + E [ Ex.dx Ey. dy] x + y at oigin V Ex.8 In abov poblm, find th gavitational potntial at a point whos co-odinats a (5, 4) (in J/kg) (A) 8 (B*) 8 (C) 9 (D) zo Sol. V J/kg Ex.9 In th abov poblm, find th wok don in shifting a paticl of mass kg fom oigin (, ) to a point (5, 4) (In J) (A) 8 (B*) 8 (C) 9 (D) zo Sol. W m (V ƒ V i ) (8 ) 8 J 6. GAVITATIONA POTENTIA & FIED FO DIFFEENT OBJECTS I. ing. V / xo (a + ) & E (a ˆ / + ) cosθ o E x Gavitational fild is maximum at a distanc, ± a and it is a Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 4 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

5 Gt Solution of Ths Packags & an by Vido Tutoials on FEE Download Study Packag fom wbsit: & II. A lina mass of finit lngth on its axis : (a) Potntial : V (b) Fild intnsity : ln (sc θ + tan θ ) E sin θ d d + d ln + + d d III. An infinit unifom lina mass distibution of lina mass dnsity λ, H θ π. IV. And noting that λ M in cas of a finit od Gλ w gt, fo fild intnsity E d Potntial fo a mass-distibution xtnding to infinity is not dfind. Howv vn fo such mass distibutions potntial-diffnc is dfind. H potntial diffnc btwn points P and P spctivly at distancs d and d fom th infinit od, v Gλ ln Unifom Solid Sph (a) Point P insid th shll. < a, thn V (a ) & E, and at th cnt V a a (b) Point P outsid th shll. > a, thn V d d a & E and E Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 5 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

6 Gt Solution of Ths Packags & an by Vido Tutoials on FEE Download Study Packag fom wbsit: & Ex. V. Unifom Thin Sphical Shll VI. (a) Point P Insid th shll. < a, thn V (b) Point P outsid shll. > a, thn V Unifom Thick Sphical Shll (a) Point outsid th shll M M V G ; E G (b) Point insid th Shll V E (c) Point btwn th two sufac V ; E a & E & E Calculat th gavitational fild intnsity at th cnt of th bas of a hollow hmisph of mass M and adius. (Assum th bas of hmisph to b opn) Sol. W consid th shadd lmntal ing of mass, dm Fild du to this ing at, M π sinθ (dθ) (π ) Gdmcosθ de (s fomula fo fild du to a ing) o, de sin θ cos θ dθ Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 6 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

7 Gt Solution of Ths Packags & an by Vido Tutoials on FEE Download Study Packag fom wbsit: & Ex. Sol. Hnc, o, E π E / π / de sin θ cos θ dθ Calculat th gavitational fild intnsity and potntial at th cnt of th bas of a solid hmisph of mass m, adius. W consid th shadd lmntal disc of adius sinθ and thicknss dθ Its mass, dm M π M o dm sin θ dθ Fild du to this plat at O, π ( sin θ) (dθ sin θ) GdM( cosθ) de (sinθ) (s fild du to a unifom disc) sinθ( cosθ)dθ o de π / π / π / E de cos θ cosθ + o E sinθ( cosθ) dθ Now potntial du to th lmnt und considation at th cnt of th bas of th hmisph, dv o, dv V GdM (cosc θ cot θ) sin θ(cos cθ cot θ)dθ (sinθ) π / (sin θ cosθsinθ)dθ (s potntial du to a cicula plat) cos cosθ + o, v Alit : Consid a hmisphical shll of adius and thicknss d M M d Its mass,dm (π d) o, dm π Sinc all points of this hmisphical shll a at th sam distanc fom O. Hnc potntial at O du to it is, dv Gdm d V dv 7. GAVITATIONA POTENTIA ENEGY Gavitational potntial ngy of two mass systm is qual to th wok don by an xtnal agnt in assmbling thm, whil thi initial spaation was infinity. Consid a body of mass m placd at a distanc x fom anoth body of mass M. Th gavitational foc of attaction btwn thm is givn by, m F. θ π / Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 7 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

8 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on Now, t th body of mass m is displacd fom point. C to B though a distanc 'd' towads th mass M, thn wok don by intnal consvativ foc (gavitational) is givn by, Spcial Cass: (i) Ex. Sol. m m dw F d d dw d Gavitational potntial ngy, m U Fom abov quation, it is cla that gavitational potntial ngy of two mass systm incass with incas in spaation () (i.. it bcoms lss ngativ). (ii) Gavitational P.E. bcoms maximum (o zo) at. (iii) If th body of mass m movs fom a distanc to ( > ), thn wok don o chang in gavitational P.E. is givn by du m d m d m Sinc >, so chang in gavitational P.E. of th body is ngativ. It mans, whn th body is bought na to th ath, P.E. of th ath-mass systm dcas. (iv) Whn th body of mass m is movd fom th sufac of ath (i.., ) to a hight h (i.., + h), thn chang in P.E. of th ath-mass systm s givn by du m h + m m h + + h/ m h mh Using binomial xpansion du Sinc g thn du mgh Gavitational potntial diffnc is dfind as th wok don by an xtnal agnt to mov a unit mass fom on point to th oth point in th gavitational fild. Accoding to th dfinition, E is th foc xpincd by a unit mass at A. Th diction of this foc is towads th body of mass M. Now th wok don to mov th unit mass fom A to B is givn by dw F. d x Edx cos 8º Edx This wok don is qual to th gavitational potntial diffnc (dv). dv Wh is calld potntial gadint. dx Calculat th vlocity with which a body must b thown vtically upwad fom th sufac of th ath so that it may ach a hight of, wh is th adius of th ath and is qual to m. (Eath's mass 6 4 kg, Gavitational constant G 6.7 nt-m /kg ) Th gavitational potntial ngy of a body of mass m on ath's sufac is U () m wh M is th mass of th ath (supposd to b concntatd at its cnt) and is th adius of th ath (distanc of th paticl fom th cnt of th ath). Th gavitational ngy of th sam body at a hight fom ath's sufac, i.. at a distanc fom ath's cnt is U ( ) m chang in potntial ngy U ( ) U() m m m Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 8 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

9 Gt Solution of Ths Packags & an by Vido Tutoials on This diffnc must com fom th initial kintic ngy givn to th body in snding it to that hight. Now, FEE Download Study Packag fom wbsit: & Ex. Sol. supos th body is thown up with a vtical spd v, so that its initial kintic ngy is mv. Thn mv m m o v. Putting th givn valus : v (6.7 nt m /kg ) (6 6 (6.4 m) 4 kg).7 4 m/s. Distanc btwn cnts of two stas is a. Th masss of ths stas a M and 6 M and thi adii a a & a sp. A body is fid staight fom th sufac of th lag sta towads th small sta. What should b its minimum initial spd to ach th sufac of th small sta? t P b th point on th lin joining th cnts of th two plants s.t. th nt fild at it is zo G.6M Thn, ( a ) (a ) 6 G.6M Potntial at point P, v P (a ) a 4 a 5. a a a Now if th paticl pojctd fom th lag plant has nough ngy to coss this point, it will ach th small plant. Fo this, th K.E. impatd to th body must b just nough to ais its total mchanical ngy to a valu which is qual to P.E. at point P. i.. o, G(6M)m m mv mv a 8a P v 8 5m o, v a 8a a 45 4a o, v min 5 a 8. GAVITATIONA SEF-ENEGY Th gavitational slf-ngy of a body (o a systm of paticls) is dfind as th wokdon by an xtnal agnt in assmbling th body (o systm of paticls) fom infinitsimal lmnts (o paticls) that a initially an infinit distanc apat. Gavitational slf ngy of a systm of n paticls Potntial ngy of n paticls at an avag distanc '' du to thi mutual gavitational attaction is qual to th sum of th potntial ngy of all pais of paticl, i.., mim U s G all pais ij j i i This xpssion can b wittn as U s n G i j If consid a systm of 'n' paticls, ach of sam mass 'm' and spatd fom ach oth by th sam avag distanc '', thn slf ngy o U s G n i n j j i m i j j n j j i m m Thus on th ight handsid 'i' coms 'n' tims whil 'j' coms (n ) tims. Thus m U s Gn (n ) i ij j Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 9 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

10 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on Gavitational Slf ngy of a Unifom Sph (sta) U shll G ( 4π d ρ) 4 π ρ G (4 πρ) 4 d, wh ρ U sta G (4 πρ) 4 d G (4πρ) U sta 5 M 4 π 5 5 4π G ρ 5 9. ACCEEATION DUE TO GAVITY : It is th acclation, a fly falling body na th ath s sufac acquis du to th ath s gavitational pull. Th popty by vitu of which a body xpincs o xts a gavitational pull on anoth body is calld gavitational mass m G, and th popty by vitu of which a body opposs any chang in its stat of st o unifom motion is calld its intial mass m Ι thus if E is th gavitational fild intnsity du to th ath at a point P, and g is acclation du to gavity at th sam point, thn m Ι g m G E. Now th valu of intial & gavitational mass happn to b xactly sam to a gat dg of accuacy fo all bodis. Hnc, g E Th gavitational fild intnsity on th sufac of ath is thfo numically qual to th acclation du to gavity (g), th. Thus w gt, g wh, M Mass of ath adius of ath Not : H th distibution of mass in th ath is takn to b sphical symmtical so that its nti mass can b assumd to b concntatd at its cnt fo th pupos of calculation of g.. VAIATION OF ACCEEATION DUE TO GAVITY (a) Effct of Altitud Acclation du to gavity on th sufac of th ath is givn by, g Now, consid th body at a hight 'h' abov th sufac of th ath, thn th acclation du to gavity at hight 'h' givn by g h ( ) + h g h h + ~ g whn h <<. Th dcas in th valu of 'g' with hight h g g h g gh h 'g' % g gh.. Thn pcntag dcas in th valu of Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

11 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on (b) Effct of dpth Th gavitational pull on th sufac is qual to its wight i.. mg mg 4 G π ρm o g 4 π G ρ...() m Whn th body is takn to a dpth d, th mass of th sph of adius ( d) will only b ffctiv fo th gavitational pull and th outwad shall will hav no sultant ffct on th mass. If th acclation du to gavity on th sufac of th solid sph is g d, thn g d 4 π G ( d) ρ...() By dividing quation () by quation () d g d g IMPOTANT POINTS (i) At th cnt of th ath, d, so g cnt g th cnt of th ath is zo.. Thus wight (mg) of th body at g gd d (ii) Pcntag dcas in th valu of 'g' with th dpth. g (c) Effct of th sufac of Eath Th quatoial adius is about km long than its pola adius. W know, g quato to th pol. Hnc gpol > g quato. Th wight of th body incas as th body takn fom th (d) Effct of otation of th Eath Th ath otats aound its axis with angula vlocity ω. Consid a paticl of mass m at latitud θ. Th angula vlocity of th paticl is also ω. Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

12 Gt Solution of Ths Packags & an by Vido Tutoials on FEE Download Study Packag fom wbsit: & Accoding to paalllogam law of vcto addition, th sultant foc acting on mass m along PQ is F [(mg) + (mω cosθ) + {mg mω cosθ} cos (8 θ)] / [(mg) + (mω cosθ) (m gω cosθ) cosθ] / mg ω + g ω cos θ cos g θ / At pol θ 9 g pol g, At quato θ g quato g ω g. Hnc g pol > g quato If th body is takn fom pol to th quato, thn g g ω g. Hnc % chang in wight ω mg mg g m ω mg mg ω g. ESCAPE SPEED Th minimum spd quid to pojct a body fom th sufac of th ath so that it nv tuns to th sufac of th ath is calld scap spd. A body thown with scap spd gos out of th gavitational pull of th ath. Wok don to displac th body fom th sufac of th ath o ( ) to infinity ( ) is givn by dw W m d m m d m W m t v b th scap spd of th body of mass m, thn kintic ngy of th body is givn by mv m v g. km s. Impotant Points. Escap spd dpnds on th mass and siz of th plant. That is why scap vlocity on th Jupit is mo than on th ath.. Escap spd is indpndnt of th mass of th body.. Any body thown upwad with scap spd stat moving aound th sun. Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

13 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on MOTION OF SATEITES AND KEPE AWS A havnly body volving aound a plant in an obit is calld natual satllit. Fo xampl, moon volvs aound th plant th ath, so moon is th satllit of th ath. Thi motions can b sttudid with th hlp of kpl's laws, as statd : I. aw of obit : Each Plant movs aound th sun in an lliptical obit with th sun at on of th foci as shown in figu. Th ccnticity of an llips is dfind as th atio of th distanc SO and AO SO i.. AO SO SO a a Th distanc of closst appoach with th sun at F is AS. This distanc is calld pig. Th gatst distanc (BS) of th plant fom th sun is calld apog. Pig (AS) AO OS a a a ( ) Apog (BS) OB + OS a + a a ( + ) II. aw of Aas : Th lin joining th sun and a plant swps out qual aas in qual intvals of tim. A plant taks th sam tim to tavl fom A to B as fom C to D as shown in figu. (Th shadd aas a qual). Natually th plant has to mov fast fom C to D. Th law of aas is idntical with th law of consvation of angula momntum. Aal vlocity aa swpt tim Hnc ω constant. (dθ) dt dθ constant dt III. aw of piods : Th squa of th tim fo th plant to complt a volution about th sun is popotional to th cub of smimajo axis of th lliptical obit. i.. Cntiptal foc Gavitational foc m v m v Now, spd of th plant is Cicumfnc of th cicula obit π v Tim piod T Substituting valu in abov quation 4π T o T 4π Sinc 4π T is constant, T o constant Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

14 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on Ex.4 A satllit is launchd into a cicula obit 6 km abov th sufac of th ath. Find th piod of volution if th adius of th ath is 64 km and th acclation du to gavity is 9.8 m/sc. At what hight fom th gound should it b launchd so that it may appa stationay ov a point on th ath's quato? π Sol. Th obiting piod of a satllit at a hight h fom ath's sufac is T g Ex.5 Sol. π( + h) + h thn, T g H, 64 km, h 6 km /4. Thn T π ( + ) 4 + g 4 / wh + h π( J) / g 6 Putting th givn valus : T m (.5) 9.8 m / s / 79 sc.97 hous Now, a satllit will appa stationay in th sky ov a point on th ath's quato if its piod of volution ound th ath is qual to th piod of volution of th ath ound its own axis which is 4 hous. t us find th hight h of such a satllit abov th ath's sufac in tms of th ath's adius. t it b n. thn T π( + n) + n g (575 sc) ( + n) / (.4hous) ( + n) / Fo T 4 hous, w hav (4 hous) (.4) hous) ( + n) / 6 π g ( + n)./ mt / sc 9.8 mt / sc ( + n) / 4 o ( + n) / 7 o + n (7) / o n 5.6 Th hight of th go-stationay satllit abov th ath's sufac is n km.59 4 km. In a doubl sta, two stas (on of mass m and th oth of m) distant d apat otat about thi common cnt of mass. Dduc an xpssion of th piod of volution. Show that th atio of thi angula momnta about th cnt of mass is th sam as th atio of thi kintic ngis. Th cnt of mass C will b at distancs d/ and d/ fom th masss m and m spctivly. Both th stas otat ound C in thi spctiv obits with th sam angula vlocity ω. Th gavitational foc acting on ach sta du to th oth supplis th ncssay cntiptal foc. G(m)m Th gavitational foc on ith sta is. If w consid th otation of th small sta, th cntiptal d foc (m d mdω ω ) is m ω and fo bigg sta i..sam G(m)m d Gm o ω d d m ω π Thfo, th piod of volution is givn by T ω Th atio of th angula momnta is ( Ι ω) ( Ι ω) big small Ι Ι π d Gm big small d (m) d m, Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 4 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

15 Gt Solution of Ths Packags & an by Vido Tutoials on FEE Download Study Packag fom wbsit: & sinc ω is sam fo both. Th atio of thi kintic ngis is which is th sam as th atio of thi angula momnta. ( Ιω ( Ιω ) big ) small. SATEITE SPEED(O OBITA SPEED) Th spd quid to put th satllit into its obit aound th ath is calld obital spd. Th gavitational attaction btwn satllit and th ath povids th ncssay cntiptal foc. m mv ( + h) ( + h) v o, v o ( + h) Whn h << thn v g ( + h) g ( + h) v ms 7.9 km s Tim piod of Satllit Tim piod, T ( h) Cicumf nc of th obit π + obital spd v ( + h) g π But v T ( + h) g Hight of th satllit abov th ath's sufac Tim piod of satllit is givn by, T T 4π ( + h) ( + h) g T 4π g ( + h) ( h) π + g T o ( + h) g 4π h T 4π g Ι Ι big small ( h) π + g, Engy of a Satllit P.E. of a satllit of mass m volving aound th ath in a cicula obit of th ath is givn by U mv m and K.E. mv m m o mv. Hnc K.E. m m m m Total Engy E U + K.E. + o E Sinc total ngy is ngativ, so it implis that satllit is bound to th ath. If satllit is clos to m th sufac of th ath thn total ngy E. Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 5 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

16 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on 4. GEO-STATIONAY SATEITES O GEO-SYNCHONOUS SATEITES (i) Th tim piod of th satllit aound th ath must b qual to th otational piod of th ath (i.. 4 hous.) (ii) Th diction of motion of th satllit must b sam as that of th ath. i.. fom wst to ast. Th hight of th gio-stationay satllit fom th sufac of th ath can b calculatd fom th T g quation h 4 π Now T 4 hous 4 6 s, m, g 9.8 ms 6 ( 4 6) ( 6.4 ) h 6.4 4π o h 59 m 59 km. Uss of Atificial Satllits Som impotant uss of atificial satllits a : (i) Thy a usd as communication satllits to snd mssags to distant placs. (ii) Thy a usd as wath satllits to focast wath. (iii) Thy a usd to xplo th upp gion of th atmosph. (iv) Thy a usd to tlcast T.V. pogams to distant placs. (v) Thy a usd to know th xact shap of th ath. 5. AUNCHING OF AN ATIFICIA SATEITE AOUND THE EATH Th satllit is placd upon th ockt which is launchd fom th ath. Aft th ockt achs its maximum vtical hight h, a sphical mchanism givs a thust to th satllit at point A (fig.) poducing a hoizontal spd v. Th total ngy of th satllit at A is thus. m E mv + h Th obit will b an llips (closd path), a paabola, o an hypbola dpnding on whth E is ngativ, zo, o positiv. In all cass th cnt of th ath is at on focus of th path. If th ngy is too low, th lliptical obit will intsct th ath and th satllit will fall back. Othwis it will kp on moving in a closd obit, o will scap fom th Eath, dpnding on th valus of v and. (a) (b) Hnc a satllit caid to a hight h (<< ) and givn a hoizontal spd of 8 km/sc will b placd almost in a cicula obit aound th ath (fig.) If launchd at lss than 8 km/sc, it would gt clos and clos to th ath until it hits th gound. Thus 8 km/sc is th citical (minimum) spd. Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 6 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

17 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on (a) Obits and Spd: Fo a body on th ath s sufac, pojctd hoizontally with a spd v, th tajctoy dpnds on th valu of its spd v. Vlocity (v) Tajctoy (i) ss than th obital spd v < g (i) Body tuns to th ath (ii) Equal to obital spd v g (ii) Body acquis a na th ath cicula obit (iii) Btwn obital and scap (iii) Body acquis an iliptical obit with spd g < v < g th ath as th na focus (iv) Equal to scap spd v g (iv) Body just scaps th ath s gavity along in a paabolic path. (v) Gat thn scap spdv g (v) Body scap s th ath s gavity in a hypbolic path. Ex.6 Sol. A ockt stats vtically upwad with spd v. Shown that its spd v at hight h is givn by v v hg h +, wh is th adius of th ath and g is acclation du to gavity at ath's sufac. Hnc dduc an xpssion fo maximum hight achd by a ockt fid with spd.9 tims th scap vlocity. Th gavitational potntial ngy of a mass m on ath's sufac and that a hight h is givn by U () m and U ( + h) m + h U( + h) U() m + h mh ( + h) mhg h + [ g ] This incas in potntial ngy occus at th cost of kintic ngy which cospondingly dcass. If v is th vlocity of th ockt at hight h, thn th dcas in kintic ngy is mv mv mhg h +, o v v gh h + mv mv. Thus, t h max b th maximum hight achd by th ockt, at which its vlocity has bn ducd to zo. Thus, substituting v and h h max in th last xpssion, w hav v gh + max h max o v h max v g o h + max v gh max o h max Now, it is givn that v.9 scap vlocity.9 (g) h max (9.9)g (9.9) g g.6 g g v v g 4.6 Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 7 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

18 FEE Download Study Packag fom wbsit: & Gt Solution of Ths Packags & an by Vido Tutoials on Ex- 7 Fo a paticl pojctd in a tansvs diction fom a hight h abov Eath s sufac, find th minimum initial vlocity so that it just gazs th sufac of ath path of this paticl would b an llips with cnt of ath as th fath focus, point of pojction as th apoj and a diamtically opposit point on ath s sufac as pig. Sol. Suppos vlocity of pojction at point A is v A & at point B, th vlocity of th paticl is v B. thn applying Nwton s nd law at point A & B, w gt, mv A ρ A m ( + n) Wh ρ A & ρ B a adius of cuvatu of th obit at points A & B of th llips, but ρ A ρ B ρ(say). Now applying consvation of ngy at points A & B m + + h mv A m + mv m (mvb mv ( + h) A ) ρ m ( + h) ( + h) ρ o, ρ V + h + A ( + h) ( + ) wh distanc of point of pojction fom ath s cnt + h. Astonomical Data Body Man adius, m B Mass, kg mv m & ρ Man dnsity, Piod of otation kg/m about axis, days Sun Eath Moon B Tko Classs, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phon : , pag 8 Succssful Popl plac th wods lik; "wish", "ty" & "should" with "I Will". Inffctiv Popl don't.

19