8 - GRAVITATION Page 1

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1 8 GAVITATION Pag 1 Intoduction Ptolmy, in scond cntuy, gav gocntic thoy of plantay motion in which th Eath is considd stationay at th cnt of th univs and all th stas and th plants including th Sun volving ound it. Nicolaus Copnicus, in sixtnth cntuy, gav hliocntic thoy in whi h th Sun is fixd at th cnt of th univs and all th plants movd in pfct cicls aound it. Tycho Bah had collctd a lot of data on th motion of plants but did bf analyzing thm. Johanns Kpl analyzd Bah s data and gav th laws of plant y motion known as Kpl s laws. 8.1 Kpl s Laws Fist Law: Th obits of plants a lliptical with th Sun at on of thi two foci. Scond Law: Th aa swpt by a lin, joining th Sun to a plant, p unit tim ( known as aal vlocity of th plan t ) is constant. Thid Law: Th squa of th piodic tim ( T ) of any plant is dictly popotional to th cub of th smimajo axis ( a ) of its lliptica obit. 8. Nwton s univsal law of gavitation Evy paticl in th univs attacts towads it vy oth paticl with a foc dictly popotional o th poduct of thi masss and invsly popotional to th distanc btwn thm. This is th statmnt of Nwton s univsal law of gavitation. Two paticls of masss m 1 and m having position v ctos 1 and spctivly a shown in th ig. By Nwton s law of gavitation, th foc xtd on paticl 1 by paticl is givn by m F 1 G 1m ^ 1 1 l 1 l ^ 1, wh 1, wh distanc btwn th paticls and G univsal constant of gavitation.

2 8 GAVITATION Pag S I unit of G is Nm /kg and its dimnsional fomula is M 1 L T. Its valu is th sam vywh in th univs at all tims and is Nm /kg. It was Cavndish who fist dtmind its valu xpimntally. Th foc xtd on paticl by paticl 1, F 1, is th sam in magnitud but opposit in diction to F 1. Thus, F 1 G m1m ^ 1. Th focs F 1 and 8. Gavitational acclation and vaiations in it F 1 a as shown in th figu. Th acclation of a body poducd by th gavitational foc f th Eath is dnotd by g. Th gavitational foc, F, xtd by th Eath having mass, M and adius, on an objct having mass m and situatd at a distanc ( fom th cnt of th Eath, is F G mm m F g GM ( ) Fo an objct on th sufac of th Eath ( ), th acclation du to gavity is, g GM ( ) Th valu of g vais with hight nd dpth fom th sufac of th Eath and also with th latitud of th plac as discussd blow Vaiation in g with altitud: Using quation 1 ), th acclation du to gavity at a hight h fom th sufac of th Eath is givn by g ( h ) GM ( + h ) ( Q + h ) ( ) Div ding quation ( ) by quation ( ), w gt g ( h ) 1 g ( h ) g ( + h ) h 1 + If h <<, thn by Binomial appoximation, g h 1 + g h 1 + ( 4 ) g ( h ) g h 1 ( 5 ) Eqn. ( 4 ) is valid fo any hight h, but qn. ( 5 ) can b usd only whn h <<.

3 8 GAVITATION Pag 8.. Vaiation in g with dpth fom th sufac of th Eath: Th figu shows an objct of mass m at a dpth d blow th sufac of th Eath. Its distanc fom th cnt of th Eath is d. It can b povd that th matt in th out shll of thicknss d xts no gavitational foc on th objct. Only th matt insid th solid sph of adius xts gavitational foc on it. Assuming th Eath to b a solid sph of unifom dnsity ρ, th mass of th shadd sph of adius is 4 M π ρ 4 G π ρ GM' g ( ) 4 G π ρ g ( ) 4 π G ρ g ( ) g ( ) g ( ) g ( ) Th figu on th ight shows th gaph f g ( ). Th gavitational acclation is zo at th cnt of th Eath and incass lina y up o th sufac of th Eath. It achs a maximum valu of 9.8 m /s on th sufac of th Eath, i fo and is invsly popotional to th squa of th distanc fom th cnt of th Eath fo >. Th acclation du t gavity at a dpth d fom th sufac of th Eath is givn by g ( d ) g ( ) ( d ) g Vaiation in ffctiv g with latitud: Consid a body of mass m at P, locatd on th sufac of th Eath at latitud λ, as shown in th figu. As th body movs ov a cicula path of adius with lina spd v and angula vlocity ω, it mv xpincs a cntifugal foc mω in th diction, PQ. Its componnt in th diction P is mω cos λ. d [ g g ( ) ]

4 8 GAVITATION Pag 4 As th gavitational attaction mg and th componnt of cntifugal foc mω cos λ a in th opposit dictions, th sultant foc acting on th body is mg mω cos λ. If g is th ffctiv gavitational acclation at P, thn mg mg mω cos λ Fom th figu, cos λ g g 1 ω g cos λ g g ω cos λ ( i ) At th quato, λ 0 and cos λ 1. Hnc, th cntifugal acc ation is ω and is maximum. Hnc, th ffctiv acclation du to gavity g is minimum. ( ii ) At th pols of th Eath, λ 90 and cos λ 0. H nc, h ffctiv acclation du to gavity g ( g ) is maximum at th pols. M ov th adius of th Eath at th pols is lss than th adius at th quato. This also sults in th incas in th valu of g. 8.4 Gavitational potntial and gavitational potntial ngy na th sufac of th Eath Th wok don in binging a unit mass fom infinity to a givn point in gavitational fild, against th gavitational fild is dfind as th gavitational potntial ( Φ at that point. Th unit of gavitational potntial is J/kg ( joul/kg ) and its dimnsional fomula is M L T. Th gavitational foc on an objct of unit mass at P, as shown in th figu is GM F, wh ^ is th unit vcto in th diction of. If t displacmnt of th objct und this foc, away fom th cnt of th Eath, is d, h wok don is dw. GM ^ ^ F d. ( d ) GM d ( Q ^ ^ 1 ) Thus, th total wok ( W ) don in binging a unit mass fom infinity to a point situatd at a distanc fom th cnt of th Eath against th gavitational fild which is dfind as th gavitational potntial ( Φ ) at that point is

5 W Φ GM d GM 8 GAVITATION Pag 5 GM Th gavitational potntial fo a point on th sufac of th Eath ( ) is Φ GM Th wok don in binging an objct of mass m, fom infinity to a givn point in gavitational fild is dfind as th gavitational potntial ngy ( U of th combind systm of objct and th Eath at that point. U GM m Fo an objct of mass m lying on th sufac of th gavitational potntial ngy, U GMm Th gavitational potntial and th gavitatio al potntial ngy of a body of mass m du to th Eath s gavitational fild a zo t infini y. Whn a body movs fom infinity to a point in th gavitational fild, its potntial ngy dcass and kintic ngy incass. 8.5 Escap ngy and Escap spd GM Th potntial ngy of a body of mass m on th sufac of th Eath m GM If th body is givn m + ngy in th fom of kintic ngy, it can scap fom th gavitational fild of th Eath and go to infinity. This minimum ngy is calld th binding ngy of th body. It is also calld th scap ngy and th cosponding spd is calld scap spd ( v 1 mv GMm Escap spd, v GM GM ath, g km /s Th scap spd is indpndnt of th mass of th body and can b in any diction. If th stationay body on th sufac of th Eath is impatd spd qual to o mo than th scap spd, it will scap fom th gavitational fild of th Eath fov.

6 8 GAVITATION Pag Satllits A satllit is an objct volving aound a plant und th ffct of its gavitational fild. Its obital motion dpnds on th gavitational attaction of th plant and th initial conditions. Satllit can b natual o atificial. Th Moon is a natual satllit. Sputnik was th fist atificial satllit put into its obit by ussia in India has launchd A abhatta and INSAT sis satllits. Satllits a usd fo scintific, ngining, commci spy ng and militay applications. Suppos a satllit of mass m is launchd in a cicula obit aound th Eath at a distanc fom its cnt. Th ncssay cntiptal foc is povidd by th gavitational pull of th Eath. mvo mm G obital spd of th satllit, v 0 GM Th distanc tavld by th satllit in on volut n in tim qual to its piod T π. π v 0 T T 4 π 4 π T v GM o Thus, Th squa of th piod of th plant is dictly popotional to th cub of its adius. This is Kpl s thid law with fnc to cicula obit. Gostationay satllit: A satllit of th Eath having obital tim piod sam as that of th Eath, i.., 4 hous and moving i quatoial plan is calld a gostationay ( o gosynchonous ) satllit as it appas stationay whn viwd fom th Eath. Putting G Nm /kg, M kg and T s in th quation T 4 π, w gt 4,60 km. GM th hight of th gostationay satllit abov th sufac of th Eath is, h 4,60 6,400 5,860 km. Pola satllit: Th pola satllit obits in a nothsouth diction as th Eath spins blow it in an astwst diction. Thus, it can scan th nti sufac of th Eath. Th satllits which monito wath, nvionmnt and th spy satllits a in low flying pola obits ( km ).