m1 m2 M 2 = M -1 L 3 T -2

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Gyles Wilkins
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1 GAVITATION Newton s Univesal law of gavitation. Evey paticle of matte in this univese attacts evey othe paticle with a foce which vaies diectly as the poduct of thei masses and invesely as the squae of the distance between them. Conside two bodies of mass m1 and m2 sepaated by a distance. Then the foce of attaction F between them is, F α m1 m2 F α 1 2 m1 m2 m1 m2 F α 2 O F = G 2 whee G is a constant which has the same value eveywhee and is known as Univesal constant of gavitation o gavitational constant. Chaacteistics of gavitational foce:- *It is always attactive *It is independent of the intevening medium. *It is a cental foce- it acts along the line joining the centes of the two bodies. *It is consevative. *It is found to be tue fom inteplanetay distance to inteatomic distance. Definition of G. If m1= m2 =1 kg and = 1 m, then G = F newton. Thus gavitational constant may be defined as the foce of attaction between two bodies each of unit mass, sepaated by a distance of unity. Unit of G. F 2 G = = N m 2 m1 m2 kg 2 and ***Value of G = 6.67 x N m 2 / kg 2. F 2 and Dimension of G is G = = M L1 T -2 L 2 m1 m2 M 2 = M -1 L 3 T -2 Acceleation due to gavity. O elation between g and G. Foce between eath and a body nea it is called gavity. The unifom acceleation poduced in a feely falling body due to gavitational pull of the eath is known as acceleation due to gavity. Conside a body of mass m, placed on the suface of eath of mass M and adius. Then foce of gavity acting on the body, F = m g (1) Also, accoding to Newton s gavitational law, F = G M m (2) Fom (1) and (2) ; m g = G M m 2 O g = G M 2 This is the elation between g and G. Note: g is independent of the mass of the body. So diffeent objects on the suface of eath, iespective of thei masses, expeience the same acceleation towads the cente of the eath. Mass and density of the eath:- Since, g = G M 2, the mass of the eath, M= g 2 G

Assuming the eath to be a homogeneous sphee, Volume of the sphee = 4 3 π 3 Density of the eath, D= M V = g 2 x 3 3 g G x 4π 3 = 4π G Vaiation in the acceleation due to gavity of the eath:- 1.

2 Assuming the eath to be a homogeneous sphee, Volume of the sphee = 4 3 π 3 Density of the eath, D= M V = g 2 x 3 3 g G x 4π 3 = 4π G Vaiation in the acceleation due to gavity of the eath:- 1. Vaiation of g due to the shape of the eath The value of g on the suface of the eath is g= g = G M 1 2 g α 2 Since the pola adius is less than the equatoial adius of the eath; theefoe the value of g is moe at the poles than at the equato. 2.Vaiation of g with altitude( height):- Let g be the acceleation due to gavity on the suface of the eath and g the acceleation due to gavity at a height h above the eath s suface. We have g = GM GM (1) and g = (+ h) (2) eqn (1) / (2) g g' = (+ h) 2 2 = ( 1+ h ) 2 g g = ( 1+ h / ) 2 = g ( 1+ h ) -2 = g ( 1-2h ) Thus acceleation due to gavity deceases with altitude. 3.Vaiation of g with depth:- Assume the eath to be homogeneous sphee ( having unifom density) of adius and mass M. Let ρ be the density of eath. Conside a body of mass m lying on the suface of eath at a place whee g is the acceleation due to gavity. g = GM G x 4 3 π 3 2 = 2 = 4 3 π G ρ Let g be the acceleation due to gavity at a depth h below the suface of the eath then, G 4 GM ' 3 π ( - h )3 4 g' = ( - h ) 2 = ( - h ) 2 ρ = 3 π G (- h) ρ g' g = 4 3 π G (- h) ρ 4 3 π G ρ = ( - h) h g = g ( 1- ) At the cente of the eath (h = ) then g = 0. Thus the value of acceleation due to gavity deceases with the incease of depth. Gavitational field of eath Gavitational field of eath is the space aound the eath whee its gavitational influence is felt.

3 Intensity of gavitational field at a point: It is defined as the foce expeienced by a body of unit mass placed at that point. Gavitational field intensity of eath: ( I ) If a body of mass m is placed on the suface of eath of mass M and adius, then the gavitational foce expeienced by the body, F = G M m 2 Intensity of gavitational field ( I )= F m = GM 2 = g Thus intensity of gavitational field nea the suface of eath is equal to the acceleation due to gavity at the place. Gavitational Potential Gavitational potential at any point inside a gavitational field is defined as the wok done in taking a unit mass fom infinity to that point. Conside a body of mass m situated at O as in figue.the body has a gavitational field suounding it on all sides. The foce F acting on a unit mass placed at a point A, at a distance x fom O is given by F = G m x 1 G m x 2 = x 2 Along AO Now let the unit mass be displaced though a small distance dx fom A to B. If dw is the wok done by the gavitational foce in moving the unit mass fom A to B, GM dx dw = F dx = x 2 Theefoe the total wok done in taking the unit mass fom infinity to P,. The negative sign shows that the gavitational foce is attactive in natue. Gavitational potential enegy due to eath:- Let M be the mass of the eath and its adius.conside two points A and B distance 1 and 2 fom the cente of the eath (1, 2 >)

4 Potential enegy of mas m placed at A = - G M m 1 Potential enegy when it is placed at B = - G M m 2 Incease in P.E when it is moved fom A to B = - G M m 2 - ( - G M m 1 = G M m [ ] 2 If the suface of the eath is taken as zeo potential enegy level then, h is obtained by putting 1 = and 2 = +h ; ie. P.E = G M m [ h ] = G M m [ + h - ( + h) ]= G M m [ If h << then, (neglect h) P.E = [ G M m h ) P.E of a mass at a height h ( + h ) 2 ] = mgh. (We have g = GM ] 2 ) Geostationay o Synchonous satellite A geostationay satellite is an atificial satellite obiting the eath so that its time peiod is synchonous (same) with that of the eath. The obit in which they stay is called paking obit o geostationay obit. Uses: Communication, weathe foecasting etc. Obital velocity of a satellite. The velocity with which a satellite moves in its closed obit is called obital velocity. Conside a satellite of mass m moving with a velocity v aound the eath of mass M and adius. Let is the obital adius of the satellite. Hee the centipetal foce fo otation of satellite is given by gavitational foce. i.e m v 2 = G M m 2 O v 2 = GM v= GM If the satellite is at a height h above the suface of eath, then, = + h (1) v= GM ( +h ) (2) But g = GM 2 GM = g 2 If the satellite is close to the eath, (+ h) Obital velocity, v = GM = g This obit in which the satellite evolve is called minimum obit. The velocity coesponding to minimum obit is called fist cosmic velocity. The value of fist cosmic velocity is 7.92 km/s. Time peiod of a satellite:- It is the time taken by the satellite to evolve once aound the eath. If a satellite moves with a velocity v in an obit of adius, then time peiod T = 2π g

Total enegy of an obiting satellite;- Let M be the mass of eath and m mass of satellite. Let be the adius of obit of the satellite and v the velocity of the satellite, P.E of satellite = - G M m K.

5 Total enegy of an obiting satellite;- Let M be the mass of eath and m mass of satellite. Let be the adius of obit of the satellite and v the velocity of the satellite, P.E of satellite = - G M m K.E of satellite = ½ mv 2 Total enegy of obiting satellite, T.E = P.E + K.E = - G M m 2 The K.E and total enegy ae equal in magnitude. The total enegy is negative and half the potential enegy. Escape Velocity. (v e ) Escape velocity is defined as the least velocity with which a body must be pojected vetically upwads so that it may escape the gavitational pull of eath. Conside a body of mass m placed on the suface of eath of mass M and adius. Let the body be pojected upwads with a velocity ve so that it escapes fom the gavitational field of eath. Then KE of the body nea the suface of eath = ½ m ve 2 PE of the body on the suface of eath, P E = - G M m Total enegy of the body nea the suface = ½ m ve 2 - [ G M m ] At infinity P.E = K.E = 0 The escape velocity of eath is 11.2 km/s. *The escape speed does not depend on the mass of the object and the diection in which the pojectile is fied. Keple s laws of planetay motion Law 1. Evey planet evolves aound the sun in an elliptical obit with sun at one of its foci. Law 2. The adius vecto joining sun and the planet sweeps equal aeas in equal intevals of time. That is aeal velocity of the planet is constant. Law 3. The squae of the peiod of a planet is diectly popotional to the cube of semi majo axis of the obit.

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects