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1 Section 4 Gavity and the Sola System The oldest common-sense view is that the eath is stationay (and flat?) and the stas, sun and planets evolve aound it. This GEOCENTRIC MODEL was poposed explicitly by Ptolemy of Alexandia in the nd centuy AD BC. Howeve, as ealy as cica 70 BC, Aistachus of Samos had poposed that the Eath evolved ound the Sun. This HELIOCENTRIC MODEL did not gain widespead ageement Achimedes among othes did not accept it. So the Geocentic Model pevailed until COPERNICUS ( ) evived the Heliocentic Model as a means of esolving the poblem of planetay Retogade Motion. and in 005 Paths of planet Mas in sky
2 Galilei GALILEO ( ), using telescopes fo the fist time to view the heavens, obseved the phases of Venus and the Moons of Jupite, which both suppoted the Heliocentic Model. Johannes KEPLER ( ) used expeimental data on planetay obits obtained by Tycho BRAHE ( ) to make a theoetical analysis. He deduced thee Laws (simply fom obsevation, not fom any undelying theoy). Keple s Laws 1. The planets move on elliptical obits with the Sun at one focus of the ellipse.. The planet sweeps out equal aeas in equal times. (While called the Law of Aeas, it is about the vaiations of speed of the planet aound its obit). 3. The squae of the peiod of a planet (the length of its yea) is popotional to the cube of the adius of its obit. Regading Law No.1, note that an ellipse is chaacteised by is semi-majo axis a, by its semi-mino axis b, and it eccenticity e. These ae elated by b e 1 a Fo a cicula obit, a = b and e = 0. The obit has a futhest point fom the Sun, called the Aphelion and a neaest point, the Peihelion. Fo an obit aound the Eath, these ae called the Apogee and Peigee. The majo axis joins these points.
3 Fo the Eath, e = Fo Venus it is still smalle, Only fo Pluto (not cuently a eal planet) it is as high as Fo comets, it is close to 1. focus x y Cented at oigin: 1 b a Pola, oigin at focus: 1 ecos l l is the semilatus ectum, b l a Regading Law No., note that the aea swept out in a shot time dt is da ½ d So da d constant ½ ½ ½v dt dt That is, L mv constant and the toque, dl 0 dt If the toque is zeo, any foce is a cental foce. Regading Law No.3, we shall see late that the deivation of 3 P is vey staightfowad fo a cicula obit with invesesquae gavitation. It is less easy fo elliptical obits, fo which the semi-majo axis a is the key paamete eplacing the adius: 3 P a
4 Law of Univesal Gavity Many wokes tied to find what foce-law would poduce motion accoding to Keple s Laws. Isaac NEWTON ( ) established that an INVERSE-SQUARE LAW would do it that is, he found that the adial acceleation in a Kepleian obit vaies as the invese squae of the distance to the sun. Using his mechanics, F = ma, he had at once the invese-squae foce-law. Newton was still a long way fom establishing univesal gavitation as the cause. Fo this, he needed a theoetical beakthough and an expeimental veification. Theoetical Beakthough: Newton was the fist to ealise that the Laws of Physics which ae established on Eath might also apply in the heavens in what the Ancients had called the supelunay pat of the Univese. Then his invese-squae foce might be the foce familia on Eath known as gavity. He knew the acceleation due to gavity at the suface of the Eath say, 10 m s -1, and the adius of the Eath say, 6380km. He could calculate the centipetal acceleation of the Moon towads the Eath in its obit: v a M km 7.3 days m s So his cucial test was whethe the acceleations go as the atio of the squaes of the distances:
5 E M.7810 am ae This beautiful confimation enabled him to popose the Law of Univesal Gavitation, 4 F GMm Newton could only make ough estimates of the value of G, the Univesal Gavitational Constant, as the mass of the Eath could only be estimated. Finding G moe accuately is an impotant expeiment that has often been called Weighing the Eath. See the tosion balance expeiment of Heny CAVENDISH ( ). It is still the least accuate of all the fundamental constants, being G = ( ) N m kg This makes gavitation vey weak. The atio between the gavitational attaction and elecostatic epulsion between a pai of potons is
6 Gavity above the Eath s Suface. M E R E h Altitude h M E = kg R E = m G = N m kg GMm F mg GM E g R h E 9.81ms 8.18ms at h 0 at h 600km Deivation of Keple s 3 d Law: Conside a satellite in a cicula obit at adius aound a planet of mass M Centipetal foce = gavitational foce, i.e. Intoduce the peiod Reaanging, T mv GMm T v 3 GM 4, then GM 4 T. This is Keple s 3 d Law fo the special case of a cicula obit. The calculation fo an elliptical obit can be done, and yield the same expession with eplaced by the semi-majo axis, a.
7 Gavitational Potential Enegy Gavitational foce is a vecto quantity, diected along the line joining M and m. GMm F GMm GMm F ˆ 3 It is an attactive, cental, consevative foce. A consevative foce is one unde which the wok done in moving an object fom A to B does not depend on the path taken between A and B. L A L B W And W AB W W L L L W B A L F.d L L F.d 0 F.d F.d O, equivalently, a consevative foce is one unde which the wok done in moving an object aound a closed loop is zeo.
8 Fo a small displacement d, the change in potential enegy of the body is du = dw, so that du F. d du F d du d ˆ du d We theefoe have the gavitational potential enegy: GMm U Total Enegy of Gavitational System Conside two point masses m and m, moving at velocities v and v unde thei mutual gavitation, the total mechanical enegy is E KE PE ½ mv ½mv Gmm Let m m, then if we use the Cente-of-Mass fame of efeence we can ignoe the kinetic enegy of m. Now conside a cicula obit of m (a planet) aound m (the Sun). We have centipetal foce equal to gavitational foce, mv Gmm Gmm ½mv ½ Gmm E KE PE ½
9 It is no accident that the total enegy is negative. This distinguishes closed obits (the planet is captued by the Sun s gavitational field), cicula o elliptical, fom open obits. Open obits ae paabolic fo E = 0 and hypebolic fo E > 0. Escape Velocity: Fo a body to escape the gavitational field of anothe (e.g. a comet to leave the Sola System, o a ocket to escape the Eath ballistically), it must have sufficient speed so that E = KE + PE is zeo o positive. The escape velocity is defined by E = 0: Gmm E ½mv esc 0 Gm vesc Putting in the values fo Eath s mass and adius, the escape velocity fo a body at the Eath s suface is appox m s -1. Note that it does not matte what the diection of the velocity is (though it would be as well if it wee not to esult in a collision) the diection meely detemines which open obit the body follows. Obital Speed: Fo a satellite in a cicula obit, mv GMm GMm ½mv ½ GM v We cannot choose obital speed and adius sepaately; fo a given adius thee is a given speed.
10 Obital Peiod: The cicumfeence of the obit is, so the peiod T is T v GM GM in ageement with Keple s 3 d Law and its deivations above. Geosynchonous Obits: To obtain an obital peiod aound the Eath of one day (useful fo telecommunications), we solve 3 GM m 47 km 639 miles This coesponds to an altitude of km ~ km. Fist poposed by Athu C. Clake in an aticle in Wieless Wold in 1945, thee ae now hundeds of geosynchonous satellites in a ing aound the Equato. Time delay: At c, time to geosynchonous satellite and back is ~ m / m s 1 ~ 0.5 s This is vey noticeable on live satellite links on TV and in phone calls outed by satellite. Fo this eason, optical fibe links ae pefeed. 3
m1 m2 M 2 = M -1 L 3 T -2
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