Chapter 13 - Universal Gravitation
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1 Chapte 3 - Unesal Gataton In Chapte 5 we studed Newton s thee laws of moton. In addton to these laws, Newton fomulated the law of unesal gataton. Ths law states that two masses ae attacted by a foce gen by Gmm F, whee G 6.67 x 0 - N m /kg (not g 9.8 m/s ). Fo sphecal masses, s the dstance between the centes of the masses. The weght of a mass, m, on the suface of eath s then m W F, whee s the adus of the eath. Snce we can also wte W mg, then xample: g. Calculate g fom the aboe fomula. (6.67x0 N m / kg )(5.98x0 kg) g 9.8m / s 6 (6.38x0 m) At a dstance of 6.38 x 0 6 m aboe the suface of the eath, g 9.8/().5 m/s and a peson s weght would be ¼ that on the suface. Keple s laws of planetay moton Well befoe Newton s tme, Johannes Keple fomulated hs thee laws of planetay moton. Keple deduced these laws by caefully studyng data on planetay moton obtaned by Tycho Bahe. Isaac Newton was able to explan Keple s laws fom the laws of moton and the law of unesal gataton. Keple s laws ae ) Planets moe n ellptcal obts wth the sun at one of the focal ponts. ) A lne fom the sun to a planet sweeps out equal aeas n equal tmes. 3) The squae of the obtal peod of the planets s popotonal to the cube of the aeage dstance fom the planet to the sun.
2 Keple s st law: It can be shown mathematcally fom Newton s law of gataton that an object wll obt a second masse object n an ellptcal path unless the enegy of the obtng object s so geat that the obt s not closed. An ellpse s an oblong closed cue wth two focal ponts, as shown to the ght. The ellpse can be taced out by F F followng the path such that the sum of the dstances fom the focal ponts to any pont on the cue s constant. That s, + constant. In the case of a planet obtng the sun, the sun s at one of the focal ponts of the ellpse. A ccle s a specal case of an ellpse whee the two focal ponts ae the same. Keple s nd law: The fgue to the ght s meant to llustate Keple s nd law. A planet goes n an ellptcal path aound the sun. The tme to moe fom A to B s the same as the tme to moe fom C to D. Fom A to B the planet moes faste than fom C to D so that the aea swept out by the lne fom the planet to the sun s the same fo both tme nteals. Keple s nd law s a dect consequence of conseaton of angula momentum and the fact that the foce of attacton s dected alone the lne connectng the two bodes. The aea swept out by the ecto fom the sun to the planet n a small tme dt s da d dt ω dt whee and ae the components pependcula to the ecto connectng the two bodes. d A d Thus, da dt ω The angula momentum of the planet about the sun s L Iω m ω, o, ω L m
3 Thus, da dt L m The angula momentum of the planet, L, s constant snce the sun does not exet a toque on the planet. Ths s because the foce exeted by the sun s along the lne connectng the sun and the planet. Thus, da/dt constant, whch means that the planet sweeps out equal aeas n equal tmes. Keple s 3d law: The 3 d law can most easly be obtaned fo a ccula obt as follows. F mac sm m (π / T ) m π m T whee M s mass of sun, m mass of planet, T peod of obt of planet, and sun-planet dstance. Solng the aboe fo T, we hae T π s 3 Ths equaton would apply fo any object obtng a fxed body. Fo example, fo a satellte obtng the eath, M s would be eplaced by M, the mass of the eath. (Note: In the case of an ellptcal obt, Keple s 3 d law stll apples as aboe except that the adus of the ccle s eplaced by the sem-majo axs of the ellpse.) xample: What would be the peod of a satellte n obt just aboe the suface of eath? (Of couse, such an obt could not be sustaned because of atmosphec esstance.) π 3 π 6 3 T ( ) x ( )( ) x x T 5,070 s 85mn What s the speed of a satellte n a ccula obt? 3
4 o, F m m Fo such a low eath obt, (6.67x0 )(5.98x0 ) 7,900m / s 7, 700mph x0 Questons: How does the peod of obt change as the adus nceases? How does the satellte speed change wth nceasng adus? How does the peod o obt depend on the mass of the satellte? Gatatonal potental enegy: Fom the gatatonal foce fomula, one can obtan a geneal expesson fo the potental enegy, U, of two attactng masses. In geneal, the change n potental enegy s the negate of the wok done by the conseate foce, so f f Gm m Gm m Gm m U U f U Fd. d + f By conenton, the potental enegy n unesal gataton s taken to be zeo when. Thus, Gm m ( ) U Ths expesson s moe geneal than the expesson U mg y, whch s ald fo alues of y that ae small compaed to eath s adus. Fo fnte sepaatons, U fo unesal gaty s always negate; howee, n applcatons we only woy about changes n U, whch can be poste o negate. xample: What s the escape speed of an object fom a planet?
5 By escape speed, we mean the mnmum speed to launch an object such that t nee etuns to the suface of the planet. Ths would eque that t go an nfnte dstance fom the planet whee t eentually comes to est. We use conseaton of enegy. esc 0 f K + U K f + U f m m m esc m f f esc Fo eath, the escape speed s (6.67x0 )(5.98x0 ) esc.x0 m / s ( 5,000mph) x0 (The actual escape speed s lage because of atmosphec esstance.) 5
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