Fudametal Joual of Mode Physics Vol. 8, Issue, 015, Pages 141-145 Published olie at http://www.fdit.com/ GRAVITATIONAL FORCE IN HYDROGEN ATOM Uiesitas Pedidika Idoesia Jl DR Setyabudhi No. 9 Badug Idoesia e-mail: adikaaisetyawa@upi.edu Abstact This is techical pape to poe mathematically that Newto s Uiesal Law of Gaity ca be eshaped fom elliptical obit geomety i [1]. Thus, we ca deie ew fomula fo calculatig gaitatioal foce i a case of Hydoge atom. It eables us to quatize gaity fo eey electo obit. Keywods ad phases: hydoge atom, Newto s uiesal law of gaity. Receied July 3, 015 1. Itoductio The poblem of gaity i atomic scale is still mysteious i sciece. By all of iteactios i atue, oly gaity could ot be fully udestood by scietists. This pape is tyig to sole how gaity woks i quatum scale, especially i a case of hydoge atom.. Reisited Newto s Uiesal Law of Gaity Let us stat fom stadad Newto s Uiesal Law of gaity as follow: Mm F = G. (1) Ou cuet itepetatio fo stadad Newto fomula is that of if thee ae two massie body m 1 = M ad m = m sepaated with the distace, the based o 015 Fudametal Reseach ad Deelopmet Iteatioal
14 equatio (1), The bigge the masses of the objects, the bigge gaitatioal foce betwee them. This is ot wog itepetatio, but actually, this is ot fudametal leel to explai why gaity seems supeio at maco ad it seems ifeio at mico scale. The moe we dig equatio (1), the moe ew isight we get i this pape 3. Reshapig Newto s Uiesal Law of Gaity fom Elliptical Obit Fom [1], we hae elocity fo elliptical obit as follows: By squaig both sides, we hae GM () + GM 0 GM (3) + GM 0 Diidig it with, we get = GM. + GM 0 (4) Multiplyig it with m, we get cetipetal foce as follows: GMm F s =. (5) + GM 0 Now, suppose we will moe a object fom the eath suface to ifiity distace. We hae fom [1] that total eegy of a two-body gaitatioal system is elated by The, we get GMm 0.5 m = 0, (6) GMm 0.5 m (7)
GRAVITATIONAL FORCE IN HYDROGEN ATOM 143 Sice it is o eath suface, we ca wite equatio (8) ito GM (8) GM E (9) 0 I which E is adius of the eath fom the cete, 0 is escape o iitial elocity fom the eath suface. follows I geeal situatio, I mea ot oly o eath suface, we ca ewite (9) as Substitutig (10) ito (5) we get If = 0, the GM 0 (10) 0 F GMm s = ( + ). (11) 0 GMm GMm F s = = ( + 0 ) F g (1) I which cetipetal foce is equal to Newto s uiesal law of gaity. Sice = 0, the we ca substitute (10) ito (1) as follows 4 GMm m0 F g = =, (13) 4GM GM 0 1 0 EK F g = m 0. (14) GM I which, the foce of gaity is popotioal to kietic eegy ad it is iesely popotioal to distace. It meas, the foce of gaity ca be iewed as kietic eegy that is eeded to moe a object fom the suface of a gaitatig body to the cetai poit o obit pe uit distace.
144 4. Gaitatioal Fomula fo Hydoge Atom Let s stat fom the poblem of Hydoge atom, it is ey simple atom which oly cosist of oe poto ad oe electo. We kew that the foce that keepig the electo to stay i its obit is Coulomb s foce o electo static foce. We will use Boh Model as follows: Figue 1. Hydoge atom. Based o [], we hae fomula fo the possible adius of that ca be allowed i Hydoge Obit as follows: (15) me Fom the figue 1, equatig (15) with adius of = 0 i equatio (14), we get: 0 GMme 16) We ca expess (16) ito kietic eegy fomula as follows: EK GMm e (17) Substitutig (17) ito (14), we get fial fomula fo the magitude of gaitatioal foce i Hydoge obit as follows: F g EK GMm e GMm = = = K (18) ħ with
GRAVITATIONAL FORCE IN HYDROGEN ATOM 145 K e =, = 1,, 3,. 8πε0 I which equatio (18) is a quatized fomula fo calculatig gaitatioal foce fo Hydoge atom. K is kietic eegy fo electo (see []), G is uiesal costat of gaity, M is the mass of poto, m is the mass of electo, ħ is the educed plack costat, ad is quatum umbe. 5. Result ad Discussio Based o (18), fo eey kietic eegy K of the electo obit, thee is gaitatioal foce which wok to attact electo to stay i its obit besides Coulomb s Foce. But, we must emembe that kietic eegy EK i equatio (17) has diffeet meaig with kietic eegy K i equatio (18). EK i equatio (17) has the same meaig with equatio (14). Refeeces [1] A. Aisetyawa, The imagiay ad eal elocity of a obitig body based o diffeet types of coics sectio, axi:131.0967 013. [] Kae ad S. Keeth, Fisika Mode (Mode Physics); Taslated by Has J. Wospakik, UI- Pess, Sofia Niksolihi, Jakata, 199.