On the Eötvös effect

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Jonah Pitts
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1 On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion the Eötvös tem could ise ntully without supplementy ssumptions. The Eötvös foce nd the Coiolis foce e the veticl nd hoizontl pojections of foce geneted by the cicul motion. Unde these cicumstnces we cn conceive the Eötvös effect like veticl Coiolis effect. In ddition we hve deduced the Eötvös tem fom centifugl foce, clssic hypothesis. The cosine function ppes only due to spheicl coodintes nd expess the vition of centifugl foce with ltitude. Intoduction Duing the ely 9s scientific tem fom Posdm Institute of Geodesy pefomed gvity mesuements on moving ships in the Pcific, Indin nd Atlntic Ocens. Thei esults wee stonishing. The mesued vlues of the intenl weigh of the gvimete wee lowe when the bot moved estwd nd highe when it moved westwd. These esults wee then explined by the Hungin physicist Lond Eötvös fte nothe set of mesuements cied out in the Blck Se on two ships, in 98. The esults wee in vey good geement with those clculted with fomul: u v u cos () R Hee is the eltive cceletion, Ω is the ngul velocity of the Eth, u is the velocity (eltive to the Eth) in ltitudinl diection (west-est), v is the velocity in longitudinl diection (noth-south), Φ is the ltitude whee the mesuements e tken nd R is the dius of the Eth. The fist ight tem in the fomul () coesponds to the Eötvös effect. The second tem epesents the equied centifugl cceletion fo the ship to follow the cuvtue of the Eth. It is independent of both the Eth s ottion nd the diection of motion. Unde noml conditions this tem is negligible. The fomul () is the esult of simple esoning. The ttction foce of the Eth is the esultnt of two foces: the gvittionl foce, ccoding to Newton s lw nd the centifugl foce cused by the Eth ottion. Since the msses on the Eth s sufce e unifom distibuted nd the ngul velocity t which the Eth ottes e constnt, the weight of the objects t est on the Eth s sufce is constnt. In the cse of moving objects the sitution is diffeent. Since the Eth s ottion is fom west to est, the centifugl foce cting on moving object is gete if its motion on Eth s sufce is towds the est thn towds the west. Theefoe the specific weight of moving estwds object is decesing, while the specific weight of moving westwds object is incesing, []. Howeve, this intuitive nd simple theoy is not the only one to explin the Eötvös effect. In the following section such theoy cn be given on the bsis of non-inetil motion.

2 The motion in non-inetil efeence systems Figue Conside the efeence systems ( S ), (S) nd (S ) which e in the sme plne (xy). The system (S ) is the cente of mss efeence system of two co-moving msses (the Eth nd smlle body plced in P). The system (S) (Eth system) is not inetil becuse it is otting with espect to the fixed system ( S ). Assume lso tht the system (S ) it is otting with espect to (S) nd this ottion is descibed by the eqution: ωt t z In this cse on point mss P (with mss m ) e cting two foces: m FIS Fot The fist ight tem is n ttction foce due to the inetil motion: FIS m u 3 m m, nd whee G u. The second ight tem is due to ottion of (S ) with espect to ( S ): F ω ω ω ω ot m Consequently the motion of point mss P is descibed by the eqution:

3 () m u - ω ω ω ω m m 3 In ode to be consequent with the poblem we should now wite eqution () in moe detil by using spheicl coodintes. The motion we study is on spheicl sufce of the Eth, so it is noml to descibe this motion in spheicl coodintes. Nevetheless such desciption goes nowhee. The fct we must wite the pseudo vecto ω in spheicl coodintes elimintes it completely fom intemedite clculi nd fom finl esults. And this is something we don t wnt to occu. It is essentil fo ou study to expess ou esults s function of ω. The mthemticl desciption of non-inetil motion would be inconceivble without it. At ω= we hve inetil motion nd specific eqution fo it. This is the eson why to simplify the poblem we must evlute the bove eqution in cylindicl coodintes, []. Accodingly we hve: ρ v ρ η ρ η ω z nd: ρ η z ω η ω ω z η ρ ω z ρ η η By eplcing these ptil esults into eqution () nd mking the pojections onto pol xes we obtin: ρ (3) (4) The motion in spheicl coodintes Fist of ll notice tht if ω= equtions (3) nd (4) depict n inetil motion. The pojection onto ngul diection, eqution (4), compise the tem coesponding to Coiolis foce. The pojection onto dil diection, eqution (3), contins the tems coesponding to centifugl foces. It is noticeble tht the lst ight tem of eqution (3) esembles to Eötvös tem (see fomul ()). If we denote ω=ω (the ngul velocity of the Eth) nd (dθ/dt) =u (the velocity eltive to the Eth, in west-est diection) nd neglect the contibutions of centifugl foces, then we hve the Eötvös tem. The only poblem is the fct tht in elity the Eötvös effect emege in the motion on sphee nd

4 ou esults e into cylindicl coodintes. Equtions (3) nd (4) e coect only if the Eth hs cylindicl fom. In this cse, no doubt, the ottion of it nd of body, in cicles, upon its sufce, genetes both Eötvös nd Coiolis effects. The z component of the motion does not count, so these effects e the sme, unde condition tht veticl motion to be null. In the cse of Eth s motion ound the Sun, in the context of glxy ottion, this desciption is tue. Pehps the glxy coss section ne cylinde is due to the Eötvös effect. But in the cse of motion upon spheicl sufce, equtions (3) nd (4) do not hold. At this stge of ou study is wothless to convet these two equtions into spheicl coodintes. We don t obtin the sme esults s we initilly wote these equtions in spheicl coodintes. And this is conceptully wong. Let now look fo nothe wy to infe the eqution (3) nd (4). These equtions e descibing cicul motion. It is simple then to wite cceletion in pol coodintes nd follow the logicl steps in ode to find the equtions of the sme cicul motion. It follows: ( ) (5) nd: ( ) ( ) (6) The pojections of foces cting on point mss P will look, fte simplifiction nd simple clculi: which e, without question, the sme equtions (3) nd (4). This obsevtion is vey impotnt becuse it helps us to imgine wy to find out the coect fom of the equtions of motion in spheicl coodintes in simple mnne. The existence of pevious section is justified not only by the deduction fom othe hypothesis thn centifugl foce vition of Eötvös effect but to pove tht equtions (5) nd (6) e coect. The mnne in which ω ws included in these equtions cn be extpolted fo the coespondent equtions in spheicl coodintes. Consequently we hve fo the dil component of cceletion: ( ) sin The pojected coespondent foce cting on point mss P, fte elementy clculi will be: sin sin sin (7) Assume tht dil velocity of the motion is constnt, theefoe its deivtive will be null. If ω is the ngul velocity of the Eth then we cn neglect lso the coespondent centifugl foce. It emin the non-null tems due to non-inetil motion nd the gvittionl cceletion If we keep only the tems of inteest then we cn wite: sin sin E

5 If we e tking into ccount tht, fo hoizontl velocity of the body on the Eth s sufce, its veticl velocity nd the eltion between ltitude nd elevtion ngle, the expessions e: u sin v 9 then we find n expession simil to (), in which the symbols hve the sme significtions: u v E u cos (8) The centifugl effects e most of the time negligible, so this could not cuse seious poblems to the finl evlution. Accoding to (7) nd (8) the esulting foce cting on moving object to the Eth s sufce it is smlle if its motion is estwds thn westwds. Theefoe the specific weight of moving estwds object is decesing, while the specific weight of moving westwds object is incesing, with the sme vlues s those clculted with (). Conclusions We obtin the Eötvös tem by mking diffeent ssumptions. The study of non-inetil motion in cylindicl coodintes give ise to this possibility. So-clled Eötvös foce nd Coiolis foce e the dil nd ngul pojections of foce geneted by the cicul motion. Thus the Eötvös effect it is no moe the esult of the centifugl foce vition. This model not fits to the motion on spheicl sufces, it is moe ppopite to descibe the evolution of cosmic bodies. We hve shown then tht the simil esults cn be obtined if the centifugl foce vition hypothesis hs been consideed. The model is much simple nd dive to conclusion tht the centifugl foce vition hypothesis fits the study of non-inetil motion in spheicl coodintes too. The esults e vey conclusive in this mtte, I think. The cosine function seems to be the smll diffeence between the two models esults. It ppes ntully due to vition of centifugl foce with ltitude. Refeences [] R. Eötvös, Ann. d. Phys. 59 (99), ; [] C. I. Boş, Tenso 53 (993), 7.

RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1

RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1 RELAIVE KINEMAICS he equtions of motion fo point P will be nlyzed in two diffeent efeence systems. One efeence system is inetil, fixed to the gound, the second system is moving in the physicl spce nd the