Set up Invariable Axiom of Force Equilibrium and Solve Problems about Transformation of Force and Gravitational Mass

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1 Applied Physics Reserch; Vol. 5, No. 1; 013 ISSN E-ISSN Published by Cndin Center of Science nd Eduction Set up Invrible Axiom of orce Equilibrium nd Solve Problems bout Trnsformtion of orce nd Grvittionl Mss Kexin Yo 1 1 Institute of Mechnicl Engineering of Shnxi Province, Xi n Metering Institution, Xi n, P. R. Chin Correspondence: Kexin Yo, Institute of Mechnicl Engineering of Shnxi Province, Room 1-7-1, Stff Building, Xin Metering Institution, No. 1, Lodong South Rod, Xi n, , P. R. Chin. Tel: E-mil: yyydwpq@163.com Received: August 9, 01 Accepted: October 10, 01 Online Published: Jnury 9, 013 doi: /pr.v5n1p7 URL: Abstrct Invrible xiom of force equilibrium is set up ccording to the common knowledge tht force equilibrium is independent of observers. The trnsformtion formuls of force in different inertil system re derived nd the vlidity of these trnsformtion formuls is ffirmed vi deducing the electric-field distribution of high speed moving chrge. And through deduction, conclude tht the grvittionl mss of n object is not equl to its inertil mss. Keywords: force equilibrium, electric field, specil reltivity theory, trnsformtion, grvittionl mss 1. Introduction Specil reltivity theory elucidtes the trnsformtion reltion of the length, time nd mss between two inertil systems, but does not specify the trnsformtion reltion of cting force. This pper determines the trnsformtion reltion of forces ccording to the invrible xiom of force equilibrium, new theory proposed by the uthor, mking the length, time nd mss become the fundmentl trnsformtion reltions of the specil reltivity theory, nd deducing tht grvittionl mss hs nothing to do with its moving speed ccording to the trnsformtion reltion of force.. Invrible Axiom of orce Equilibrium When 1kg object rests on spring blnce, the pointer of the spring blnce will stop t the position of 1kg scle. or the person reltively sttionry to the spring blnce, this scle indictes 1 kg, nd for other persons in motioning stte will lso see tht the spring blnce indictes 1 kg. It mens tht grvity nd spring force mintin blnce t 1kg scle position nd this blnced stte does not chnge in different inertil systems. This objective fct is common knowledge to ll. Nturlly we cn further deduce tht set of vrious forces mintining blnced stte, which will not chnge in ny inertil system. This cknowledged principle is clled invrible xiom of force equilibrium. 3. Vrition of orce in Different Inertil Systems igure 1. Motion of Equilibrium orce nd orce Conversion 7

2 Applied Physics Reserch Vol. 5, No. 1; 013 In igure 1, () t upper left represents n isosceles tringle support bc (b=c) nd three negtive chrges Q, Q b, Q c were plced on the three vertexes of the support. We mde Q b = Q c. It is lredy known tht the height of tringle k bc, plcing positive chrge +Q 0 t point O on k, the grvittion cted upon Q 0 by Q, Q b, Q c re, b, c respectively. As fr s sclr quntity is concerned, b = c, the resultnt force of, b, c is s shown in ( ) in igure 1. These three forces mintin blnced stte, there is As b = c in vlue, mking b = c =, thus there is cos cos b c cos If bc support nd Q, Q b, Q c s well s Q o tht mintin force equilibrium motion long k t speed v, ccording to the specil reltivity theory, the length k in direction of motion is shorted to k s shown in (b) in igure 1. l nd l respectively represent k nd k nd x nd x respectively represent k nd k, ccording to the specil reltivity theory, there is l l 1 (c in the formul is velocity of light); x x 1 v / c. The length of verticl direction of motion remins unchnged, thus there is b c =bc, b k =bk=h nd forces, b, c ccordingly chnge into, b, c. The composition of forces is bok, b = c s shown in (b ) in igure 1. According to invrible xiom of force equilibrium, fter four electric chrges with interction forces mintining blnce s shown in () in igure 1 is trnsferred to the new inertil system s shown in (b) in igure 1, the interction forces still mintin blnce, nmely the resultnt force of, b, c is zero, therefore there is cos b To correspond with the previous formul nd mke = b, then the bove formul is revised to cos Suppose the conversion rtio of nd is k (the specific vlue of k is to be determined), nmely By compring with cos, it is known tht cos k cos rom () nd (b) in igure 1, it is known tht k cos Substitute x x 1 v / c into cos, there is x x h ; cos x x h cos x x 1 1 h Substitute the bove cos nd cos reltionl expressions into k cos cos, there is k 1 x / x h v 1 / c k 1 v cos / c 1 73

3 Applied Physics Reserch Vol. 5, No. 1; 013 igure. orce motioning prllel to V is irrelevnt to V Here we use specil exmple to nlyze rtio k. In igure, A is n infinite flt plte with electric chrge, which produces uniform electric field with field intensity of E. the force cting on the positive electric chrge is =EQ; Q motions long the line of electric force t speed v, s chrged flt plte A is verticl to v, A does not shrink nd the field intensity remins unchnged E =E. =E Q=EQ=, K= /=1. Substitute K=1 into the bove formul, there is 1 v cos / c 1 (1) cos k cos cos cos cos cos 1 1 v cos / c cos 1 cos 1 1 v cos / c () The ormuls (1) nd () express the force in sttionry stte nd included ngle θ formed by force nd horizontl direction. After motioning t speed v in horizontl direction, force produced force nd included ngle θ. 4. Electric ield Distribution of Speed Chrged Prticle Electrodynmics (Electrodynmics Teching nd Reserch Group of Tsinghu University, 1964) derived the electric field distribution formul of high speed chrged prticle s shown in ormul (3) E k 1 v / c 1 v sin / c 3 r Where Q indictes the crrying cpcity of chrged prticle; v indictes motion speed of Q; r indictes the distnce from point E to Q; θ indictes the included ngle between r nd v; k indictes constnt quntity of electrosttic force. We cn use force conversion formul to deduce the electric field force distribution ormul (3); If Q keeps sttic, the electric field force cting on chrge is ; when Q motions t speed v, the electric field force cting on the chrge is. As the electric field force is directly proportionl to the field intensity, the electric field force conversion formul cn be written s field intensity conversion formul: Q (3) E E 1 v cos / c 1 (4) 74

4 Applied Physics Reserch Vol. 5, No. 1; 013 Where E indictes the field intensity of nd E indictes the field intensity of. If numertor nd denomintor in ormul (4) multiply 1 1 cos v / c Since E E 3 1 1cos, there is 1 1cos 1 E cos 1 1 cos 1 1sin cos, the bove formul cn be converted to E 1 E 3 1 cos sin 1 1 cos (5) igure 3. Deduction of Electric ield Distribution of Motionl Chrged Prticle igure 3 indictes the shrinking grph of Q in motioning. r nd r respectively indicte the chnge of distnce before nd fter motion of Q; l nd l respectively indicte the chnge of length in V direction before nd fter motion of Q. According to specil reltivity theory, there is l l 1 v c ; θ nd θ respectively indicte the included ngle between r, r nd V. rom the figure, it is known tht r l h ; r l h. Since l l 1 v c, there is Since There is l / l h cos r l h l l 1 r l h l h 75 r r 1 cos (6) rom the figure it is known tht sin h / r ; sin h / r, thus sin r sin / r. Substitute ormul (6) into this formul, there is: sin sin (7) 1 cos Substitute E kq / r nd ormuls (6), (7) into ormul (5), there is E k 1 Q 3 r 1 sin This is front electric field distribution ormul (3). It shows tht ormul (3) cn be used to clerly express the distribution of electric field, but inconvenient to be used to clculte electric field fter motion of Q, becuse θ nd r must be firstly clculted. But the use of electric field force conversion ormul (4) cn directly cquire the nswer.

5 Applied Physics Reserch Vol. 5, No. 1; Inertil Mss is not Equl to Grvittionl Mss igure 4. Universl grvittion nd electric field force equilibrium re irrelevnt to motion In igure 4, two objects with the mss of M nd two chrges of different polrity with sme electric quntity in inertil system Z constitute force equilibrium system. The universl grvittion of two objects is M GM / r (G-constnt quntity of universl grvittion); the electrosttic ttrction of two chrges of different polrity is Q kq / r. Under the circumstnce of force equilibrium, certinly there is M Q. Now inertil system Z, reltive to inertil system Z, motions t speed v. According to the invrible xiom of force equilibrium, the force equilibrium stte of inertil system Z in igure 4 is still blnced for inertil system Z. But ccording to the force conversion ormul (1) nd for Z, the grvittion between two Qs in inertil system Z should be: 1 v cos / c kq Q 1 r 1 1 Q (8) In bove formul, since r v, there is cos 0 r r According to specil reltivity theory, when M motions t speed v, M should convert to M M/ 1 v /c, thus for Z, the universl grvittion between two Ms in inertil system Z should be: M GM M M G (9) r r 1 1 v c Since Q M, consequentilly there is M > Q. rom inertil system Z, it cn be found tht the grvittion between two Ms is greter thn tht between two Qs. When force loses blnce, two Ms will collide, which obviously does not comply with invrible xiom of force equilibrium nd is impossible to occur. This indictes tht the bove deduction is wrong. If we use grvittionl field to nlyze the universl grvittion, we will lso obtin result of disequilibrium. Becuse by inferring to the electric field distribution ormul (3) of motionl chrged prticle, when the object is motioning, the field intensity in its verticl motion direction will convert to M / 1 nd M M/ 1 v /c, the universl grvittion of two objects should be the following M 76 3 GM r 1 1 M lso there is M > Q. How to correct the bove improper deduction? It cn be found tht Q Q 1 is derived from force conversion ormul (3), which conforms to the invrible xiom of force equilibrium, thus it should not be revised. Therefore, we hve to mke mendment from the spect of deduction of the universl grvittion. The nlysis shows tht it comes possible only by tking M s Q tht hving nothing to do with motionl speed. Under this circumstnce, the distribution of grvittionl field is completely similr to the electric field nd the chnge of field intensity of motionl M nd Q is completely similr, there certinly is M = Q. Therefore, rtionl deduction is tht the mss of n object hs dulity. In terms of the universl grvittion, it expresses s

6 Applied Physics Reserch Vol. 5, No. 1; 013 grvittionl mss M 0, which is constntly equl to the rest mss of n object hving nothing to do with the motion of n object. The expression of the universl grvittion should be s follows: M km M r (10) In terms of quntity nd energy of motion, it expresses s inertil mss nd its vlue hs nothing to do with motionl speed v. According to specil reltivity theory, there is inertil mss: M M0 1. Obviously, M M 0 Strictly speking, inertil mss is not equl to grvittionl mss nd the equivlence principle of principle of reltivity is untenble when n object moves t high speed. Of course, when the movement speed of n object is not high, it cn be deemed tht M=M 0 nd equivlence principle is tenble. But when the movement speed of n object pproches the velocity of light, for exmple, v =0.99c, then there is M>7.M 0, nd inertil mss clerly differs from grvittionl mss. It is deemed tht this difference is reflected on the movement trck of prticles under the effect of grvity. Up to now there is no experiment to prove the existence of this difference. It is suggested to testify the existence of this difference by relevnt experts. 6. Conclusion According to the invrible xiom of force equilibrium, the bove contents, combining the length trnsformtion of the specil reltivity theory, nlyzes nd determines the trnsformtion reltion between two inertil systems. The trnsformtion formul derives correct result by deducing the electric field distribution of high-speed chrged prticles. This deduction verifies the trnsformtion formul of force in this pper is correct. Bsed on this new theory, this pper derives the conclusion tht grvittionl mss of n object hs nothing to do with its motioning speed. If this conclusion cn be proved by experiments, it will hve theoretic significnce for physics. References Electrodynmics Teching nd Reserch Group of Tsinghu University. (1964). Bingjing, Tsinghu University, p.08. Guo, S. H. (010). Electrodynmics. Chin: Chin Higher Eduction Press. Guo, Y. Z., & Li, Z. X. (011). undmentls of Applied Mechnics. Chin: Chin Higher Eduction Press Zhng, Z. H. (005). undmentls of Principle of Reltivity. Beijing: TsinghuUniversity Press. 77