Indeterminate Form in the Equations of Archimedes, Newton and Einstein.

Size: px

Start display at page:

Download "Indeterminate Form in the Equations of Archimedes, Newton and Einstein."

Kathlyn Fox
6 years ago
Views:

1 Indeterinate For in the Equations of Archiedes, Neton and Einstein. Ajay Shara Fundaental Physics Society. His Mercy Enclave Post Box 17 GPO Shila 1711 HP India Eail Abstract A very peculiar siilarity is observed beteen equations of Archiedes, Neton and Einstein. As under soe condition the equations based upon the established principles/las lead to indeterinate for i.e.. The reason is that in the case of Archiedes principle, equations becae feasible in 1935 after enunciation of the principle in 1685, hen Neton defined g in The Principia. Such specific studies do not ean any coent or conclusion of the established status of the ell-knon las. 1. Indeterinate for of volue leads to a generalization of Archiedes principle in this particular case. 1. Copletely suberged floating balloons lead to indeterinate volue of fluid in the Consider a balloon filled ith a ediu of density D floating in ater of density D. The volue of the sheath of tje balloon /vessel is v (ass =) and the volue of the ediu filled inside the balloon is V ( say ood, etal and gases). According to Archiedes principle the upthrust experienced by the balloon is equal to the eight of fluid displaced [1-2]. The body displaces fluid equal to its on volue. V D g + g = (V+v) D g (1) Or = (V+v) D -V D (1) Fro Eq.(1) different values of V, D, D and v can be ritten as ( vd ) V = ( D d ) (2)

2 ( + VD ) D = ( V + v) (3) and D = ( V + v) D (4) V v = V D ( D D ) (5) No e can try to calculate the volue (V) of a fluid filled in a balloon (ood, etal and gases ), the sheath of the balloon has a volue v and ass, such that the density of fluid filled inside is equal to that of ater (D = D ). The obvious volue should turn out equal to V, hich is the actual value. Hence substituting, ass fro eq.(1) in eq.(2), e get ( vd ) vd V= ( D D ) = (6) hich is the indeterinate for i.e. volue of ediu filled in the balloon (ood, etal and gases) becoes undefined but in the actual experiental setup, the volue is V consisting of etal, ood and gases. Thus RHS of Eq.(6) becoes devoid of units and diensions hich are not defined. Although division by zero is not peritted, yet it soothly follos fro equations based upon Archiedes principle as per its definition. Also in this case, the nuerator of the equation also becoes zero. Also v, D, D cannot be calculated fro Eqs.(3-5) due to eq.(6). Thus Archiedes principles in its original for; in this particular case is not applicable. L Hospital Rule is not applicable here, as it is an exact equation, NOT a function or liit or differentiation of the function. No such ethod can give the exact value of V. Thus Archiedes principles in its original for in this particular case, is not applicable. It can be solved by generalizing the 2265 years old Archiedes principle. It ay be understood in vie of the fact that atheatical equations based upon Archiedes principle becae feasible after 1935 years of enunciation of the principle hen Neton published The Principia and defined acceleration due to gravity in This liitation can be easily explained if Archides principle is generalized for this particular case i.e. upthrust experienced by balloon is proportional to the eight of fluid displaced U gen α (V+v) D g or U gen = f (V+v) D g (7)

3 No eqs.(1-2) becoe = f (V+v) D -VD (8) ( fvd ) V = ( fd d ) (9) Under the siilar condition (D = D ) f 1 V =V (1) ( ) V = f 1 No, consistent results are obtained. Also no, correct values of v, D and D are obtained. Unlike eq.(6), in this case, no division by zero is involved, also the nuerator is non-zero, hence a consistent and logical result is obtained. Correctly in this case, the internal volue of the balloon is V hich is filled ith ood, etal and gases. The physical significance of the co-efficient of proportionality, f can be understood as it ay be regarded as accounting for the shape of body. According to original for of the principle, the body floats hen its density is equal to that of the ediu (D b = D ) and other factors, including shape, have no role to play. This analysis does not ean any coent or judgeent on the established status of the Archiedes principle, as this analysis is valid under certain conditions only. 2. Indeterinate for of ass in Neton s Second of Motion hen it reduces to first la of otion According to The Principia [2] the second La of otion is The alteration of otion is equal to the otive force ipressed; and is ade in the direction of the right line in hich that force is ipressed. Matheatically force, F =a = ass x acceleration This la also gives the inertial ass of the body. F M = = (11) a The second la of otion is the real la of otion as the first and third las of otion can be derived fro it. If no external force F acts (a=), then the body either reains at rest, v= or oves ith unifor velocity v =u ( hich is the first la of otion). So the second la of otion reduces to the first la if F=. Under this condition (hen the second la of otion reduces to the first la, F=, v=u) the inertial ass is given by

4 M = F = a (12) hich is an inconsistent and unphysical result in this particular situation only. It is siilar to eq.(6). L Hospital rule is not applicable here. 3. Indeterinate for of frequency in Einstein s equation of Doppler principle hen v=c In the paper hich is idely knon as the special theory of relativity, Einstein [3] put forth a equation of relativistic variation of frequency of light v 1 cosφ c ν = ν (13) 2 v 1 2 c here ν is the frequency easured by an observer in a frae oving ith velocity v, and ν is the frequency in the rest frae. φ is the angle beteen the source and observer. If the value of φ is, and the observer oves ith the speed of light i.e. v=c then ν = (14) hich is again a siilar result, and frequency becoes undefined. If v c then ν. Hence the siilar results. 4. Conclusions Thus equations of the scientific legends i.e. Archiedes, Neton and Einstein lead to indeterinate for. In the case of Archiedes principle, the indeterinate for arises fro the applications in copletely suberged balloons, hereas in the case of Neton s and Einstein s equations it follos directly. The reason is that at the tie of foration of the equations, all possibilities ere not taken in account, hence there are liitations in the equations in soe cases. Anyho there is soe siilarity in the equations of Einstein, Neton and Archiedes. This analysis does not iply any coent or judgeent on the established status of the doctrines, as this analysis is valid under certain conditions only. The las of the scientific legends are not applicable under soe conditions as discussed, otherise the las hold good. If Archiedes principle is generalized, it leads to logical and consistent results.

5 Acknoledgeents Author is highly indebted to to Dr T Raasai and Dr Stephen Crothers for encourageents and critical discussions. References 1. Shara, A., Speculations in Science and Technology, 2, (1997) 2. Neton, I. The Principia: Matheatical Principles of Natural Philosophy (Trans. I. B. Cohen and A. Whitan). Berkeley, CA: University of California Press, Einstein,A, Annalen der Physik 17, (195).

USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS. By: Ian Blokland, Augustana Campus, University of Alberta

1 USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS By: Ian Bloland, Augustana Capus, University of Alberta For: Physics Olypiad Weeend, April 6, 008, UofA Introduction: Physicists often attept to solve