Archive for the 'Newtonian Mechanics' Category

Feb 09 2017

E = mc^2

It occurred to me in January or February 2008, during my first foray into Quantum Mechanics, that the reason there is no 1/2 factor in front of mc^2 in Einstein’s formula E=mc^2, – like there is in the Newtonian formula for kinetic energy K. E. = (1/2)mv^2, is that there are gravitons inside a fundamental particle that are bouncing back and forth against gravitational pressure on the outside, which doubles the energy.

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Jun 24 2014

Macroscopic vs. Atomic G

Dan Fordice sent me two articles about newer experiments that were set up to measure the gravitational constant.  One of the articles referenced a paper by Tino et al. where the constant is determined using one mass type of hundreds of kilograms of tungsten, and the other being laser cooled rubidium atoms.  The apparatus involving the tungsten masses looks like it may be the same apparatus as was used for the Schwarz et al. experiment from 1998 [1].  The Tino et al. value for G is given as 6.667 x 10-11 m3kg-1s-2 [2] with statistical uncertainty and systematic uncertainty given in the paper.

When we have a macroscopic mass where atoms are chemically bonded, and masses are held together by various means, a gravitational field acting on one atom can have a component of force on another atom that is chemically bonded to it.  A single atom free of bonding to other atoms, on the other hand, has fewer instantaneous electron orbital path vectors than a macroscopic mass of several kilograms when we consider it as a whole.  Therefore one would expect that the value of G when measured on individual atoms would be lower than a conventional value of 6.672 x 10-11 m3kg-1s-2 [3].

The gravitational constant based on one third the mass of the proton is 6.6807 x 10-11 m3kg-1s-2, but does it ever get this high in reality?  Planets in orbit around the sun would get close to this value.

G = 6.672 x 10-11 m3kg-1s-2 is probably still a good value to use when considering macroscopic masses on the surface of the earth, or in the atmosphere, or in orbit around the earth.



[1]  Schwarz, Robertson, Niebauer, Faller, “A Free-Fall Determination of the Newtonian Constant of Gravity”, Science, 282, 2230-2234; 1998:

[2]  G. Lamporesi, A. Bertoldi, L Cacciapuoti, M. Prevedelli, G.M. Tino, “Determination of the Newtonian Gravitational Constant Using Atom Interferometry”,, 2013

[3]  Tipler, Paul A., Physics, Worth Publishers, Inc., 1976, inside back cover

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Sep 01 2011

Fermilab Magnets

With the Fermilab Tevatron shutting down this month, I wonder if its magnets could be used for a space debris vacuum.  The problem pops up in the news periodically, and did again today:

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Jan 12 2011

Gradient of a Scalar Potential

In Thomas’ Calculus we find the definition: “If F is a vector field defined on D and F = del f for some scalar function f on D, then f is called a potential function for F.” ([1], pg 921)

Kaplan states that the “gravitational field is the gradient of the scalar f = kMm/r” ([2], Prob. 4, pg 180), which for General Relativity is indeed the case.  With gamma ray energy exchange however, the scalar is f = k1M/r, in form much like the scalar potential k2q/r with electrodynamics.  This tells us that both gravitational and Coulomb forces are of independent action, and the corresponding fields are gradients of scalar potentials, which can be considered as more evidence for unification.

The vector fields we are looking at have units of N/kg for gravity, and of course E = N/C for an electric field.  A gravitational field calculation is most applicable when the object in the field is small compared to, or far away from, the source of the field, such as Jupiter compared to the sun.

Aside from the King James Version of the Bible, Wilfred Kaplan’s Advanced Calculus is still my favorite book.  By the way, we used Thomas’ Calculus book at the University of Wisconsin in 1976 and 1977.


[1] Thomas, George B., as revised by Weir, Maurice D. and Hass, Joel, Thomas’ Calculus, Twelfth Edition, Addison-Wesley of Pearson Education, Inc., 2010, 2005, 2001

[2] Kaplan, Wilfred, Advanced Calculus, Fifth Edition, Addison-Wesley of Pearson Education, Inc., 2003

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Dec 14 2010

Independent Action

In one of the FGT YouTube videos of February 2008 it was mentioned that unlike a tug of war where the tension in the rope is the same throughout when the rope is stationary, gravity is one mass acting independently on another.  In the case of two masses of spherical shape the force is known to be F12 = Gm1m2/r122, which we can equate to m1a or m2a, depending on whether we want to calculate the acceleration of m1 or m2, and where r12 is the distance between the centers of the two masses.  With planetary masses that are close to spherical, and the objects they pull, this formula has been used most reliably, while it has been known for a long time by learned mathematical and scientific people that shape matters, and using center of mass with the standard physics book formula can cause increased error in some instances.

As an analysis that has already been done, let us use Kline Chapter 16, Sections 5, 6, and 7.  Starting with Section 5, “Gravitational Attraction of Rods”, the example given is that of a “rod 6 feet long and of mass 18 pounds which is uniformly distributed” and “so thin that we think of it as extending in one dimension only.  Three feet from one end of the rod and along the line of the rod is a small object of mass 2 pounds which we shall regard as located at one point.”   Kline first calculates the force that the rod exerts on the 2-pound mass as though the entire mass of the rod were concentrated at its center, according to the standard F12 = Gm1m2/r122 formula, which comes out as equaling G poundals.  Then he does the calculation properly using an integration over the rod, showing that the force that the rod exerts on the 2-pound mass is actually (4/3) G poundals.  In Section 6, “Gravitational Attraction of Disks” [1], again a difference from the standard formula is shown which for comparison includes Exercise 1 from that section.  Finally, in Section 7 it is shown that the standard physics book formula can be used with spheres.

Since the title of the referenced book includes “An Intuitive and Physical Approach”, intuition may tell us that, since the rod was integrated along its length to obtain a correct answer, every piece of a nearly one dimensional rod in the limit of a Riemann sum exerts a gravitational force independently on a separate point mass.

With General Relativity, unproven to date, a mass produces surface curvature in space, which may contact a series of points on other surfaces, due to another mass, through a tensor product [2] that reduces to a force vector.  This may be interpreted as space-time surface interaction, and not necessarily complete independent action.  At least two multi-million dollar projects are built and running ([3], [4]) and at least one is being developed [5] in attempts to prove General Relativity.

Special Relativity, on the other hand, E = mc2, has long been proven correct, and the gravitational theory presented on this web site would not work without it.  Some physicists in their writings seem to make a transition from Special to General Relativity as though the two are somehow linked, and in reality they bear no relationship to each other.


[1] Kline, Morris, Calculus, An Intuitive and Physical Approach, John Wiley and Sons, Inc., 1967, 1977; Dover (1998) unabridged republication.

[2] Rainich, George Yuri, Mathematics of Relativity, John Wiley and Sons, Inc., 1950, Chapter 4




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Nov 13 2007

The Comets

Comet McNaught, C/2006 P1, reached apogee around the Sun at the time of the local publicity on the gravity theory with the newspaper articles and an interview on television.  I think of the designation for this comet as follows:
C:  for my wife Cynthia;
2006:  the year the comet was first spotted, even though pictures from late 2005 showed it;
P:  for momentum, the increased electron momentum making the Coulomb force the final mediator of the gravitational force, and the theory a unification theory;
1:  for the quantum intrinsic spin number of a photon, though some physicists say that gravity must have an intrinsic spin of 2 to be always attractive.
Comet Holmes, on the other hand, partially blew up on October 24, 2007.  This was the two year anniversary of the day the main calculation was started.  The main calculation was finished November 12, 2005, a Saturday because I don’t get paid to be a physicist.
About the intrinsic spin, think of gravity as two separate actions after a graviton has been generated.  One is the absorption of a graviton by an electron in an atomic orbital, and the second is the resultant centripetal Coulomb force.  On average, electrons in a body would have greater mass when traveling in the direction of where the highest flux density of gravitons is coming from.  Since an electron gains mass by absorbing a massless photon, which is possible due to special relativity, an intrinsic spin of 1 works out.
Of the two hundred or so emails sent out to scientists, thank you to the small few who responded.  I am also very appreciative to the three PhD physicists who each read and analyzed my paper at some point along the way, and gave me feedback.
Just think, in the free world some of your tax dollars may still be going to support string theory!

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