h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

You are quite right. Cones are the natural shape to use, and my earlier comment on the areas not adding up was completely irrelevant. I must have had a rush of blood to the head!

It is not as though the force acted exclusively at any one surface or layer, but throughout a volume. I am inclined to think the geometry supports the fraction of a third of the radius in calculating the volume of a conic section of a sphere more precisely than a third of the height in a flat based cone, but that is not really a problem that need concern us when the typical difference is measured in centimetres.

So it's on with the algebra...

In the short time since my last posting, one of my family familiar with the procedure for it, and before he gets too busy tomorrow with a new appointment, has installed my latest thinking for a gravitational field on my website.

It includes a couple of illustrations, I hope clarifying my argument, together with its reduction to algebraic terms evaluated for conditions on the Earth's surface. It may still not be accessible to the generality of h2g2 readers, and you may be better able to put it into their language.

Consideration of the implications will depend on the reaction it gets.

The address of the website is: www.gravitational.co.uk

I've just taken a quick look! It's very good. For some reason I have to scroll left and right to read all of it, but on the smallest font size the problem is minimal. I was going to get onto the algebra, but this solves it for me. Well, let's do it anyway, and see whether what the result agrees with your explanation. The web page is very clear, by the way.

It has gone very quiet this last week. I hope this is a sign you have found no problems with the line of reasoning.

One thought I have had is it might need to be made clearer that between them the two diagrams show how the inverse square law works in two dimensions in Kepler's law for arcs swept and in three dimensions for gravity. I don't know whether one follows as logically from the other as Newton claimed, but coincidentally seeing how common the ratio is.

How does it strike you?

No, the silence is just because work has increased in the office! I will be back on this as soon as I find time! I understand the basic inverse square now, but I have not been able to look at the exact derivations.

Still submerged in folders, I'm afraid. The waves will hopefully subside soon!

I'm stalled. Is the next step to express the volume of the cone in terms of the Rr/d derived earlier?

The volume of a cone is 1/3'pi'['r'squared]h.

Then the volume of a cone in a sphere of radius D is going to equal 1/3'pi'['Rr/d'squared]xD, which in the general case is (since R and are both potentially = 1) proportional to 1/3'pi'['d'squared] - I think. Does that make sense to you? It looks like an inverse square. How does one get to gravity from there?

Whoops. There is a confusion of terms here because R and r and not the same in the two equations. Disregard that final paragraph.

I meant to write: Then the volume of a cone is going to equal 1/3'pi'['Rr/d'squared] x the radius, which in the general case is (since the radius can be set to equal '1') proportional to 1/3x['pi'/'d'squared], even proportional to 1/'d'squared - I think. Does that make sense to you? It looks like an inverse square, varying with 'd'squared, the distance the other sphere is away. So that's the volume of the cone created. How does one get to gravity from there?

I see I'm falling prey to the usual inability to express the science clearly in words. Here's another try.

There was a confusion of terms two postings up because (D and d) were not the same in the two equations.

I meant to write: the volume of a cone 'created' in one sphere by the influence of another a distance 'd' away, is going to equal 1/3'pi'['Rr/d'squared] x the radius (R or r), which in the general case is (since the radius can be set to equal '1', and is a constant anyway) proportional to 1/'d'squared - I think.

Does that make sense to you? It looks like an inverse square, varying with 'd'squared, the distance between this sphere and the other one. Thus the volume of the cone 'created' by the shielding effect varies according to the inverse square. How does one get to gravity from there?

Hallo again!

The inverse square as calculated relates to partial volume, whereas Newton's inverse square appears to relate to total surface area and so to the whole mass contained within it. It wasn't until nearly two centuries later that a value was put on the gravitational constant G, and that was a composite figure incorporating geometrical elements. Rather than trying to backtrack through this calculation it's easier to start again.

In constructing the equation on my website I started by equating M/d^2 to M/V x V/d^2, in which V is proportional to d - for diameter - cubed, and replacing mass divided by the inverse square with average density (within the cones) times d.

Introducing density simplifies the calculation by taking a value for a notional sphere having a diameter such that one of unit radius is sited at the surface. This unit radius is the reciprocal of that of the notional sphere (which is 'D' for distance). The equation remains valid whatever the actual dimensions, though several factors may affect the precise value of G.

Inserting the known value of the Earth's diameter, and typical density and acceleration at its surface, gives a local value of the 'constant' in straightforward newtons per kilogram, though researchers have found measurements of it varying significantly even daily. Does this sort out the symbols to complete the picture for you?

Best wishes for Christmas, with more levity than gravity!

Thank you very much! And Merry Christmas to you as well , and a Happy New year!
I do hope to be able to say something about gravity as well, over this period...

What do you know about Diamagnetic Levitation?!

I had an interesting first direct reply to my article 'Newton's False Conjecture' a few days ago from Firehawk (in Indiana, I believe) calling my attention to HFML, a Dutch research group's, website (dating from 1999!). It starts off quite entertainingly before getting really serious.

He suggested that the mechanism described there for their experiments in levitation might provide a key also to gravitation. No more than that, and initially I was deeply sceptical of any connection between the two.

Delving into the various papers until I was left floundering in the mathematical analysis, though, I found the idea in principle was not without merit provided the distinction is made between a net force of gravity and a gravitational field operating at a sub-atomic level.

If you have a spare moment you might like to have a look at it, over a pint of black coffee!