Co- authors Wanted for Journal Paper (related to "Jello-O... " thread)

I am not sure I understand this. I assume that the "centroid" and the "center of mass" are one and the same. Is this correct? I have my doubts if you did integrate z over the eggoid but if the wider part of the eggoid was expressed as you did, then think it is true that they are the same. Perhaps that is exactly what you said by integrate over the eggoid.

One thing not clear to me is I think that if a = b, then the center of mass is at (0, 0, 0) Cartesian even for the ellipsoid. Yet if I were to replace your "centroid" expression's a with your b, I do not see haow zero would result. Can you put in form of an equation, to better define it for me? is it what I have several times called "C"?

Must stop for a at least an hour - wife says dinner is getting cold.

 

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I haven't tried that yet. I was still waiting for the "official" function from MacM, but I can certainly run integrals for your interest also.

So what exactly do you want me to integrate? If you have a constant density then c rho^k is a constant. That obviously would be the easiest function to start with. I would start with a sphere and a point at an arbitrary distance from the sphere and then integrate the force vector over the sphere. Is that what you had in mind or did you want me to calculate the force on an interior point?

-Dale

 

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I hope you have a great amount of patience because you are going to get an "official" function from MacM. MacM never described any mathematical concept of doing the integration in this manner. MacM doesn't even know how density changes the psuedo-mass/psuedo-volume method either.

I suppose you could assume a constant density, but then you shouldn't even need a function with density, just assume a constant c so that F = c L, where L is the length of the line through the sphere.

Suppose you only have one sphere with constant density. If you integrate to find the net force at each point in the sphere and add those net forces to get yet another total net force on the entire sphere, the result should be zero.

However, if density is not constant, nor symmetrical, there should be a non-zero total net force. I would suspect that a dynamical analysis would show that the sphere tends toward a symmetrical density.

I would suggest setting up the integration for just a single sphere so that you have a form to work with.

If you integrate over the sphere, you'll have the following:

∫∫∫ F(...) dφ dθ dρ

where F(...) is the net force at each point. To get F(...), you have to integrate over all orientations in space:

F(...) = ∫∫ f(...) dβ dα

where α is the horizontal angle and β is the vertical angle.

Now f(...) itself will be an integral as well if we assume density to not be constant. So,

f(...) = ∑ ∫ cρ_d^k dλ

where λ is the differential length along the line to be integrated and the summation indicates that you do the integral for as many lines as there are. Notice that I changed the density to ρ_d to identify it separately from ρ in the spherical coordinates integration.

If we assume density is constant, then:

f(...) = ∑ c L

 

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You got it.

Your equivalent volume eggoid equation is interesting. Basically, what it does is adjust the eccentricity so that the volume of the two objects are equal regardless of distortion or "a". I think it is a good idea overall. I don't know exactly where you are heading, but you may also want to set a == R. If you do that, then the z axis is the same length for both the sphere and the eggoid. Then the eccentricity is only adjusted in order to compensate for the distortion.

I'm sure you already noticed, but I calculated your C in the next post. However, I don't think that it affects the results since there is no inherent r^2 dependence. There is only geometric dependence and we can just as easily measure the geometry from the surface as from the centroid.

Yes, the symmetry will allow exactly what you describe, but only for points on the z axis. That is nice because it becomes a scalar integral rather than a vector integral. Off axis I would need to do the full vector integral.

So, you are interested in the net "flux" force at an external point located on the z axis an arbitrary distance from a constant-density eggoid. Correct?

-Dale

 

Yes, the centroid is the center of mass for a constant-density object. And the center of mass is the center of gravity for an object in a uniform gravitational field. I was just being explicit about what I calculated. Since I did not include a density function it was a centroid.

You are a little wrong here. Any ellipsoid will have the centroid at the origin, regardless of the length of any of the principal axes. It is only the distortion parameter D that will move the centroid off the axis. Remember a>0 so C == 0 iff D == 0. In that case there is no distortion and the eggoid is actually a spheroid. I believe that the expression is correct.

C == 2 a^2 D /(5 + a^2 D^2)

-Dale

 

Aer, I see what you are saying.

I think I will just do two integrals. One to find the force on a point outside a uniform-density sphere and one to find the force on a point inside a uniform-density spherical shell. Those two will allow me to find the force at any point inside a solid sphere.

I know that the constant density expression may be a little simplistic, but the math is difficult enough geometrically that I think it is a good place to start.

-Dale

 

I am not following your logic here. I was under the impression that we wanted the total force on the sphere. What is the point of being able to find the force at any point inside a solid sphere? You should be able to do that just by integrating over all orientations in space, (the α and β I defined in the prior post).

 

I am game for any integral you want me to run. (You are right, your approach for the interior would be easier than mine).

So what do you want? The total force on a sphere from what? A second sphere? I think I can do that integral using a constant density.

-Dale

 

I am having a little difficulty (not fully confident we still have the freedom to set a = R, but I think we do.). I have lots of problems with your: "then the z axis is the same length for both the sphere and the eggoid" as I demonstrate later this post (My bold below.)

In some sense the a/b ratio and the eccentricity are the same thing at least for the ellipsoid. When I get rid of b by an expression for it involving R I have set the size of the eggoid in my equivalent volume equation (rewritten to eliminate the "= 1" right side.):

(S^2)a{1 + (a^2) * D^(2/5)} = (R^3){1 -(Z/a)^2}

If we set a = R as you suggest then:

(S^2){1 + (R^2) * D^(2/5)} = R^2{1 -(Z/R)^2}
or
(S/R)^2 = {1 -(Z/R)^2} / {1 + (R^2) * D^(2/5)}
Note that the numerator on the right is always less than unity and the demonator is always greater than unity. Thus I would conclude that (S/R) is always less than one. That is at all points on the eggoid, the distance of the surface from the z-axis is less than the radius of the sphere, yet it has the same volume. So it sticks out from the sphere at the extremes of z.

Also note we can make it even simpler by defining our unit of length so that R = 1. Then:

S^2 = (1 - Z^2) / {1 + D^(2/5)} and the denominator being a constant makes me think: "Hey what do I need DaleSpam for?" - JUST KIDDING !!!!

Perhaps it would be useful to graph a 2D plain containing the z-axis. - I.e. a plot of S vs Z for D =0.01 & D = 0.1 if any one reading can do this and locate the centroid (center of mass) on these plots. I am assuming it is shifted from the origin towards the fatter end. (The right drawing in first post of this thread and the text below the equation 3a there both show that the fatter end is in the z negative region.)

I do not follow you here, pehaps because you do not know where I am going. (more later this post)

unfortunately, I need all interior points if I am to consider the total force acting on the eggoid.

No I am not currently interested in any exterior points. I am focused on the stability of a single, uniformly dense object which is assumed to initially be a sphere, but then has a very slight distortion making it "egg shaped." I want to know if the distortion tends to grow (an instability) or gives rise to restoration of the sphere.

I think MacM, Aer, and I (probably you also) think making the force along any line passing thru the differential volume (that you integrate over the entire eggoid volume) is directly proportional to the difference between the two distances along that flux line to the surface. Consequently, if you can do the integrals, we have a problem defined. We need only make sure that the CoMs of sphere and slightly distorted eggoid are co located and evaluate the net work done by the motion of the sphere as it deforms at constant volume to the shape of the eggoid. If the sphere must do work on the flux field, it will not deform (at least not to the assumed shape) but if the net work is done by the field it will deform to some shape, but not necessarily the one we assumed. I. e. It will not remain a sphere. That is where I am "going." I want to know the net work done in a slight deformation.

You did not yet make it completely clear to me is the transform or shift from original z to z' by z' = z -C where:
C = 2 a^2 D /(5 + a^2 D^2) ?

Sorry to be so slow and plodding, but I want to be sure we are on the same page. We are making real progress. PS what is the significance of your use of "==" instead of only "=" ?

 

Ahh, I haven't included any description of anything other than a single sphere. The basic setup only involves the first sphere since you have to integrate over all points in the first sphere whereas you do not do this for the second sphere. This is where the geometry get's rather tricky. You must calculate the α and β angles in which the second sphere will have influence on any given point. I imagine you would do something similar to MacM's CoS calculation. You have to take a line from the point in the first sphere to the surface of the second sphere such that the line is tangent to the second sphere and do the same to the opposite side. Compute α₀ and α₁ from these tangent lines. Then when you compute f(...), it will take the following form:

f(...) = ∑ c L = c (-L₀ + L₁ + L₂*δ )

where δ = 1 if α is between α₀ and α₁ and δ = 0 for all other cases. Basically it is a conditional statement in mathematical form. We could alternatively just say L₂ = 0 when α is not between the two computed α's

Additionally, for each α between α₀ and α₁, you would have to compute β₀ and β₁ where the conditions on δ apply here also, that is δ = 1 only when β is between β₀ and β₁. This seems like a complicated way of going about this, but I don't see any other way.

As for computing L₂ (the length of the line through the second sphere), that is a geometry task that I haven't really considered. Finding L₀ and L₁ is easy enough for the first sphere, so I imagine figuring out what L₂ is shouldn't be too difficult. Any ideas? Does any of this make sense? It would probably be easier to draw than it is to put in words.

 

Good. Aer and I both want to consider (for now) only a single solid object.

Thus there is no "second sphere."
I am not sure of Are's position as he was critical of my assumption of uniform density, but for now, I agree with DaleSpam that this is the most reasonable thing to assume.

Aer - you have not responded to:

Would you please do so. thanks

 

OK, so the force at an arbitrary point in space outside of a solid, uniform-density sphere. I think I can do that. In fact, I think this is simply a special case of the integral that Billy wants me to do. A sphere is an eggoid with e = 1 and D = 0.

-Dale

 

Not full time for a complete answer. Gotta go work. To find the maximum extent of Z you set S == 0 and therefore find the Z intercepts. Thus the whole left hand side is 0. Then you get Z^2 == R^2 or Z == +/- R.

Could you tell me if there is any trick to loading graphics. I had errors last time I tried.

Sorry if this is confusing, i was rushed.

-Dale

 

To DaleSpam:
Somethng is wrong somewhere, I think.

With your help, I have derived the following simple equation for the z-axis symmetric eggoid with volume of (4/3)π. (I do not think it necessary to state, but note, just for clarity, that we have set the unit of length so that the sphere we will also consider has radius R = 1.)

S^2 = (1 - Z^2) / {1 + D^(2/5)}

Where “D“ is slightly greater than zero ( 0< D < < 1 );
Where “S“ is the distance from the z-axis of a point on the surface of the eggoid;
Where “Z” is also on the surface and with z coordinate corresponding to that “S.”

We will always assume some particular value for Distortion constant, D, so we can define D‘ = 1+ D^(2/5) and I will do this, when convenient. Then:

(D’)S^2 = 1 - Z^2
Or
Z^2 + (D’)S^2 = 1, which is easily recognized as the equation for a sphere centered on the origin, if D’ = 1, but I am not so sure that the volume is (4/3)&pi; when D’ > 1 and this my first of two worries, but perhaps it is as I can not find anything wrong.

(I think D‘ = 1.1 will be large enough to see that it is an eggoid if sphere and eggoid graphs are superposed, as was illustrated in the right drawing of the first post this thread. Perhaps D‘ = 1.001 or smaller should be used in the final analysis for the stability of an initially assumed spherical shape, if negligible numerical errors result.)

Note that the eggoid described by the very simple equation immediately above is “unshifted.” That is, its centroid is on the negative z-axis, (if it is only a new version of the “eggoid 1” of the first post.) not the origin, but at:

z = - {2D / (5 + D^2)} and here it is convenient to use the original D, not D‘.
Which is approximately z ~= - 2D/5.
(This comes from your: C == 2 a^2 D /(5 + a^2 D^2) with a = R =1.)

When I first achieved my very simple eggoid equation, [Z^2 + (D’)S^2 = 1], I was troubled as the circular diameter (2S) at all values of +z is the same as at -z. (Symmetric about any z = 0 line.) I find this interesting, and think it may be OK, because if we go along the z-axis equal distances from the centroid, then when the corresponding “S1” at more negative z, say z twice the centroid-to-origin distance or approximately at z = - 4D/5 is compared to the “S2” corresponding exactly to z = 0. Then we have:

(D’){(S2)^2} = 1 = ~ [(4D/5)^2 + (D’){(S1)^2}]
Or
(S2)^2 = ~ [{(4D/5)^2}/D’ + (S1)^2]

Which clearly states S2 > S1, because the first term on the right has value approximately of {(16D^2)/25} but there is something wrong somewhere, I think.

S2 was the radius of the surface circle at z = 0 or point with z more positive than the centroid and S1 was the surface circle radius with z more negative than the centroid. That is telling me the “fat end” of the eggoid is at the positive z and the “sharp end” of the eggoid is at negative z.

This is troublesome because the original ellipsoid and eggoid equations of first post, from which all this came, and the text below them was:

Eq. (2) Ellipsoid (z-axis symmetric):
(Z/a)^ 2 + 2(R/b)^2 = A
And
Eq. (3a): Eggoid 1 (z-axis symmetric but “un-shifted” &“un-normalized”):
(Z/a)^2 + 2(Rf/b)^2 = B.
Where “B” is a constant, not necessarily equal to A, and we are now using the “Linear f,” or:
f(z) = 1 + Dz.

The form of Eq.(3a) differs from the final Eq.2 only by a near unity factor or function, “f,” which slightly exceeds 1 for z > 0 but is slightly less than 1 where z < 0 and 1 at z = 0. Eq.(3a) is thus an “eggoid“ (a shape like an egg) because if the final version of Eq.2 and Eq.(3a) have the same values of Z, and B is set equal to A, then the value of R in Eq.2 must equal the value of Rf in Eq.(3a). In this B = A case, where f > 1 the R of Eq.(3a) is smaller than the R of Eq.2. - That is, the eggoid’s more pointed end is in the +z hemisphere, and by the same reasoning, its fatter end is in the -z hemisphere…

I hope (and suspect) that you turned the eggoid around when you did: “To get the eggoid geometry I multiplied r(z) (1 + D z).” If that is the case, then the centroid is in the +z, not the -z part of space.

Comments?

 

to DaleSpam:

Yes it is true the S = 0 IFF Z = R in:

(S/R)^2 = {1 -(Z/R)^2} / {1 + (R^2) * D^(2/5)}

but my prior comments below still seem true, so I am confused.

Note that the numerator on the right is always* less than unity and the demonator is always greater than unity. Thus I would conclude that (S/R) is always less than one. That is at all points on the eggoid, the distance of the surface from the z-axis is less than the radius of the sphere, yet it has the same volume. So it sticks out from the sphere at the extremes of z.
___________________________________
* except for the circle in the z = 0 plain, when the numerator is unity.

 

Oh this is good news indeed. Sorry, I haven't really been following what you have been doing, maybe I'll take a second look at it.

Only critical in the sense that your constant density sphere isn't a model of anything in reality. Of course it is ok to assume constant density and see what comes of it. But from my understanding of your argument, you've been assuming a constant density and then saying that the theory is no good because it doesn't agree with real world results. All I am saying is that of course it is not going to agree because your model isn't very good.

That would be because I didn't see it. I would suggest making a separate post with any direct questions to me.

Technically, &rho; can be a function of anything. It appears you are using cylindrical coordinates whereas I've only been working in spherical coordinates. If &rho; is constant throughout the sphere, then c&middot;&rho;^k will become just a constant. If the density is spherically symmetrical, then &rho; will be a function of r only, &rho;(r). If the density is bottom heavy, then &rho; will be a function of z where z is the vertical axis &rho;(z). You could also assume bottom heavy and heavy center so that &rho; will be a function of z and r, &rho;(r,z).

I suspect you already know all of this, but I don't see why you have any questions about the simple function c&middot;&rho;^k. c & k are just constants. I do have to say though, that I do not know that this is the correct form, another example I gave before is c^{k&middot;&rho;}

 

Sorry, but I don't think we are on the same page. Perhaps we are not even reading the same book.

Our goal here is to model a push gravity mathematically, correct?

What is gravity? According to Newton, gravity is a force on a body. So we want to calculate this force using a separate model from Newton, specifically the push gravity that is the topic of this thread. Now, thus far, I thought we had agreed to only assume a single sphere and to calculate what, if any, gravity the sphere has on itself.

I've hypothesised that the calculation will show that the net force on a constant density spherically symmetric body will be 0. Also, I've said that a spherically symmetrical density of a spherically symmetric body will also have a net force of 0.

However, if the density is some other form, say bottom heavy as I introduced in the discussion with Billy T, then the net force on the body will be non-zero. I've also said elsewhere that the dynamic analysis of such a system will show that the body will tend towards a spherically symmetrical density as well as a spherically symmetrical shape.

So how do we show this mathematically? We have to integrate over all points in the sphere and calculate the net force on that point. So now I ask, how do your points outside of the sphere apply: "OK, so the force at an arbitrary point in space outside of a solid, uniform-density sphere."

 

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OK - If we revisit the model I had set up for calculating the total force of the sphere, we have"

So,

∑ c L = c(L₁ - L₀)

And from geometry,

L₁ - L₀ = 2&rho;&radic;(cos&sup2;&phi;&middot;sin&sup2;&theta;&middot;sin&sup2;&alpha; + sin&sup2;&phi;sin&sup2;&beta

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So, we have the following:

<table border=0 cellspacing=0 cellpadding=0><tr><td class=limit>r</td><td class=limit>2&pi;</td><td class=limit>&pi;/2</td><td class=limit>2&pi;</td><td class=limit>2&pi;</td><td class=limit></td></tr><tr><td class=expression>&int;</td><td class=expression>&int;</td><td class=expression>&int;</td><td class=expression>&int;</td><td class=expression>&int;</td><td class=expression>c·2ρ√(cos²φ·sin²θ·sin²α + sin²φ·sin²β) dβ dα dφ dθ dρ</td></tr><tr><td class=limit>0</td><td class=limit>0</td><td class=limit>-&pi;/2</td><td class=limit>0</td><td class=limit>0</td><td class=limit></td></tr></table>

And given our assumption of constant density, the evaluation of the above expression should be 0.

 

If you want more details of the planned paper, read the first posts of this thread, but I plan to post the first "progress report" within about an hour, which is a compact summary of what has been achieved; however, it will not tell the overall plan as the first and later post do.

You have misunderstood my approach in my "density issue post" arguments and some others. They do NOT relate to "physical reality," but to MacM and other's claim that push gravity can reproduce Newtonian gravity results.

For example 1: Even though I carefully explained how it would be possible to actually do the "marble in orbit" at 1AU with orbit diameter equal to Earth diameter test (Two "solar sailors," one above and one below the plain of ecliptic with marble in the ecliptic glued to the light weight string between them) you chose to attack it as impossible. All I was proving is that, because Newtonian gravity passes thru Earth as if it were not there but push gravities are at least slightly absorbed, it is impossible for push gravity to reproduce Newtonian gravity results for BOTH the marble in the "midnight" and "noon" positions. (Four cases compared when Earth "present" and "absent" are considered.)

For example 2:You only made trivial objections and comments ("define ideal gas” etc.) when I compared: (1) The required density dependence for two pendulum clocks, identical except their geometrically identical pendulums are lead and aluminum with (2) The required density dependence of a star collapsing to become a neutron star, which starts to collapse TOWARDS a black hole (but I stopped well before the black hole formed to avoid GR consideration.)
In case (1) If there is any dependency upon density the clocks do not keep the same time.
In case (2) the flux intercepted is decreasing with the R^2 and the star's pressure is initially increasing as P = &rho; *T or with (R^-3)*T while there is space between the stellar ions, and then much more rapidly as this space disappears. (The compression of a "quasi solid.") and later once the neutrons begin to form, the pressure is increasing extremely rapidly. Yet all the while the flux incident on the star is decreasing rapidly. Thus in case (2) this steadly decreasing flux can supply the compressive force IFF the flux absorption is a strong function and rapidly changing function of the density.

I would very much have liked to have had your comments on the substance of these arguments. Even MacM, I think, understands they are powerful problems for uniKEF and this is his reason for refusing to answer even after my asking 14 times for the functional form of the density dependency.

I also note that NO form of your &rho;^k etc is adequate for the collapsing star (case 2). Thus, I think ALL of your functional approaches discussed in the remainder of your post will fail if applied to the realistic case of a collapsing star. I was glad to read your last sentence, and fully agree. (The above and that is some of the many reasons why I think all push gravities fail.)

 

MacM has already told you his position regarding what I'll label as the "eclipse effect" which has nothing to do with the relationship between two bodies, but is only applicable when analyzing 3 or more bodies.

I don't believe MacM has ever said that his theory will predict the same thing as Newton when you analyze 3 such bodies in an appropriate configuration such that you have the eclispse effect. In fact, I recall him stating that his theory will predict something different. I fail to see your point in continuing this silly argument of yours.

Are you trying to prove one instance of MacM's theory differing from Newton? Because you wouldn't have to really do much of anything except take one of the examples MacM has given in which his theory differs from Newton's theory of gravity.

My point in bringing up the definition of an ideal gas was to show that your use of the ideal gas law doesn't apply even if the star doesn't completely collapse. The ideal gas law is only an approximation which has an exponentially increasing error as the gas is compressed. Even while the star is in a gaseous state, the ideal gas law has an error. It is not a trivial objection as you state.

Also, your reference to R^2 above seems to be ignoring my correction of your analysis that the volume of the sphere must be used, not some cross-sectional area of the sphere. The volume is proportional to R^3, not R^2.

If your arguments had any substance, perhaps you would get more favorable responses.

I never claimed that &rho;^k is the correct form, but I do not agree with your conclusion anyway.