Calculate Gravitational Forces Among Objects: jKarlos' Guide

[jKarlos]

I'd like to know how to calculate the gravitational forces among several objects, better I'd like to be able to draw some representation of field itself. The objects are at rest, in the vacuum and very far apart from any other objects.

So, my first approach was

For 2 particles with M₁ and M₂ at D distance

What is the distance d1 from M₁ or d₂ from M₂ ( D = d₁ + d₂ ) where the F₁₂ = F₂₁ ?

Since the force between the 2 particles is given by

F₁₂ = G M₁M₂ / d₁²

and D = d₁ + d₂

making F₁₂ = F₂₁

would allow me to calculate d₁ as

d₁ = d₂ * sqrt ( M₁/M₂ )

is it?and how go on if I have more particles involved ( M₃, M₄, M_n ... ) ?

Thanks,

jKarlos

 

[JANm]

Hallo jKarlos
Had a very nice supply but it is eaten.
Essention is that force of M_1 and M_2 at distance d is not calculated with your d_1 or d_2.
F=GM_1*M_2/d^2
I can learn you much about this problem, so ask what you want.
greetings Janm

 

[ideasrule]

What exactly are you trying to calculate? Just the force on each particle, or are you trying to calculate how particles can be arranged so that one of them doesn't move?

 

[jKarlos]

in this case the particles are fixed they can't move, suppose a group of steel balls laying in the sand. Each ball has different sizes and masses and I'd like to be able to calculate or draw the gravitational field around them or at least the equilibrium point. Would be it correct just to calculate the forces between the all the possilble pars?

 

[JANm]

Hello jKarlos
This is a problem which can be solved potentially. In the case the balls lay on a beach it is solvable for any points on the plane twodimensionally. In the case that the balls lay on a dune this is a three dimensional potential problem.
It is the way of the first approximation one solved for our solar system: Sun is so heavy that it is exactly in the foci of the ellipses of the planetcurves.
Essential for the potential is that there is no consumption of energy. The solutions are harmonics and the potential is a function of place alone and has no timedependancy.
I think those functions are called autonomous...
Let us start on the beach: suppose you are at the place of ball_10, then you calculate the distance of ball_10 to the balls_n |n<>10, 1<n<#balls and fill in the potential to the balls by V=GM_n/r_n...
greetings Janm

 

[jKarlos]

thank you JaNm

but if I drop one small ball among three pre-existing balls in the sand, how could I calculate and express the forces this ball will be acted upon. Suppose this new small ball is so small that its effects on the others ball are neglected.

That was the reason I thought first about d1 and d2 and D = d1 + d2 because what I really want to know what are the forces at some point in the between the balls which are generating this gravitational field.

So, I have a group of massive spheres laying on the 2D plane and I'd like to draw something similar to a topographical map ( http://www.waterfordhistory.org/graphics/maps/waterford-topo-map.jpg" ) of the gravitational field formed by this group of spheres.

jKarlos

 

[JANm]

Hello jKarlos
Clever place Waterford. At the height of 665 (meters?) with a river flowing by at 350. On a sunday afternoon one can walk to the hill 689 (why is this number in red?), which gives a small creek at height 379. One can also descent from Waterford to 451 in the north east...
The balls have circular potentials, d.i. concentrical heightlines. The descent is steepest very close to them. Since the balls don't move in the sand their three potentials can be added.
In comparison to the picture of Waterford I would say the three balls are at:
1 350 the river, 2 the vallei in NE 451 3 the creek at 379 and
the ball_1 is the heaviest then comes ball_3 and then ball_2.
greetings Janm

 

[jKarlos]

-) -) -)

actually I didn't pay attention to what they were trying to represent with data map, I was just trying to explain what I'd like to draw. -)

There are some theories in the Geography field which try to model and estimate the flow among cities using gravitational concepts where the mass is represented by the population, but the main concern is the flow. I'm trying to investigate if it would be possible to use potential gravitational model to estimate how the small cities in the space between 2 big cities possibly could influenced by each one.

thank you

 

[JANm]

-) -) -) I'm trying to investigate if it would be possible to use potential gravitational model to estimate how the small cities in the space between 2 big cities possibly could influenced by each one.
thank you

First you have one of the cities. It needs infrastructure, a slow growing city gets that empirically, d.i. pragmatic. Peoples walking carring and bycicling needing road to get out of the city. There were the heightlines (I still keep klinging a little to the hill concept) are close together the descent is quick. Passive objects (with no engine of their own) follow Newtons laws (a little adjusted by Hamilton) in search of the easiest descent. Water is the best example for that. It seems to follow a path of the least resistance...
So those paths can be narrow while the velocity is high. In comparison to human logistics that is strange: we make our highways broad and provincial roads narrow. Ok one town needs its infrastructure. The other one could be a modern quickbuilded one. There the roads need to be ingeniously designed. Possibly lesser space needed while purpose of design is here foreknown. Now combining the two cities for one thing the road between them needs to be reinforced. People saying I want out of this town have a large possibility to just go to the other and vice versa, but adding the two potentials does not give a valey between them. No there is a ridge! In peoples terms a threshold. There must be some force one goes from one town to the other, sufficient to climb the ridge between them and from that roll
to the other.
The ridge just between the two cities where the acceleration to want to go to one or the other is called the Lagrange point 1 (there are four others). In your terms
GM_2/d_2^2 =GM_1/d_1^2
and I think your squareroot formula comes close:
M_2/M_1=d_2^2/d_1^2=(d_2/d_1)^2
so d_2/d_1=Sqrt(M_2/M_1)
Interpretation of this solution at distance d_I of object M_I, for I=1 to 2 the acceleration due to to mass M_I is equal to the acceleration due to the other. L1 is on the ridge (instable maximum) where objects don't know to decide which city is the more attractive...
indeed d=d_1+d2 also correct.
greetings Janm

 

[jKarlos]

thank you JaNm,

just for "intelectual enlightenment" -) -) -) some links about the use of gravitational model in the geography.

http://geography.about.com/library/weekly/aa031601a.htm"

"[URL
http://faculty.washington.edu/krumme/systems/gravity.html


I'm doing some machine learning experiments and I have a lot of data to express the city potential (population, gdp, banks, etc.) , besides that I have some other data related to airplane terrestrial routes of people and goods among cities, so "theorically" I could do some computational simulations or even propose a model to represent these fields and I was looking for some way to represent them with the gravity model and with our results and study what we are going to find.

thank you again

jKarlos

 

[JANm]

Hello jKarlos,
Inspired on your Waterford Heightmap I have made a printscreen of a program of mine, that will follow...
View attachment waterford.bmp
I see that you try to relate geografic models to the potential equation. I like the first site you mention on that subject.

Note that any function of r are valid then, because they are autonomous.
The Poles of 1/r are not necessary then. The circumference of the metal balls in the sand are needed as extra information to avoid the poles. In a program of a model distance to the centre of an object canot be calculated to the centre, not even for a molecule...

greetings Janm