Comment: I checked it in MS Excel and it seems to be false.
I would like to prove that a celestial body's gravitational force is the same as if it would have been a point.
Suppose that the density is the same all over at a certain depth into the body. (It may be less at the surface and more halfway down to the center for example). The density is given by , where r is the distance from the center of the body. An object with mass is laying at the distance d from the center of the celestial body. Suppose it's outside the celestial body, in other words, .
Suppose is the total force the body is affecting the body with, considering only the mass in the body at a distance from the center within the interval , a mass which still fulfills the density criteria*. Note that , since there is no mass within the interval , and that , though I don't want to prove it.
Now, I state that
which can be simplified as
But according to a clause (I have only heard of it) that states that the celestial body can be treated as if it where only a point but with the same mass, we would get
We now has two different formulas for :
Is this true? Correct me if I'm wrong. Or if I have made any error (probably). Or if I am using the mathematical signs wrong.
* The density critera says that the body shall have the same density all over any given distance from the center. And I just made it up.
Anyway, your theorem is certainly correct. I would recommend looking up Gauss' Law for gravitation. The main idea is that the flux of gravitational field lines through a closed surface is proportional to only the mass inside the surface. It has nothing to do with the position of the mass, nor the density function. This is a direct consequence of the inverse square nature of the Newtonian gravity law.
Oops, I think I forgot something very important in my calculations:
I forgot to consider the direction of the forces. Every force is now directed directly forward. But that is not the case. It should really look like this:
So we would get this relation instead:
And that actually does seem to be true. Hm, I wonder how to show it, I'm not that good at integrals. （´-`）.｡oO( ... )
But when dealing with sines and cosine in rational form it is useful to use "Weierstrass Substitution".
Also, you can use a powerful algorithm The Integrator.
And finally you are dealing with countinous definite integral, why not approximate by Trapezoidal rule.
> First, I have absolutely no idea what means! Some physics alchemistry.
If you read my post again I'm sure you will understand. It's because I multiply the expression by the square root of the same expression.
> But when dealing with sines and cosine in rational form it is useful to use "Weierstrass Substitution".
Yes, I looked that up, very clever way to go, I tried it but I didn't succeed though. Maybe I didn't try hard enough.
> Also, you can use a powerful algorithm The Integrator.
It didn't finish my expression in the given amounth of time, is that a bad sign?
> And finally you are dealing with countinous definite integral, why not approximate by Trapezoidal rule.
Thank you, but I want to prove the equality, so I prefer an exact expression to a value.
There is absolutely no purpose of having an exact complicated model in physics. Because all we need an approximation. (In fact there is no such thing as an ideal physics model). So approximating now, or getting it exact and then approximating makes no difference. (As long as the answer works).Thank you, but I want to prove the equality, so I prefer an exact expression to a value.
I use it in the end of my integral because that is the standard way everybody writes them. I do not not want to confuse the readers. They are useful in Multi-variable calculus when you are dealing with several functions they can be used to keep track of which function is first, second ,third ,.... As I said I see them used all over the place. And I have a philosophy if I do not understand something (the proof or definition) I usually do not use it. Which is why I am the only person on this site that integrates using the inverse chain rule instead of a u-substitution.
Historically Leibnize defined them as infinitemal quantities (but then you need to define what infintesmal means!). But mathematicians were not satified with that because it was not rigorous. I have no idea if you were dealt with mathematicans before but now you just have. Be prepared with the pain they put you through. Not because they (the mathemations) want to bother or show off but because it is necessary if you want to prove something completely.
May there has been a small missunderstanding here?
I have never written , but I did write after the equation. I don't know if it is this you meant, and maybe I should call it something else. Then the in the denominator is applied on an area turning into a volume. However, I do have a variable in it, representing a distance. If I still haven't got it right, it's probably not your fault!
TPH and I were guilty of not looking at your previous work carefully enough.