Mathematical problem - sphere density

[Zugbop]

Ok, I've been having a bit of a problem with spheres at the moment. I'm trying to work out a formula which will calculate the average density of a sphere of given radius, in terms of the density of the material from which it is made (under standard conditions). The problem is that the density increases with depth, and, I presume, differently for different materials. I have got as far as:
a = (d2^r)/r
Here, a = average density of the sphere, d = density of the material under standard conditions, and r = the depth of the material.
It shows, I think, what the average density is on a hypothetically flat (so non-spherical) surface, when density doubles with depth. I also think that the 2, representing the fact that it doubles, could be replaced with a letter for some kind of coefficient of compressibility (so, if the commpressibility is greater, more material can be put in a volume, giving a greater density).
Or I could be talking complete rubbish, you tell me. But more importantly, tell me what the equation I need is!

Phew, that was pretty damn tiring. Naptime.

Oh, and, by the way, this is my first post in the new forums. I haven't been on for a long while, and hadn't realised they had moved. LOL.

[Zugbop]

Yeah, I figured that, but I'm not sure if it's much use. The ultimate thing I'm trying to work out is the freefall acceleration due to gravity - I worked outt hat it would be (4/3)Gdr, where G = gravitational constant, d = average density, and r = radius.

[t̷s͢uk̕a͡t͜ư]

I'm no physicist, but I figure the density of a sphere is: (4/3)(pi)(r^3)(m^-1), where r is the radius and m is the mass.

It's a good thing you aren't a physicist, because what you have there is the multiplicative inverse of the density.
(Density is mass per volume.)

Disregard my first post if you saw it. I get what you're asking now.
This sounds like a calculus problem.

If you have the equation for the density for a point along r:


Then the density at some point r times the differential bit of volume at that point will give you a differential bit of mass:


Density is a function of the radius, so you'd have to either put it in terms of volume or find some substitution for dV that gives you something * dr.
In this case, the latter is much easier:


Substituting in for dV, you have:


To solve for total mass, we can use integration by parts:


If we pick u = r and dv = 2^r, we find that v (in the integration by parts substitution) is:


Applying integration by parts to that whole integral:


Evaluating the resulting expression from 0 to r gives:


So our final expression for the total mass is:


We know the total mass and the total volume, and we know the two constants involved (d and r), so we can now solve for average density:


It's plug 'n chug from there.

[darkshadow]

I'm getting:

-(3*d*(-2*(-1 + 2^r) + r*(r*Ln[2]^2 + Ln[4])))/(r^3 Ln[2]^3)

Are you sure you did it correctly Tsukatu?

Edit: I forgot to adjust for something in the Integral so nevermind.