Rearranging universal gravitational law for r?

Question

Hi, i'm trying to rearrange the equation
g = GM/r^2 for r,
however i don't have the mass M, i do have a quantity for density, and with
mass = density (p) x volume (4/3 pi r^3),
so i've managed to get the equation
g = G (p x 4/3 pi r^3)/r^2, and i'm slightly confused as to where to go from here. Would the equation end up as
g = G x p x 4/3 x pi x r, and if so, to find r would the equation end up as r = G x p x 4/3 x pi /g ?
Thanks.

I'm trying to find the radius of a planet, when i have only the acceleration due to gravity, and the density of the planet rearranging. So, no, i haven't combined 2 unrelated equations in a meaningless way.

Answer 1

U got it

Answer 2

You have combined two unrelated equations in a meaningless way. I think what you were supposed to do is take the algebraic equation you were given and solve for r. Simple algebra. You would get an equation for r as a function of G, g, and M.

Answer 3

You got it almost right. The correct formula is
r = 3/(4π) * g/(ρG)

For exampe if we use ρ = 5g/cm³ = 5,000 kg/m³ for the earth,
we will have

Re = 3/ (4 x 3.14) * 10 / (5,000 x 6.7e-11) = 7,120 km
(quite reasonable match)

Answer 4

u got it right