Calculate Earth's Mass Using Gauss Law & Density

[bananasplit]

Is it possible to find the mass of the Earth based on the Earth's volume mass density, roe = A-Br=R, where A = 1.42 x 10⁴ kg/m³, B = 1.16 x 10⁴ kg/m³, and Earth’s radius
R = 6.370 x 10⁶ m

I know that based on Gauss Law that (closed integral) g x da = -4Gmin, where g is the total electric field due to the inside and outside of the closed surface. I don't see how this is possible.

 

[Cantab Morgan]

Hi, bananasplit. Could you try to be a little clearer? I think you were trying to write an expression for the density of the Earth, $$\rho$$, as a function of distance from the center, r, but I can't read your expression. Does it have a spurious equals sign, or a missing / symbol?

If that's the case, then you have to do a volume integral, and I don't see any connection to Gauss's Law. Might you be confusing Gauss's Law with Gauss's Theorem?

 

[sir-pinski]

Yes, it is possible to find the mass of the Earth using Gauss Law and density. Gauss Law states that the flux of the gravitational field through a closed surface is equal to the mass enclosed by that surface. In other words, the total gravitational field produced by all the mass inside a closed surface is equal to the mass contained within that surface multiplied by the gravitational constant, G.

Using the given values for the Earth's volume mass density (ρ), radius (R), and the gravitational constant (G), we can calculate the total mass of the Earth using the formula:

M = (4/3)πρR^3

Substituting the given values, we get:

M = (4/3)π(1.42 x 10^4 kg/m^3)(6.370 x 10^6 m)^3 = 5.97 x 10^24 kg

This is the same value as the known mass of the Earth, which is approximately 5.97 x 10^24 kg. Therefore, it is possible to find the mass of the Earth using Gauss Law and density.

The equation you mentioned, (closed integral) g x da = -4Gmin, is actually Gauss's law for gravity, which is similar to Gauss's law for electricity. It relates the gravitational field (g) to the mass contained within a closed surface (min). However, in this case, we are dealing with the Earth's mass, which is already known, so we do not need to use this equation.

In summary, Gauss Law and density can be used to calculate the mass of the Earth by relating the gravitational field to the mass enclosed within a closed surface. By substituting the given values for density and radius, we can find the mass of the Earth, which is consistent with the known value.