Question

Question #598b2

Answer 1

At the exact centre of an idealised earth, weight would be zero.

Explanation:

At the exact centre of an idealised (ie perfectly spherical, uniform density, etc) earth, a body would be pulled radially outwards in every direction equally. The net force (weight) would be zero.

To find it I think the quickest approach uses Gauss' Law for gravity:

`intint_S mathbf {g} \cdot d mathbf {A} = -4\pi GM`

• crucially, where `M` is the total mass enclosed within the surface S.

With a spherical Gaussian surface of radius `r` concentric with the earth, this means that:

`M = M_e (r/r_e)^3` where `M_e` and `r_e` are the the earth's mass and radius.

And so the integration becomes:

`g cdot 4 pi r^2 = -4 pi GM_e (r/r_e)^3`

`implies g(r) = -(GM_e )/r_e^3 r`

`implies g(0) = 0`