Our weight is more at the pole than at the equator (because it is closer to the center).
Does that mean if we go down into the earth (some kind of bore), will our weight increase further?
I remember reading somewhere that this is not the case. Is that true?
NO!
An approximate value for gravity at a distance $r$ from the center of the Earth can be obtained by assuming that the Earth's density is spherically symmetric. The dependence of gravity of earth on depth given by $$g(r)=\frac{4\pi}{3}G\rho r$$ $$g'=g\left( 1-\frac{d}{R}\right)$$ It's clear from here that the gravity goes to zero as $d\rightarrow R$. Note $d$ represents the depth from the surface.
