Q.A hole is drilled half way to the centre of the Earth. A body is dropped into the hole. How much will it weigh at the bottom of the hole if the weight of the body on the Earth's surface is 350 N?

Modelling a planet as built of many thin, uniform concentric spherical shells, the shell theorem shows that at a depth d below the surface, only the mass of the smaller INNER sphere of radius $(R-d)$ contributes any net gravitational pull -- the outer shell (of thickness d) surrounding that point contributes exactly zero, since the point lies inside it. Assuming uniform density, this leads to the strikingly simple, EXACTLY linear relation $g_d=g\left(1-\dfrac{d}{R}\right)$, valid all the way from the surface ($d=0$, $g_d=g$) down to the very centre ($d=R$, $g_d=0$). …