Q.What is the decrease in weight of a body of mass 500 Kg when it is taken into a mine of depth 1000 Km? (Radius of earth R = 6400 km, $g = 9.8\ \text{m/s}^2$)

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 ($d=R$, $g_d=0$). …