Consider a mine of depth d drilled straight towards the centre of the Earth. The Earth is assumed to have uniform density.
(i) Using the idea that only the mass lying at a distance less than (R−d) from the centre contributes to the gravitational force, derive an exact expression for the acceleration due to gravity gd at depth d below the surface.
(ii) For d≪R, simplify the result obtained in part (i) to the linear form gd=g(1−Rd).
(iii) Show mathematically that g→0 as d→R and interpret the physical meaning of this result.
(iv) Find the value of gd/g when d=2R. Comment on how rapidly g falls with depth in comparison with altitude.