In the study of Earth’s gravitational field, acceleration due to gravity g is known to vary with position.
(i) Write the mathematical expressions for acceleration due to gravity (a) at a height h above Earth’s surface and (b) at a depth d below the surface. Define the symbols used.
(ii) Explain qualitatively, using the expressions obtained in (i), how the magnitude of g changes as (a) altitude increases and (b) depth increases. Comment on whether the variation is linear or non-linear for small h and d compared with Earth’s radius R.
(iii) For a numerical value of 10 km, decide whether the fractional change in g is larger at 10 km altitude or at 10 km depth. Justify your answer without performing detailed calculations.