5.64. Mine shaft
(a) If the earth had constant density, the gravitational force would decrease linearly with radius as you descend in a mine shaft; see Eq. (5.44). However, the density of the earth is
not constant, and in fact the gravitational force increases as you descend. Show that the general condition under which this is true is $\rho_c < (2/3)\rho_{avg}$, where $\rho_{avg}$ is the average
density of the earth, and $\rho_c$ is the density of the crust at the surface. (The values for the earth are $\rho_c \approx 3$ g/cm³ and $\rho_{avg} \approx 5.5$ g/cm³.) See Zaidins (1972).
(b) A similar problem to the one in part (a), which actually turns out to be exactly the same, is the following. Consider a large flat horizontal sheet of material with density $\rho$ and
thickness x. Show that the gravitational force (from the earth plus the sheet) just below the sheet is larger than the force just above it if $\rho < (2/3)\rho_{avg}$, where $\rho_{avg}$ is the average
density of the earth. A sheet of wood (with a density roughly equal to that of water) satisfies this inequality, but a sheet of gold doesn't. A result from Problem 5.13 will be
useful here.
(c) Assuming that the density of a planet is a function of radius only, what should $\rho(r)$ look like if you want the gravitational force to be independent of the depth in a mine shaft,
all the way down to the center of the planet?
