Halliday, Resnick & Walker — Gravitation: Problem 42

42 Figure 13-33 shows, not to scale, a cross section through the interior of Earth. Rather than being uniform throughout, Earth is divided into three zones: an outer crust, a mantle, and an inner core. The dimensions of these zones and the masses contained within them are shown on the figure. Earth has a total mass of \(5.98 \times 10^{24} \text{ kg}\) and a radius of \(6370 \text{ km}\). Ignore rotation and assume that Earth is spherical. (a) Calculate \(a_g\) at the surface. (b) Suppose that a bore hole (the Mohole) is driven to the crust–mantle interface at a depth of 25.0 km; what would be the value of \(a_g\) at the bottom of the hole? (c) Suppose that Earth were a uniform sphere with the same total mass and size. What would be the value of \(a_g\) at a depth of 25.0 km? (Precise measurements of \(a_g\) are sensitive probes of the interior structure of Earth, although results can be clouded by local variations in mass distribution.)