The moment of inertia of a planet is used to evaluate its interior structure. In this problem you will construct a simple ``two-zone'' model of the interior of Earth, assuming spherical symmetry. Take the average densities of the core and mantle to be 10,900 kg m−3 and 4500 kg m−3, respectively (neglect the thin surface crust).
(a) Using the average density of the entire Earth, determine the radius of the core. Express your answer in units of Earth's radius.
(b) Calculate the moment-of-inertia ratio (I/MR2) for the ``two-zone'' Earth (the actual value is 0.3315). The moment of inertia for a spherically symmetric mass shell of constant density ρ, having inner and outer radii R1 and R2, respectively, is given by
I≡∫vola2dm=158πρ(R25−R15).a is the distance of the mass element dm from the axis of rotation.
(c) Compare your answer in part (b) with the value expected for a solid sphere of constant density. Why are the two values different? Explain.