The moment of inertia of a sphere with uniform density about an axis through its center is 2/5 MR = 0.400MR'. Satellite observations show that the earth's moment of inertia is 0.330SMR, Geophysical data suggest the earth consists of five main regions: the inner core (T = 0 to T = 1220 Ian) of average density 12,900kg/m, the outer core (T = 1220 Ian to T = 34801an) of average density 10,900 kg/m', the lower mantle (T = 3480 Ian to T = 5700 Ian) of average density 4900 kg/m', the upper mantle (T = 5700 Ian to T = 6350 Ian) of average density 3600 kg/m, and the outer crust and oceans (T = 6350 k ID to T = 6370 Ian) of average density 2400 kg/m3.
(a) Show that the moment of inertia about a diameter of a uniform spherical shell of inner radius Rio outer radius R and density p is 1 = p(πm/15)(R5/2 – R5/1'). (Hint: Form the shell by superposition of a sphere of density p and a smaller sphere of density -p.)
(b) Check the given data by using them to calculate the mass of the earth.
(c) Use the given data to calculate the earth's moment of inertia in terms of MR.