The earth does not have a uniform density; it is most dense at its center and least dense at its surface. An approximation of its density is p(r) = A - Br, where A = 12,700kg/rri' and B = 1.50 X 10 kg/m4. Use R = 6.37 X 100 m for the radius of the earth approximated as a sphere.
(a) Geological evidence indicates that the densities are 13,100 kg/m' and 2,400 kg/m' at the earth's center and surface, respectively. What values does the linear approximation model give for the densities at these two locations?
(b) Imagine dividing the earth into concentric, spherical shells. Each shell has radius r, thickness dr, volume dV = 41rr2 dr, and mass dm = p(r) dV. By integrating from r = 0 to r = R. show that the mass of the earth in this model is M = tuR'(A - iBR).
(c) Show that the given values of A and B give the correct mass of the earth to within 0.4%.
(d) We saw in Section 126 that a uniform spherical sell gives no contribution to g inside it. Show that g(r) = m-Gr (A - iBr) inside the earth in this model.
(e) Verify that the expression of part (d) gives g = 0 at the center of the earth and g = 9.85 m/s' at the surface. (t) Show that in this model g does not decrease uniformly with depth but rather has a maximum of 47rGA/9B = 10.01 m/s' at r = 2A/3B = 5640 km.