The earth does not have a uniform density; it is most

Chapter 13, Problem 75P

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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 ?(r) = A ? Br, where A = 12,700 kg/m3 and B = 1.50 × 10?3 kg/m. Use R = 6.37 × 106 m for the radius of the earth approximated as a sphere. (a) Geological evidence indicates that the densities are 13,100 kg/m3 and 2400 kg/m3 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 = 4?r2 dr, and mass dm = ?(r)dV. By integrating from r = 0 to r = R. Show that the mass of the earth in this model is (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 13.6 that a uniform spherical shell gives no contribution to g inside it. Show that 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/s2 at the surface. (f) Show that in this model g does not decrease uniformly with depth but rather has a maximum of 4?GA2/9B = 10.01 m/s2 at r = 2A/3B = 5640 km.