In Example 12.10 (Section 12.6) we saw that inside a planet of uniform density (not a realistic assumption for the earth) the acceleration due to gravity increases uniformly with distance from the center of the planet. That is, g(r) = gr/R, where g, is the acceleration due to gravity at the surface, r is the distance from the center of the planet, and R is the radius of the planet The interior of the planet can be treated approximately as an incompressible fluid of density p.
(a) Replace the height y in Eq. (14.4) with the radial coordinate r and integrate to find the pressure inside a uniform planet as a function of r. Let the pressure at the surface be zero. (This means ignoring the pressure of the planet's atmosphere.)
(b) Using this model, calculate the pressure at the center of the earth. (Use a value of p equal to the average density of the earth, calculated from the mass and radius given in Appendix E)
(c) Geologists estimate the pressure at the center of the earth to be approximately 4 X 1011 Pa. Does this agree with your calculation for the pressure at r = O? What might account for any differences?