1.6. Exercises 21 x y θ Figure 1.11. Complementary curves M, surface (mass) density σ, and radius x by integrating over a set of ring elements. (See Figure 1.12.) Assume that point P is a distance r from the center of the shell and that y is the distance between the ring and point P . Be sure to consider the case when P is inside the shell (r x) as well as outside the shell (r x). 1.6.4. Potential energy inside the earth. Use your results from the last problem and integrate over shells of appropriate radii to show that the potential energy of a point mass m in a spherical and homo- geneous earth can be written, to within an additive constant, as V (r) = 1 2 mg R r2 , (1.53) where R is the radius of the earth, g is the magnitude of the gravi- tational acceleration at the surface of the earth, r is the distance of the point mass from the center of the earth, and ρ is the (volumetric) density of the earth.
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