Assume that the earth is a solid sphere of uniform density, with mass M and radius R = 396...

Assume that the earth is a solid sphere of uniform density, with mass M and radius R = 3960 (mi). For a particle of mass m within the earth at distance r from the center of the earth, the gravitational force attracting m toward the center is Fr = –GM, m/r2, where Mr is the mass of the part of the earth within a sphere of radius r.

(a) Show that Fr = –GMmr/R3.

(b) Now suppose that a small hole is drilled straight through the center of the earth, thus connecting two antipodal points on its surface. Let a particle of mass m be dropped at time t = 0 into this hole with initial speed zero, and let r(t) be its distance from the center of the earth at time t (Fig. 1). Conclude from Newton’s second law and part (a) that r"(t) = –k2r(t), where k2 = GM/R2 = g/R.

FIGURE 1. A mass m falling down a hole through the center of the earth (Problem 12).

(c) Take g = 32.2 ft/s2, and conclude from part (b) that the particle undergoes simple harmonic motion back and forth between the ends of the hole, with a period of about 84 min. (d) Look up (or derive) the period of a satellite that just skims the surface of the earth; compare with the result in part (c). How do you explain the coincidence? Or is it a coincidence? (e) With what speed (in miles per hour) does the particle pass through the center of the earth? (f) Look up (or derive) the orbital velocity of a satellite that just skims the surface of the earth; compare with the result in part (e). How do you explain the coincidence? Or is it a coincidence?