. Assume that the earth is a solid sphere of uniform

Chapter , Problem 12

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. Assume that the earth is a solid sphere of uniform density, with mass M and radius R D 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 D GMrm=r2, where Mr is the mass of the part of the earth within a sphere of radius r (Fig. 3.4.13). (a) Show that Fr D 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 D 0 into this hole with initial speed zero, and let r.t / be its distance from the center of the earth at time t, where we take r<0 when the mass is below the center of the earth. Conclude from Newtons second law and part (a) that r00.t / D k2r.t /, where k2 D GM=R3 D g=R. R