Analytical Mech Homework Solutions 64

Analytical Mech Homework Solutions 64 - The gravitational...

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The gravitational force on the particle is due only to the mass of the earth that is inside the particle’s instantaneous displacement from the center of the earth, r. The net effect of the mass of the earth outside r is zero (See Problem 6.2). 3 r π ρ = 4 3 M 4 ˆˆ 3 rr FG m r e k r e πρ =− G The force is a linear restoring force and induces simple harmonic motion. r F G 23 22 4 m T kG ππ ω == = The period depends on the earth’s density but is independent of its size. At the surface of the earth, 3 4 3 e ee GMm Gm mg R RR 4 3 e Gg R = 6 2 6.38 10 1 1 . 9.8 3600 e R mh r Th gm s s × ×≈ 4 r 6.4
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