15_C_3 - 3. (0, 0, b) m b - a cos D dS a cos a Fig. C-3 The...

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3. D m φ dS ( 0 , 0 , b ) ψ a a cos φ b - a cos φ Fig. C-3 The mass element σ dS at position [ a , φ, θ ] on the sphere is at distance D = p a 2 + b 2 - 2 ab cos φ from the mass m located at ( 0 , 0 , b ) , and thus it attracts m with a force of magnitude d F = km σ dS / D 2 . By symmetry, the horizontal components of d F coresponding to mass elements on opposite sides of the sphere (i.e., at [ a , φ, θ ] and [ a , φ, θ + π ]) cancel, but the vertical components d F cos ψ = km σ dS D 2 b - a cos φ D reinforce. The total force on the mass m is the sum of all such vertical components. Since dS = a 2 sin φ
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