15_5_25 - 25. The surface element d S = a d dz at the point...

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25. The surface element dS = a d θ dz at the point with cylindrical coordinates ( a , θ, z ) attracts mass m at point ( 0 , 0 , b ) with a force whose vertical component (see the figure) is d F = km σ dS D 2 cos ψ = km σ a ( b - z ) d θ dz D 3 = km σ a ( b - z ) d θ dz ( a 2 + ( b - z ) 2 ) 3 / 2 . The total force exerted by the cylindrical surface on the mass m is F = - Z 2 π 0 d θ Z h 0 km σ a ( b - z ) dz ( a 2 + ( b - z ) 2 ) 3 / 2 Let b - z = a tan t - dz = a sec 2 t dt = 2 π km σ a Z z = h z = 0 a tan t a sec 2 t dt a 3
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