By Thomas R. Kane

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**Sample text**

N n a curve, surface, or solid; any one of these is called a "figure" the total length, area, or volume of F the n elements of F the length, area, or volume of Fi 58 CENTROIDS AND MASS CENTERS; SECTION 2 . 5 -F(T) FIG. 1 a point of F» a point the position vector of P t relative to 0 the centroid of the set of points P t , i = 1, . . , n, of strengths r t , i = 1, . . ,n the position vector of P relative to 0 the centroid of F the position vector of P * relative to 0 a symbol denoting the limit approached by the quantity Q as n tends to infinity and each of the elements Fi, i = 1, 2, .

The validity of the theorem b» and / . t (a X b») can each be regarded as a sum of only two vectors. 7 A set of mutually perpendicular unit vectors m, n2, n3 is called right-handed if ni X n2 = n3. If nx X n2 = — n3, the set is called left-handed. Example: Three right-handed sets of unit vectors are shown in Fig. 7. t s^ ^ b^^ ^ η v^ ^ 1 \ n,i n3 FIG. 6) and the relationships ■h X ni = n2 X Ü2 = n3 X n3 = 0 ni X Π2 = n3, n2 X n3 = ni, n3 X ni = n2 Problem: Referring to Fig. 4, resolve F X n into its ni, n2, n3 components.

5 Hence, for Θ = x/n, V^ /-Λ V^ f / \ 1 , l s i n ( x + ir/2n) ζ cos (τθ) = £ cos (tx/n) = - ^ + § s i n ( r / 2 n ) I-I 2 2 in sin (x/2) sin y~] sin (ίθ) = ^ sin (ίτ/η) = t=l i (1+L·) sin (π/2η) = cotan (π/2η) Substitute into Eqs. (4): -—, y = —cotan (π/2η) n (5) η The centroid P of the set of points P f (i = 1, . . , n) is shown in Fig. 5e. (e) Limiting position of P : as the elements were chosen in such a way that each element automatically shrinks to a point as 2«HsJ n tends to infinity, the coordinates x* and t/* of the centroid of the curve are obtained by proceeding to the limit in Eq.