By Thomas R. Kane

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**Extra resources for Analytical Elements of Mechanics**

**Example text**

6n3 Hence, F X n = [ ( - 8 ) ( - 0 . 8. 6 = ( - 8 ) ( - 0 . 1 The parentheses in the expression a · (b X c) are un necessary because (a · b) X c is meaningless, (a · b) being a scalar. Hence one writes a b X c. , [a, b, c] = a b X c = a X b e Proof: Resolve each of the vectors a, b, c into m, n2, n3 com ponents (in, n2, n3 being a right-handed set of mutually perpen dicular unit vectors), and carry out the indicated operations. 4 If a, b and c are parallel to the same plane, or if any two of the vectors a, b, c are parallel to each other, [a, b, c] = 0 Proof: In the first case, b X c is perpendicular to a.

5 59 The theory of limits shows that limit of the quotient Υ^. Ti is equal to the quotient of the limits L | / . ηρΑ T

P n , as shown in Fig. 5c. P t is a typical point of this set of points. (c) Strengths of the points P ; (i = 1, . . , n ) : let Ni be the strength of Pi. The length of each element is wR/n, as noted in (a). Hence, all of the Ni(i = 1, . . , n) must be taken equal to each other. Take N4: = 1, i = 1, 2, . . , n (1) (d) Location of the centroid of the set of points P t (i = 1, . . , n): All of these points lie in a plane. 6), and the centroid lies in this plane. Set up rectangular cartesian coordinate axes, X and F, as shown in Fig.