Analytical Elements of Mechanics by Thomas R. Kane

By Thomas R. Kane

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4 for notation): The sum of the first mo­ ments of the n points P t ·, i = 1, . . , n, with respect to 0 is equal Pi (Hi) / P Ρ*(Σ Νί) " FIG. 4 0 Σ η i=l NiPi. 5 The cartesian coordinates of the centroid P* of a set of points P», i = 1, 2, . . , n, of strengths Niy i = 1, . . , n, are given by three expressions of the form £#<** Σ* x* = ^ 1 r= l Proof (see Fig. 2) Substitute these into p * ΣΝίΡί = LULL Σ* and write the three scalar equations corresponding to the vector equation thus obtained.

Similarly, n X b 2 and n X (bi + b2) can be obtained by rotating (b2)p and (bi + b2)p through 90 degrees in plane P . 1) l a l and multiply both sides of the resulting equation with |a|. This concludes the proof for the case n = 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.

2) are respectively equal to x,y,z: P = xni + 2/112 + zn 3 This relationship is the link between vector analysis and scalar analytic geometry. Problem: The rectangular cartesian coordinates of a point Pi are xh y\, Z\, those of a point P 2 , xi, yi, z2. , show that the distance d between Pi and P 2 is given by d = [(x, - xi) 2 + (i/2 - yiY + (z2 - zi)2]* Solution: Let pi and p2 be the position vectors of Pi and P 2 relative to the origin of the coordinate system, and let m, 112,113 be unit vectors such that (see Fig.

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