By Thomas R. Kane
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Centroid of the set S is the point P* with respect to which the sum of the first moments of the points of S is equal to zero. 1 The position vector p* of P*, relative to an arbitrarily selected reference point 0, is given by P* r = ijzi n i=l where p» is the position vector of Pi relative to 0 (see Fig. 1). Proof: The position vector of P» relative to P* is p» — p*. The sum of the first moments of the points P t with respect to P* __n is 22 - N% (Pi — p*)· If P* is to be centroid of S, this sum must be equal to zero: £ΛΓ,·(Ρ,--Ρ*) = Ο t= l Expand the left-hand member of this equation: 46 CENTROIDS AND MASS CENTERS; SECTION 2 .
M The remaining terms in 22 - ^ ^ S P i ^° sums of the type Nz + N(—z), and each such sum is equal to zero. Problem: Fig. 6 shows the strength and location of each of five points. Locate the centroid of this set of points by using symmetry considerations. ς(-2) FIG. , the plane of the paper; the plane passing through Pi and P& and normal to the plane of the paper. The centroid must lie in each of these planes. Hence it coincides with their point of intersection, the point P3. 52 CENTROIDS AND MASS CENTERS; SECTION 2 .
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. 5d, and let x t and yi be the coordinates of P t . Note that angle PiOX = ίπ/η radians. Thus, Xi = R cos (tV/n), yi = R sin (ιπ/ri) (2) Let x and y be the coordinates of a point P, the centroid of the set of points P» (z = 1, . . , n). 5) —n ^ ..
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