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centroid
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CENTROID OF COMPLEX GEOMETRIC FIGURES:



So in this articles, we are going to discuss the concepts of centroid for one dimensional as well as two dimensional objects. Let's first discuss about 1D and 2D objects, one by one, an 1D object is a line, practically a metallic rod will be considered as a linear, 1D object. Where any thin plate of negligible thickness can be considered a 2D body. Suppose we have a thin metallic rectangular plate. If it is considered as a 2D rectangular area of b X h.
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The total moments produced about Y axis will be equal to the summation of all the individual moments produced by n elemental areas. Now moment is a vector quantity and we know vectors of same kind can be added together, therefore, all the n moment vectors can be added to get a single value of Resultant Moment. We also know this resultant moment's position vector. Let the resultant moment passes through a point G. The point through which resultant moment passes through is called Center of the Area or Centroid. How can we find out the point G, whose coordinates are (X,Y). As moment of an area also obeys VARIGNON'S THEOREM OF MOMENT. Then sum of all the moments produced by individual elemental areas will exactly be equal to the moment produced by the total area, i.e. the resultant of all those elemental areas. Now if all the areas are added to have the resultant area which will pass through the centroid G such that it produces a moment of XA about Y axis and YA about X axis. But Varignon's theorem states us that, for a vector system, resultant vector produces the moment about a point, is exactly equal to the sum of all the moments produced by all elemental areas about the same point and in the same plane. Hence, we can write now that,

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