When forces are distributed over a line, an area, or a volume, it is often necessary to determine where the resultant force of such a system acts. To have the same effect, the resultant must act at the centroid of the system. | CHAPTER 48__ SECTIONS AND SHAPES TABULAR DATA Joseph E. Shigley Professor Emeritus The University of Michigan Ann Arbor Michigan CENTROIDS AND CENTER OF GRAVITY SECOND MOMENTS OF AREAS PREFERRED NUMBERS AND SIZES SIZES AND TOLERANCES OF STEEL SHEETS AND BARS WIRE AND SHEET METAL STRUCTURAL SHAPES REFERENCES CENTROIDS AND CENTER OF GRAVITY When forces are distributed over a line an area or a volume it is often necessary to determine where the resultant force of such a system acts. To have the same effect the resultant must act at the centroid of the system. The centroid of a system is a point at which a system of distributed forces may be considered concentrated with exactly the same effect. Figure shows four weights W2 W4 and W5 attached to a straight horizontal rod whose weight W3 is shown acting at the center of the rod. The centroid of this weight or point group is located at G which may also be called the center of gravity or the center of mass of the point group. The total weight of the group is W Wi w2 w3 w4 w5 This weight when multiplied by the centroidal distance x must balance or cancel the sum of the individual weights multiplied by their respective distances from the left end. In other words Wx Wfx W2l2 W3l3 W4 4 W5 5 or _ Wjl3 W2l2 W3l3 W4 4 W f x Wi w2 w3 w4 w5 A similar procedure can be used when the point groups are contained in an area such as Fig. . The centroid of the group at G is now defined by the two centroidal STANDARD HANDBOOK OF MACHINE DESIGN W1 8 U2 6 1 - I FIGURE The centroid of this point group is located at G a distance of x from the left end. distances x and y as shown. Using the same procedure as before we see that these must be given by the equations A similar procedure is used to locate the centroids of a group of lines or a group of areas. Area groups are often composed of a combination of circles rectangles triangles and .
