The center of gravity of a rigid body is the point G where a single force W, called the weight of the body, can be applied to represent the effect of the earth’s attraction on the body. Consider two-dimensional bodies, such as flat plates and wires contained in the xy plane. By adding forces in the vertical z direction and moments about the horizontal y and x axes, the following relations were derived W = dW xW = x dW yW = y dW which define the weight of the body and the coordinates x and y of its center of gravity. Chapter 5 DISTRIBUTED FORCES: CENTROIDS AND CENTERS OF GRAVITY

For a homogeneous flat plate of uniform thickness, the center of gravity G of the plate coincides with the centroid C of the area A of the plate, the coordinates of which are defined by the relations xA = x dA yA = y dA These integrals are referred to as the first moments of area A with respect to the y and x axes, and are denoted by Q y and Q x, respectively Q y = xA Q x = yA

xL = x dL yL = y dL Similarly, the determination of the center of gravity of a homogeneous wire of uniform cross section contained in a plane reduces to the determination of the centroid C of the line L representing the wire; we have x y O x y L C

 W W x y z O Y G X x y z W1W1 W3W3 G3G3 G1G1 G2G2 O The areas and the centroids of various common shapes are tabulated. When a flat plate can be divided into several of these shapes, the coordinates X and Y of its center of gravity G can be determined from the coordinates x 1, x 2... and y 1, y 2... of the W2W2 centers of gravity of the various parts using X  W =  xW Y  W =  yW

If the plate is homogeneous and of uniform thickness, its center of gravity coincides with the centroid C of the area of the plate, and the first moments of the composite area are x y z O G Q y = X  A =  xA Q x = Y  A =  yA

When the area is bounded by analytical curves, the coordinates of its centroid can be determined by integration. This can be done by evaluating either double integrals or a single integral which uses a thin rectangular or pie-shaped element of area. Denoting by x el and y el the coordinates of the centroid of the element dA, we have Q y = xA =  x el dA Q x = yA =  y el dA

C L x 2y2y y The theorems of Pappus-Guldinus relate the determination of the area of a surface of revolution or the volume of a body of revolution to the determination of the centroid of the generating curve or area. The area A of the surface generated by rotating a curve of length L about a fixed axis is where y represents the distance from the centroid C of the curve to the fixed axis. A = 2  yL

x 2y2y y A C The volume V of the body generated by rotating an area A about a fixed axis is where y represents the distance from the centroid C of the area to the fixed axis. V = 2  yA

x w O B x dx L w dW x w O B x L W W = A P C The concept of centroid of an area can also be used to solve problems other than those dealing with the weight of flat plates. For example, to determine the reactions at the supports of a beam, we replace a distributed load w by a concentrated load W equal in magnitude to the area A under the load curve and passing through the centroid C of that area. The same approach can be used to determine the resultant of the hydrostatic forces exerted on a rectangular plate submerged in a liquid.

The coordinates of the center of gravity G of a three- dimensional body are determined from xW = x dW yW = y dW zW = z dW For a homogeneous body, the center of gravity G coincides with the centroid C of the volume V of the body; the coordinates of C are defined by the relations xV = x dV yV = y dV zV = z dV If the volume possesses a plane of symmetry, its centroid C will lie in that plane; if it possesses two planes of symmetry, C will be located on the line of intersection of the two planes; if it possesses three planes of symmetry which intersect at only one point, C will coincide with that point.

The volumes and centroids of various common three- dimensional shapes are tabulated. When a body can be divided into several of these shapes, the coordinates X, Y, Z of its center of gravity G can be determined from the corresponding coordinates of the centers of gravity of the various parts, by using X  W =  xW Y  W =  yW Z  W =  zW If the body is made of a homogeneous material, its center of gravity coincides with the centroid C of its volume, and we write X  V =  xV Y  V =  yV Z  V =  zV

x y z P(x,y,z)P(x,y,z) x el y el z el z x el = x y el = y z el = z 1212 dV = z dx dy dy dx When a volume is bounded by analytical surfaces, the coordinates of its centroid can be determined by integration. To avoid the computation of triple integrals, we can use elements of volume in the shape of thin filaments (as shown). Denoting by x el, y el, and z e l the coordinates of the centroid of the element dV, we write xV =  x el dV yV =  y el dV zV =  z el dV If the volume possesses two planes of symmetry, its centroid C is located on their line of intersection.

dx x z y x el x el = x dV =  r 2 dx If the volume possesses two planes of symmetry, its centroid C is located on their line of intersection. Choosing the x axis to lie along that line and dividing the volume into thin slabs parallel to the xz plane, the centroid C can be determined from xV =  x el dV For a body of revolution, these slabs are circular.