EVERYDAY ENGINEERING EXAMPLES FOR SIMPLE CONCEPTS ENGR 3340 - Fundamentals of Statics and Dynamics Dr. Nedim Vardar Copyright © 2015 Centroids of Composite Bodies MSEIP – Engineering Everyday Engineering Examples Centroids of Composite Bodies Engage: Students will determine the first moments with respect to the x and y axes and the location of the centroid. Place students into small groups in an efficient way in order to maximize the effectiveness of the activity. Students are expected to determine the Center of Gravity of irregular objects similar to the ones given in Figure 1. Therefore, provide cardboards of different shapes to each group. They are also expected to observe critically the intersection of the plumb-lines which Figure 1. Irregular shapes for Centroids Calculation. Page 1 indicate the center of gravity of the irregular object. Explore: Students are firstly given cardboards of different combined shapes such as the one in Figure 2. They are required to calculate the position of the Centroids for the shapes using a plumb line as well as the theory they learned in the lecture. Once the students use the simple pin and plumb line method then they use the theory to locate the actual position of the Centroids for each of the combined shapes. Page Figure 2. Measuring Center of gravity with plumb line. 2 The centroid of a composite area, such as the one given in Figure 2 may be determined, experimentally, by using a plumb line and a pin to find the center of mass of a thin object of uniform density having the same shape. When an object is suspended so that it can move freely, its center of gravity is always directly below the point of suspension. The object is held by the pin inserted at a point near the object’s perimeter, in such a way that it can freely rotate around the pin; and the plumb line is dropped from the pin. The position of the plumb line is traced on the object. The experiment is repeated with the pin inserted at a different point of the object. The intersection of two or more vertical lines from the plumb line is the center of gravity for the object. Explain: The centroid of an area is analogous to the center of gravity of a body. The concept of the first moment of an area is used to locate the centroid. The Centroid of an object constructed from a combination of simple shapes can be found by summing the first Moment of Area of the shapes and dividing this sum by the sum of the areas of the individual shapes. Students are required to calculate the position of the Centroids for the shapes using the center of gravity theory given below. a) Centroid of an area Figure 3. Centroid of an area x W x dW x At x t dA x A x dA Q y first moment with respect to x Page yA y dA Qx 3 first moment with respect to y b) Composite area The centroid of a plane can be computed by dividing it into a finite number of simpler common shape of areas such as a triangle, rectangle, circle and semicircle(Figure 4). Figure 4. Centroids of Common Shapes The first moments of each area with respect to the axes should then be calculated. Following equations can be used to compute the coordinates of the area centroid by dividing the first moments by the total area. The centroid of each common area are summarized in Table 1. XA xA Page 4 Y A yA Page 5 Table 1. Centroids of Common Shape of Areas Elaborate: For many purposes in statics, it is necessary or useful to be able to consider a physical object as being a mass concentrated at a single point, its geometric center, also called its centroid. The centroid is essentially the average of all the points in the object. The center of mass or centroid of a region is the point in which the region will be perfectly balanced horizontally if suspended from that point. A composite body consists of a series of connected “simpler” shaped bodies, which may be rectangular, triangular, semicircular, etc. Such a body can often be sectioned or divided into its composite parts and if the weight and location of the center of gravity of each of these parts are known, it is possible to determine the center of gravity of the whole body from this information. The procedure to find the center of gravity of a composite body requires treating each of the different components of the body as a particle, and then the application of the equations given below; xA A yA Y A X When the body has constant density or specific weight, the center of gravity coincides with the centroid of the body. What did you learn? The centroid of a planar shape is the point on which it would balance when placed on a needle. The centroid of a solid is the point on which the solid would Page 6 "balance." Evaluate: Invite students to attempt the following problem: Example 1: For the plane area shown, determine the first moments with respect to the x and y axes and the location of the centroid. Calculate the first moments of each area with respect to the axes. • Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout. • Compute the coordinates of the area centroid by dividing the first moments by the total area. Page • 7 Calculus: • Divide the area into a triangle, rectangle, and semicircle with a circular cutout. Compute the coordinates of the area centroid by dividing the first moments by the total area. x A 757.7 10 mm X A 13.82810 mm 3 3 2 3 2 Y 36 .6 mm 8 3 X 54.8 mm Page y A 506.2 10 mm Y A 13.82810 mm 3 3

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