Center of Gravity and Centroid

Engineering Mechanics: Statics Fourteenth Edition Chapter 9 Center of Gravity and Centroid Section 9.1 Center of Gravity, Center of Mass, and the Centroid of a Body Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Center of Gravity, Center of Mass and Centroid of a Body Today’s Objective: a. Understand the concepts of center of gravity, center of mass, and centroid. b. Be able to determine the location of these points for a body. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Concept of Center of Gravity (CG) A body is composed of an infinite number of particles, and so if the body is located within a gravitational field, then each of these particles will have a weight dW. The center of gravity (CG) is a point, often shown as G, which locates the resultant weight of a system of particles or a solid body. From the definition of a resultant force, the sum of moments due to individual particle weight about any point is the same as the moment due to the resultant weight located at G. Also, note that the sum of moments due to the individual particle’s weights about point G is equal to zero. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Concept of CG The location of the center of gravity, measured from the y axis, is determined by equating the moment of W about the y-axis to the sum of the moments of the weights of the particles about this same axis. If dW is located at point ( x,y,z ) , then _ ~ x W =  x dW _ Similarly, _ ~ ~ zW =  z dW y W =  y dW Therefore, the location of the center of gravity G with respect to the x, y, z-axes becomes  x dW ~ x=  dW  y dW ~ y=  dW z=  z dW  dW Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved CM & Centroid of a Body x dW  x=  dW ~ y dW  y=  dW ~ z dW  z=  dW By replacing the W with a m in these equations, the coordinates of the center of mass can be found.  x dm ~ x=  dm  y dm ~ y=  dm z dm  z=  dm Similarly, the coordinates of the centroid of volume, area, or length can be obtained by replacing W by V, A, or L, respectively. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Concept of Centroid The centroid, C, is a point defining the geometric center of an object. The centroid coincides with the center of mass or the center of gravity only if the material of the body is homogenous (density or specific weight is constant throughout the body). If an object has an axis of symmetry, then the centroid of object lies on that axis. In some cases, the centroid may not be located on the object. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Steps to Determe the Centroid of an Area 1. Choose an appropriate differential element dA at a general point (x, y). Hint: Generally, if y is easily expressed in terms of x (e.g.,y = x 2 + 1), use a vertical rectangular element. If the converse is true, then use a horizontal rectangular element. 2. Express d A in terms of the differentiating element dx (or dy). ~ ~ 3. Determine coordinates ( x, y ) of the centroid of the rectangular element in terms of the general point (x, y). 4. Express all the variables and integral limits in the formula using either x or y depending on whether the differential element is in terms of dx or dy, respectively, and integrate. Note: Similar steps are used for determining the CG or CM. These steps will become clearer by doing a few examples. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example I (1 of 2) Given: The area as shown. Find: The centroid location (x,y) Plan: Follow the steps. Solution: 1. Since y is given in terms of x, choose dA as a vertical rectangular strip. 2. dA = = x 3 dx y dx ~ 3. x=x y x3 y= = 2 2 ~ and Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example I (2 of 2) 4. ( x =  A x dA ( A dA ) ) 1 ( ) 3 x x dx  = 0 1 (x )d x 3 0 1 5 1  x  0 = 5 1 4 1 x  4  0  1 5 =   =0.8 m  1 4    X3  3 1 7 1  x  A y dA 0  2  X dx 14 0 y= = = 1 1 3 A dA X dx  4 1 ( ) 0  1  14  =   =0.2857 m  1 4   Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example II Given: The shape and associated horizontal rectangular strip shown. Find: dA and ( x,y ) Plan: Follow the steps. Solution: 1. dA = x dy = y2dy 2. 1 - x ) (1 ( x=x+ = ~ 2 3. + x) 2 1+ y ) ( = 2 2 % y= y Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example III Locate the centroid y of the area Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Locate the centroid y of the area Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example IV Determine the area and the centroid x of the parabolic area. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Reading Quiz (1 of 2) 1. The Fill in the blank is the point defining the geometric center of an object. A. Center of gravity B. Center of mass C. Centroid D. None of the above Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Attention Quiz (1 of 2) 1. If a vertical rectangular strip is chosen as the differential element, then all the variables, including the integral limit, should be in terms of Fill in the blank . A. x B. y C. z D. Any of the above. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Attention Quiz (2 of 2) 2. If a vertical rectangular strip is chosen, then what are the values of % x and % y? A. (x, y) B. (x / 2 , y / 2) C. (x, 0) D. (x, y / 2) Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved