1. COMPOSITE BODIES (Section 9.3) Today’s Objective: Students will be able to determine the location of the center of gravity, center of mass, or centroid using the method of composite bodies.

2. APPLICATIONS The I - beam is commonly used in building structures. When doing a stress analysis on an I - beam, the location of the centroid is very important. How can we easily determine the location of the centroid for a given beam shape?

3. APPLICATIONS (continued) Cars, SUVs, bikes, etc., are assembled using many individual components. When designing for stability on the road, it is important to know the location of the bikes’ center of gravity (CG). If we know the weight and CG of individual components, how can we determine the location of the CG of the assembled unit?

4. CENTROIDS FOR SIMPLE LINES c c L/2

5. CENTROIDS FOR SIMPLE AREAS If an object has an axis of symmetry, then the centroid of object lies on that axis. c A= πr 2 c A= 1/2πr 2 A= 4r/3π c A= 1/4πr 2 A= 4r/3π x y y x

6. CENTROIDS FOR SIMPLE VOLUMES z y x z y x V=4/3πr 3 V=1/3 π h r 2 C C h h/4

7. CONCEPT OF A COMPOSITE BODY Many industrial objects can be considered as composite bodies made up of a series of connected “simpler” shaped parts or holes, like a rectangle, triangle, and semicircle. Knowing the location of the centroid, C, or center of gravity, G, of the simpler shaped parts, we can easily determine the location of the C or G for the more complex composite body. a be d

8. STEPS FOR ANALYSIS 1. Divide the body into pieces that are known shapes. Holes are considered as pieces with negative weight or size. 2.Make a table with the first column for segment number, the second column for lengtht, area, or volume (depending on the problem), the next set of columns for the moment arms, and, finally, several columns for recording results of simple intermediate calculations. 3.Fix the coordinate axes, determine the coordinates of the center of gravity of centroid of each piece, and then fill-in the table. 4.Sum the columns to get x, y, and z. Use formulas like x = (  x i L i ) / (  L i ) or x = (  x i A i ) / (  A i ) This approach will become clear by doing examples. 

9. EXAMPLE Given: The part shown. Find: The centroid of the part. Plan:Follow the steps for analysis. Solution: 1. This body can be divided into the following pieces rectangle (a) + triangle (b) + quarter circular (c) – semicircular area (d) a b c d

10. EXAMPLE (continued) 39.8376.528.0  27 4.5 9 - 2/3 54 31.5 – 9 0 1.5 1 4(3) / (3  ) 4(1) / (3  ) 3 7 – 4(3) / (3  ) 0 18 4.5 9  / 4 –  / 2 Rectangle Triangle Q. Circle Semi-Circle A y ( in 3 ) A x ( in 3 ) y (in) x (in) Area A (in 2 ) Segment  Steps 2 & 3: Make up and fill the table using parts a, b, c, and d. a bc d

11. x = (  x A) / (  A ) = 76.5 in 3 / 28.0 in 2 = 2.73 in y = (  y A) / (  A ) = 39.83 in 3 / 28.0 in 2 = 1.42 in   4. Now use the table data and these formulas to find the coordinates of the centroid. EXAMPLE (continued) C ·

12. Problem 9-58 6 in 3 in 1 2 3 7233314.62572  108 18 -54 108 63 162 3 2 -2 376376 36 9 27 123123 A y ( in 3 ) A x ( in 3 ) y (in) x (in) Area A (in 2 ) Segment 

13. CONCEPT QUIZ 1. Based on the typical centroid information available in handbooks, what are the minimum number of segments you will have to consider for determing the centroid of the given area? A)1 B) 2 C) 3 D) 4 2. A storage box is tilted up to clean the rug underneath the box. It is tilted up by pulling the handle C, with edge A remaining on the ground. What is the maximum angle of tilt (measured between bottom AB and the ground) possible before the box tips over? A)30°B) 45 ° C) 60 ° D) 90 ° 3cm 1 cm 3cm 30º G C A B