Module 7. Centre and Centroids of Gravity.pdf
- 1. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 1 Chapter 7: CENTROIDS AND CENTER OF GRAVITY CHAPTER OBJECTIVES 1.Discuss the concept of the center of gravity, and the centroid. 2. Determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape. 7.1.INTRODUCTION: CENTER OF GRAVITY---- is the point through which the whole weight of the body acts. a body is having only one center of gravity for all the positions of the body. it is represented by c.g or simply g. CENTROIDS---- the point which the total area of a plane figure(like rectangle, square, triangle, quadrilateral, circle) is assumed to be concentrated. The centroid is also c.g. or simply g. 7.2. CENTROIDS OF COMPOSITES FIGURES Center of Gravity Flat plate Wx=∑ 𝒘𝒙 Wy=∑ 𝒘𝒚 Centroids of Areas Ax=∑ 𝒂𝒙 Ay=∑ 𝒂𝒚 Centroids Of Lines: Lx=∑ 𝒍𝒙 Ly=∑ 𝒍𝒚
- 2. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 2 Centroids for Common Geometric Shapes Shape Area or Length x y Rectangle A=bd ( 1 2 )b ( 1 2 )d Triangle A= 1 2 bh 0 ( 1 3 ) h Circle A=Πr2 0 0 Semi-Circle A=( 1 2 )Πr2 0 4𝑟 3𝜋 Semicircular Arc L=Πr 2𝑟 𝜋 0
- 3. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 3 Quarter Circle A=( 1 4 )Πr2 4𝑟 3𝜋 4𝑟 3𝜋 Sector of Circle A=r2θrad 2𝑟𝑆𝑖𝑛𝜃 3𝜃𝑟𝑎𝑑 0 Circular Arc L=2rθrad 𝑟𝑠𝑖𝑛𝜃 𝜃𝑟𝑎𝑑 0 Quarter Ellipse A= 1 4 Πab 4𝑎 3𝜋 4𝑏 3𝜋
- 4. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 4 Parabolic Segment A= 2 3 bh 3 8 𝑏 2 5 ℎ Ellipse A=Πab 0 0 Half ellipse A= 1 2 Πab 0 4𝑏 3𝜋 Spandrel 𝐴 = 1 𝑛 + 1 𝑏ℎ 1 𝑛 + 2 𝑏 𝐴 = 𝑛 + 1 4𝑛 + 2 ℎ
- 5. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 5 7.3.Properties Of Angles and Channels Size Area Ẍ Ӯ units Sq.in. Web thickness 5x3x1/2 angle 6x4x1 angle 6x6x1/2angle 8x6x1 angle 3.75 in 9.00 in 5.75 in 13.00 in 1.75 2.17 1.68 2.65 0.75 1.17 1.68 1.65 10” -15.3 lb channel 12’’- 20.7 lb channel 4.47 6.03 0.64 0.70 0 0 0.24 0.28 Illustrative Example: A slender homogeneous wire of uniform cross section is bent into the form shown in figure. Determine the position of the centroid of the wire with respect to the given axes. Solution: The wire is considered equivalent to two line segments, one semicircleof 4 cm radius and the other a straight line 8 cm long.
- 6. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 6 Lx=∑ 𝒍𝒙 (𝟒𝝅 + 𝟖)𝒙 = (𝟒𝝅)(− 𝟐𝒙𝟒 𝝅 )+(8)(4) x = 0 Ly=∑ 𝒍𝒚 (𝟒𝝅 + 𝟖)𝒚 = (𝟒𝝅)𝒙𝟎 + (𝟖)(-4) 20.56y = -32 y = -1.558 cm The centroid of the wire Line on the Y-axis is 1.558 cm below the x-axis. Illustrative Example 7.2 Determine the position of the centroid of the shaded area shown in figure. Solution: As indicated by the dashed lines, the net shaded area may be considered as the sum of a quarter circle of 5 in radius, a rectangle 5 cm by 4 cm radius and a rectangle 6 cm by 9 cm, minus a triangle 7.5cm by 9 cm
- 7. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 7 Computation Of results Area(cm2) X (cm) Ax(cm3) y(cm) Ay(cm3) Quarter Circle +19.63 +2.12 +41.62 +2.12 +41.62 Rectangle 5x4 +20.00 +2.5 +50.0 -2 -40.0 Rectangle 6x9 +54.00 -3.00 -162.0 +0.5 +27.0 Triangle -33.75 -3.5 +118.12 -1 +33.75 Total +59.88 +47.745 +62.37 Taking the sums from the tabulation, gives the following results: x= ∑ 𝑎𝑥 𝐴 x= 47.74 59.88 x=0.80 cm y= ∑ 𝑎𝑦 𝐴 y= 62.37 59.88 y=1.04cm
- 8. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 8 EXERCISE 7: Problem7.1.Determine the coordinates of the centroid of the area shown in fig. with respect to the given axes. Problem7.2. Locate the centroid of the shaded area shown in Fig.
- 9. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 9 Problem7.3. Locate the centroid of the bent wire shown in Fig. the wire is homogeneous and of uniform cross- section. Problem7.4. Find the coordinates of the centroid of the shaded area as shown.
- 10. ES11 – STATICS OF RIGID BODIES | PREPARED BY: ENGR. RUTH ANN D. MANINGDING 10 References: Barry Onouye, Kevin Kane, C 2012, Statics and Strength of Materials for Architecture and Building Construction 4th edition, Pearson Education, Inc., Prentice Hall, One Lake Street ,Upper Saddle River, New Jersey 07458 R. C. HIBBELER,C 2013, Engineering Mechanics Statics 13th edition, Pearson Education, Inc. Pearson Prentice Hall Upper Saddle River, New Jersey 07458 https://mathalino.com/ J.L. Meriam & L.G. Kraige, C 2002, Engineering Mechanics Volume 1 Statics Fifth Edition, John Wiley & Sons, Inc.,New York Singer,Ferdinand L,C 1954,Engineering Mechanics, 2nd edition,Harper & Row, New York, Evanston & London