CENTROID

1. CENTROID CENTRE OF GRAVITY Centre of gravity : of a body is the point at which the whole weight of the body may be assumed to be concentrated. A body is having only one center of gravity for all positions of the body. It is represented by CG. or simply G or C. Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 1

2. CENTRE OF GRAVITY Consider a three dimensional body of any size and shape, having a mass m. If we suspend the body as shown in figure, from any point such as A, the body will be in equilibrium under the action of the tension in the cord and the resultant W of the gravitational forces acting on all particles of the body. Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 2

3. CENTRE OF GRAVITY Cord Resultant W is collinear with the Cord Assume that we mark its position by drilling a hypothetical hole of negligible size along its line of action Resultant Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 3

4. CENTRE OF GRAVITY To determine mathematically the location of the centre of gravity of any body, we apply the principle of momentsto the parallel system of gravitational forces. Centre of gravity is that point about which the summation of the first moments of the weights of the elements of the body is zero. Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 6

5. CENTRE OF GRAVITY We repeat the experiment by suspending the body from other points such as B and C, and in each instant we mark the line of action of the resultant force. For all practical purposes theselines of action will be concurrent at a single point G, which is called the centre of gravity of the body. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 4

6. B CENTRE OF GRAVITY Example: C www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 5

7. CENTRE OF GRAVITY The moment of the resultant gravitational force W, about any axis the algebraic sum of the moments about the same axis of thegravitational forces dW acting on all infinitesimal elements of the body. = if, we apply principle of moments, (Varignon’s Theorem) about y-axis, for example, The sum of moments of its components about y-axis The moment of the resultant about y-axis = www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS Where W = 7

8. CENTRE OF GRAVITY where = x- coordinate of centre of gravity Similarly, y and z coordinates of the centre of gravity are and ----(1) www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 8

9. CENTRE OF MASS , ----(1) , With the substitution of W= m g and dW = g dm (if ‘g’ is assumed constant for all particles, then ) the expression for the coordinates of centre of gravity become ----(2) , , Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 9

10. CENTRE OF MASS The density ρ of a body is mass per unit volume. Thus, the mass of a differential element of volume dV becomes dm = ρ dV . If ρ is not constant throughout the body, then we may write the expression as , and ----(3) Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 10

11. CENTRE OF MASS , , ----(2) Equation 2 is independent of g and therefore define a unique point in the body which is a function solely of the distribution of mass. This point is called the centre of mass and clearly coincides with the centre of gravity as long as the gravity field is treated as uniform and parallel. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 11

12. CENTROID , and ----(3) When the density ρ of a body is uniform throughout, it will be a constant factor in both the numerators and denominators of equation (3) and will therefore cancel. The remaining expression defines a purely geometrical property of the body. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 12

13. When speaking of an actual physical body, we use the term “centre of mass”. The term centroidis used when the calculation concerns a geometrical shape only. Calculation of centroid falls within three distinct categories, depending on whether we can model the shape of the body involved as a line, an area or a volume. Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 13

14. The centroid “C” of the line segment, LINES: for a slender rod or a wire of length L, cross-sectional area A, and density ρ, the body approximates a line segment, and dm = ρA dL. If ρ and A are constant over the length of the rod, the coordinates of the centre of mass also becomes the coordinates of the centroid, C of the line segment, which may be written as , , Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 14

15. The centroid “C” of the Area segment, AREAS: when the density ρ, is constant and the body has a small constant thickness t, the body can be modeled as a surface area. The mass of an element becomes dm = ρ t dA. If ρ and t are constant over entire area, the coordinates of the ‘centre of mass’ also becomes the coordinates of the centroid, C of the surface area and which may be written as , , www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS Contd. 15

16. The centroid “C” of the Volume segment, VOLUMES: for a general body of volume V and density ρ, the element has a mass dm = ρ dV . If the density is constant the coordinates of the centre of mass also becomes the coordinates of the centroid,C of the volume and which may be written as , , www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 16

17. Centroid of Simple figures: using method of moment ( First moment of area) • Centroid of an area may or may not lie on the area in question. • It is a unique point for a given area regardless of the choice of the origin and the orientation of the axes about which we take the moment. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 17

18. The coordinates of the centroid of the surface area about any axis can be calculated by using the equn. (A) x = (a1) x1 + (a2) x2 + (a3) x3 + ……….+(an) xn = First moment of area Algebraic Sum of moment of elemental ‘dA’ about the same axis Moment of Total area ‘A’ about y-axis = where (A = a1 + a2 + a3 + a4 + ……..+ an) www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 18

19. AXIS of SYMMETRY: It is an axis w.r.t. which for an elementary area on one side of the axis , there is a corresponding elementary area on the other side of the axis (the first moment of these elementary areas about the axis balance each other) • If an area has an axis of symmetry, then the centroid must lie on that axis. • If an area has two axes of symmetry, then the centroid must lie at the point of intersection of these axes. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS Contd. 19

20. For example: The rectangular shown in the figure has two axis of symmetry, X-X and Y-Y. Therefore intersection of these two axes gives the centroid of the rectangle. da da x x da × x = da × x Moment of areas,da about y-axis cancel each other da × x + da × x = 0 Contd. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 20

21. AXIS of SYMMETYRY ‘C’ must lie on the axis of symmetry ‘C’ must lie on the axis of symmetry ‘C’ must lie at the intersection of the axes of symmetry www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 21

22. EXERCISE PROBLEMS Problem No.1: Locate the centroid of the shaded area shown Ans: x=12.5, y=17.5 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 22

23. Ans: x=474mm, y=474mm EXERCISE PROBLEMS Problem No.2: Locate the centroid of the shaded area shown D=600 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 23

24. Ans: x=34.4, y=40.3 EXERCISE PROBLEMS Problem No.3: Locate the centroid of the shaded area w.r.t. to the axes shown www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 24

25. EXERCISE PROBLEMS Problem No.4: Locate the centroid of the shaded area w.r.t. to the axes shown 10 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS Ans: x= -5mm, y=282mm 25

26. EXERCISE PROBLEMS Problem No.5 Locate the centroid of the shaded area w.r.t. to the axes shown 30 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS Ans:x =38.94, y=31.46 26

27. EXERCISE PROBLEMS Problem No.6 Locate the centroid of the shaded area w.r.t. to the axes shown y x www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS Ans: x=0.817, y=0.24 27

28. Ans: x= -30.43, y= +9.58 EXERCISE PROBLEMS Problem No.7 Locate the centroid of the shaded area w.r.t. to the axes shown www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 28

29. Ans: x= 0, y= 67.22(about base) EXERCISE PROBLEMS Problem No.8 Locate the centroid of the shaded area. 20 www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 29

30. 2 Ans: x=5.9, y= 8.17 EXERCISE PROBLEMS Problem No.9 Locate the centroid of the shaded area w.r.t. to the base line. www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 30

31. Ans: x=21.11, y= 21.11 EXERCISE PROBLEMS Problem No.10 Locate the centroid of the shaded area w.r.t. to the axes shown www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 31

32. Ans: x= y= 22.22 EXERCISE PROBLEMS Problem No.11 Locate the centroid of the shaded area w.r.t. to the axes shown www.bookspar.com | Website for Students | VTU NOTES | QUESTION PAPERS 32