Engineering Mechanics (Statics) (Centroid) Dr. Hayder A. Mehdi
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1 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Centroid of an Area: If an area lies in te x plane and is ounded te curve = f (x), as sown in te following figure ten its centroid will e in tis plane and can e determined from integrals equation. Centroid of a Line: If a line segment (or rod) lies witin te x plane and it can e descried a tin curve = f (x), as sown in te following figure, ten its centroid is determined as elow. 1 Page of 25 Pages
2 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Ten, dl Or 2 Page of 25 Pages
3 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Example (01): Locate te centroid of te rod ent into te sape of a paraolic arc as sown in te following figure. Solution: 1) Te differential element is sown in aove figure. It is located on te curve at te aritrar point ( x, ). 2) Te differential element of lengt dl can e expressed in terms of te 3) differentials dx and d using te Ptagorean teorem. x 2 dx d 4) 2 5) 3 Page of 25 Pages
4 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Example (02): Solution: Polar coordinates will e used to solve tis prolem since te arc is circular. A differential circular arc is selected as sown in te figure. Tis element lies on te curve at ( R, ). 4 Page of 25 Pages
5 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Example (03): Determine te distance measured from te x axis to te centroid of te areaoftetrianglesownintefollowingfigure Consider a rectangular element aving a tickness d, and located in an aritrar position so tat it intersects te oundar at ( x, ), as sown in te aove figure. Te area of te element is da = x d 5 Page of 25 Pages
6 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi 6 Page of 25 Pages x x x x d d d d da da d da x d da x x x A A ) (. Example (04): Locate te centroid for te area of a quarter circle sown in te following figure.
7 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Differential Element. Polar coordinates will e used, since te oundar is circular. We coose te element in te sape of a triangle, as sown in te aove figure. (Actuall te sape is a circular sector; owever, neglecting iger-order differentials, te element ecomes triangular.) Te element intersects te curve at point ( R, ). Area and Moment Arms. Te area of te element is te centroid of te (triangular) element is located at 7 Page of 25 Pages
8 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Example (05): Locate te centroid of te area sown in te following figure. Differential Element. A differential element of tickness dx is sown in aove figure. Te element intersects te curve at te aritrar point ( x, ), and so it as a eigt. Area and Moment Arms. Te area of te element is da = dx, centroid is located at x ~ x ~ 2 8 Page of 25 Pages
9 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Second solution using orizontal strip Differential Element. Te differential element of tickness d is sown in te aove figure. Te element intersects te curve at te aritrar point ( x, ), and so it as a lengt (1 - x). Area and Moment Arms. Te area of te element is da = (1 - x) d, and its centroid is located at In te following te area and centroid of tese regular area are given as flows. 9 Page of 25 Pages
10 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi 10 Page of 25 Pages
11 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi 11 Page of 25 Pages
12 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (01): 12 Page of 25 Pages
13 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (02): 13 Page of 25 Pages
14 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (03): 14 Page of 25 Pages
15 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (04): x A xda ~ A da L x x. a sin dx L 0 L L 0 x a sin dx L 2 15 Page of 25 Pages
16 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (05): 16 Page of 25 Pages
17 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (06): 17 Page of 25 Pages
18 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (07): From prolem (06) Find te centroid of te saded area A ~. da A da 18 Page of 25 Pages
19 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Prolem (08): Determine te coordinates of te centroid of te area tat lies etween te straigt line x =2/3 and te paraola x2 = 4, were x and are measured in inces [see Fig. (a)]. Use te following metods: (1) Using a orizontal differential area element; and (2) Using a vertical differential area element. Solution: Part 1: Horizontal Differential Area Element For te area we ave x` x` 19 Page of 25 Pages
20 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Part 2: Vertical Differential Area Element Te area of te region is ` ` 20 Page of 25 Pages
21 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Centroid of Composite Area: Tecentroidcanefoundusingsummationofmomentofareasdivided summation of areas Example (06): Locate te centroid of te plate area sownintefollowingfigure. Te first step is divided te aove composite area to te asic area as follows First Division Or Second Division Area 3 is sutraction from Area 2 21 Page of 25 Pages
22 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Use an one from aove to fined te centroid we are use ot divisions Division (1) Segment Area (A) x ~ ~ x ~.A ~.A summation Division (2) Segment Area (A) x ~ ~ x ~.A ~.A summation Page of 25 Pages
23 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Example 7: Using te metod of composite curves, determine te centroidal coordinates of te line in Fig. (a) tat consists of te circular arc 1, and te straigt lines 2 and 3. Solution: It is convenient to organize te analsis in taular form, as follows: Segement Lengt (in) x` (in) Lx` (in 2 ) `(in) L` (in 2 ) Terefore, te coordinates of te centroid of te composite curve are: xl ~ x in L L ~ in L Page of 25 Pages
24 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Example 8: Using te metod of composite areas, determine te location of te centroid of te saded area sown in Fig. (a). Solution: Sape Area (mm 2 ) x` (mm) Ax` (mm 3 ) ` (mm) A` (mm 3 ) 1(Rectangle) (Semicircle) (Triangle) Page of 25 Pages
25 Engineering Mecanics (Statics) (Centroid) Dr. Hader A. Medi Te centroid of te composite area are: x A. x ~ A A. ~ A x x x x mm 308mm 25 Page of 25 Pages
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