CENTROID OF A STRAIGHT & CURVED LINE:

CENTROID OF A STRAIGHT LINE:

(a) Suppose we have straight lines of length (L) in a coordinate system.  Let the lines are named as AB, BC and CA as shown in the figure. The centroids of these lines will at their mid-points viz. G₁, G₂, G₃.  The lines may be vertical, horizontal or inclined.

(b) For a curved line, a centroid G(X_g,Y_g)can be defined by the equations,

X_g = (1/L)∫x.dL   ------ (i)

Where dL = elemental length and L= total length of the line.

Y_g = (1/L)∫y.dL   ------ (ii)

Suppose we have a quarter circular arc in a co-ordinate system as shown in the figure. Total length of the arc AB = (πR)/2 . We take an arbitrarily small length of the arc CD and denote it as dL.

So, dL = Rdθ  ------ (iii)    [ as s=Rθ ]

Where R = Radius of the quarter circular arc. Let the co-ordinate of the point D be D(x,y) where   
x = Rcosθ -----(iv) and y = Rsinθ -----(v)

Hence   X_g = (1/L)∫x.dL  ;  

here L = (πR)/2

        x = Rcosθ 

     dL = Rdθ

     X_g = (2/πR)   ₀∫^π/2Rcosθ.Rdθ  =>  X_g = (2/πR) R²  ₀∫^π/2cosθ.dθ = 2R/π
      _Yg = (2/πR)   ₀∫^π/2Rsinθ.Rdθ  =>  _Yg = (2/πR) R²  ₀∫^π/2sinθ.dθ = 2R/π


                                                                           

                                                                                      
Hence, for a quarter circular arc of radius R will be G(2R/π,2R/π)


CENTROID OF A LINE CONSISTS OF SEVERAL SIMPLE LINES :