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STEPS TO DERIVE CENTROIDS

centroids of 1D, 2D and 3D



CENTROIDS OF LINES (1D)

STEPS TO DERIVE THE CENTROID OF A STRAIGHT LINE.

(i) Draw the line in a coordinate axis from origin (O).

(ii) If the length of the line be (L), and the angle made by the line with X axis be (θ), and if G(Xg,Yg) be the centroid, then

Xg = (L/2)* cos θ and
Yg = (L/2)* sin θ

STEPS TO DERIVE THE CENTROID OF CURVED LINE.

(i) Draw the curve in a coordinate system. Take an elemental angle of dθ at an angle θ from positive axis. Let the small angle dθ makes an arc of (dL). If the radial distance of (dL) is R, then we can write
dL = Rdθ

(ii) If the coordinate of (dL) be (X,Y) then we can write

X = R cosθ and Y = R sinθ

(iii) If G(Xg,Yg) be the Centroid of the curve, then

Xg = (1/L).∫(XdL)
= (1/L).∫(R cosθ.Rdθ)

Yg = (1/L).∫(YdL)
= (1/L).∫(R sinθ.Rdθ)

STEPS TO FIND CENTROIDS OF COMPOSITE LINES:

(a composite line consists of several straight or curved lines.)

(i) Draw the curve in a coordinate system. Draw the dimensions too.

(ii) Divide the composite line into several parts of straight or curved lines. Lebel them as part-1, part-2, part-3, .......part-n. Let the corresponding lengths are L1, L2, L3, .... Ln. Let the centroids are G1(X1,Y1), G2(X2,Y2), G3(X3,Y3), ...... Gn(Xn,Yn).

(iii) Let the centroid of the composite line be G(Xg,Yg). Hence,
Xg = (L1X1 + L2X2 + ... +LnXn)/(L1 + L2 + .... + Ln)

Yg = (L1Y1 + L2Y2 + .... + LnYn)/(L1 + L2 + ..... + Ln)


HOW TO DERIVE THE VALUES OF Xg & Yg FOR BASIC GEOMETRIC FIGURE:

TO FIND Xg


i) Draw the figure in a Coordinate System.

ii) Draw a thin strip of area of thickness (dX) parallel to Y axis and at a distance (X) from Y axis.

iii) Find the height of the strip. Either the height will be constant or the height is a function of (X), that can be calculated from the equation of the figure.

iv) Calculate the elemental area of the strip, and named as dA. Hence, dA = hdX

v) integrate the expression XdA, but dA = hdX. Therefore, we shall integrate hXdX over the total area.

vi) Xg = (1/A)∫XdA


STEPS TO FIND Yg

i) Draw the figure in a Coordinate System.

ii) Draw a thin strip of area of thickness (dY) parallel to X axis and at a distance (Y) from X axis.

iii) Find the length of the strip (z). Either the length will be constant or the height is a function of (Y), that can be calculated from the equation of the figure.

iv) Calculate the elemental area of the strip, and named as dA. Hence, dA = zdY

v) integrate the expression YdA, but dA = zdY. Therefore, we shall integrate zYdY over the total area.

vi) Yg = (1/A)∫YdA


HOW TO FIND THE CENTROID OF A COMPOSITE AREA

(a composite area consists of several straight or curved lines.)

(i) Draw the figure in a coordinate system. Draw the dimensions too. Every dimensions will be measured with respect to origin of the coordinate system

(ii) Divide the composite area into several parts of basic geometric areas. Lebel them as part-1, part-2, part-3, .......part-n. Let the corresponding areas are A1, A2, A3, .... An. Let the centroids are G1(X1,Y1), G2(X2,Y2), G3(X3,Y3), ...... Gn(Xn,Yn).

(iii) Let the centroid of the composite area be G(Xg,Yg). Hence,
Xg = (A1X1 + A2X2 + ... +AnXn)/(A1 + A2 + .... + An)

Yg = (A1Y1 + A2Y2 + .... + AnYn)/(A1 + A2 + ..... + An)



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