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CENTROID OF AN AREA

Engineering Mechanics EME-102




Geometrical Center of an area (A) is often termed as Centroid or Center of an Area.

Suppose we have an area A in a certain X-Y coordinate system, we divide the area into n parts and named them as A1, A2, A3 .... An. Let the coordinates of those tiny elemental areas are as (X1,Y1), (X2,Y2), (X3,Y3) ..... (Xn,Yn).

As area can be represented by a vector, hence, Area A can be treated as the resultant of the tiny elemental vectors A1, A2, A3 .... An. Let the direction of the resultant vector passes through the point G(Xg,Yg) on the plane of the area. The point G(Xg,Yg) is called the CENTROID of the area A. (The direction of any area is along the perpendicular to the area drawn at the centroid of the area).

Like other vectors, an area has a moment about an axis and be represented by the product of the radial distance between the area and the axis and the area itself. So if an elementary area A1 has a coordinate (X1,Y1) it means the area is at a distance X1 from the Y axis and Y1 from the X axis. Therefore the moment produced by A1 about Y axis is X1A1 and about X axis is Y1A1.

Therefore the summation of all the moments produced by each and every elemental areas about Y axis will be AiXi and about X axis will be AiYi.

Again, the resultant area A passes through the point G(Xg,Yg). Therefore the moment produced by the area A about Y axis will be AXg and about X axis will be AYg.

Like other vectors, it will obey the Moment Theorem which states the total moment produced by individual vectors will be exactly equal to the moment produced by the resultant vector about a certain axis.

Therefore,
AXg = A1X1 + A2X2 + A3X3 + ...... + AnXn

and
AYg = A1Y1 + A2Y2 + A3Y3 + ....... + AnYn

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


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