Locate the centroid of the plane area shown.Fig. P5.73

Divide the plane area into three areas I, II and III respectively as shown in the figure.

Tabulate the areas and co-ordinates of the centroid of each component as shown below:

Calculate the x coordinate of the centroid by using the equation:

Here, is the x coordinate of the centroid, is the algebraic sum of the moments of the areas of the individual components about the yz plane and is the algebraic sum of the areas of the individual components.

Substitute for and for .

Calculate the \(y\) coordinate of the centroid by using the equation:

$$ \bar{Y}=\frac{\sum \bar{y} A}{\sum A} $$

Here, \(\bar{Y}\) is the \(y\) coordinate of the centroid and \(\sum \bar{y} A\) is the algebraic sum of the moments of the areas of the individual components about the \(x z\) plane.

Substitute \(277020 \mathrm{~mm}^{3}\) for \(\sum \bar{y} A\) and \(10422 \mathrm{~mm}^{2}\) for \(\sum A\).

$$ \begin{aligned} \bar{Y} &=\frac{277020}{10422} \\ &=26.6 \mathrm{~mm} \end{aligned} $$

Therefore, the centroid of the plane area is \((\bar{X}, \bar{Y})=(19.27 \mathrm{~mm}, 26.6 \mathrm{~mm})\).