Question

The centroid of a composite plane figure is found by

• A. Dividing the sum of the areas by the sum of their centroids
• B. Dividing the sum of the moments of the areas about an axis by the total area
• C. Adding the centroids of individual figures
• D. Multiplying the total area by the sum of centroids

Hint:

Centroid of Composite Area. \(\bar{x = (\sum x_i A_i) / (\sum A_i)\) and \(\bar{y = (\sum y_i A_i) / (\sum A_i)\). This is equivalent to dividing the sum of the first moments of area by the total area.

Answer 1

The Correct Option is B

The centroid of a composite area (an area made up of several simpler shapes) is the geometric center.
Its coordinates (\(\bar{x}, \bar{y}\)) are found by considering the composite area as a sum of simpler areas (\(A_i\)) with known centroids (\(x_i, y_i\)).
The principle is based on the first moment of area.
The x-coordinate of the centroid (\(\bar{x}\)) is found by summing the first moments of each individual area about the y-axis (\(x_i A_i\)) and dividing by the total area (\(A_{total} = \sum A_i\)).
Similarly, the y-coordinate (\(\bar{y}\)) is found by summing the moments about the x-axis (\(y_i A_i\)) and dividing by the total area.
$$ \bar{x} = \frac{\sum x_i A_i}{\sum A_i} = \frac{\text{Sum of moments of area about y-axis}}{\text{Total Area}} $$ $$ \bar{y} = \frac{\sum y_i A_i}{\sum A_i} = \frac{\text{Sum of moments of area about x-axis}}{\text{Total Area}} $$ Option (2) correctly describes this principle.