In one line
Split the figure into standard shapes, find each shape's area and centroid, then take the area-weighted average of the centroids.
Explained simply
Finding the centroid of an odd-shaped plate is like finding the average marks of a class split into groups: each group's average is weighted by how many students are in it. Here each shape's centroid is weighted by its area.
Technically, \( \bar{x} = \Sigma A_i x_i / \Sigma A_i \) and \( \bar{y} = \Sigma A_i y_i / \Sigma A_i \), which follows from the first moment of the whole area equalling the sum of the first moments of its parts.
How it works
- Choose x and y reference axes, usually along the bottom and left edges, so all coordinates are positive.
- Use symmetry: if the figure is symmetric about a line, \( \bar{x} \) or \( \bar{y} \) is known directly.
- Divide the figure into rectangles, triangles, semicircles, quarter circles and circles.
- Make a table: part, \( A_i \), \( x_i \), \( y_i \), \( A_i x_i \), \( A_i y_i \).
- Sum the columns and divide.
- State the answer as coordinates from your chosen axes and mark G on the figure.
Key formula
\( \bar{x} = \dfrac{A_1x_1 + A_2x_2 + \cdots}{A_1 + A_2 + \cdots} \), \( \bar{y} = \dfrac{A_1y_1 + A_2y_2 + \cdots}{A_1 + A_2 + \cdots} \). Areas in mm\(^2\), coordinates in mm.
Quick example
An L-section: bottom flange 100 × 20 mm (A = 2000, centroid (50, 10)) and vertical leg 20 × 80 mm on top of the left end (A = 1600, centroid (10, 60)). \( \bar{x} = (2000 \times 50 + 1600 \times 10)/3600 = 32.22 \) mm. \( \bar{y} = (2000 \times 10 + 1600 \times 60)/3600 = 32.22 \) mm. G is at (32.22, 32.22) mm from the bottom-left corner.
Common mistakes
- Overlapping parts so a region is counted twice (split the L into non-overlapping rectangles).
- Taking a part's centroid from its own edge and forgetting to shift it to the common axes.
- Not stating the reference axes, so the answer cannot be checked.
How MU asks it
This is the main Module 2 question, typically 5 to 8 marks: 'Find the centroid of the shaded area shown'. Full marks: labelled sketch with axes and part numbers, the tabular calculation, \( \bar{x} \) and \( \bar{y} \) with units, and G marked. Keep 2 decimals; use \( \pi \) consistently.