Find the centre of mass of composite laminae

Centre of Mass of Composite Laminae

A composite lamina refers to a body that is made up of multiple simple laminar shapes (for example, rectangles, triangles, circles), each with their unique centres of mass.
The centre of mass of a composite lamina can be found by considering each individual shape as a particle with mass equal to its area (since they have uniform density).
To locate the centre of mass of the composite object, each individual shape should be taken into account both in terms of its mass (area) and its position (centre of mass).

First, for each separate shape identify its centre of mass and its area. Remember that for rectangles, triangles and circles, the centre of mass lies at the geometric centre.
Then, for each shape, multiply its area by the x-coordinate and the y-coordinate of its centre of mass respectively. This will give you what’s known as “moments” about the x-axis and y-axis.
Sum all the moments about the x-axis, and also sum all the moments about the y-axis, for all the shapes. At the same time, calculate the sum of all the areas (masses) of the shapes.
The x-coordinate of the centre of mass of the composite lamina is then found by dividing the total moments about the y-axis by the total area. Conversely, the y-coordinate of the centre of mass is found by dividing the total moments about the x-axis by the total area.
The position of the centre of mass of the composite lamina is then at the coordinates (X,Y), with X and Y computed as above.

It is helpful to draw the composite body on a coordinate grid, to better visualise the centres of mass of each shape.
Different shapes may overlap. If so, consider the overlapped area as a separate “shape”, with negative mass.
The same principles apply with composite bodies in three dimensions, although the calculations are more complex as they also take into account moments about the z-axis.
Always work systematically, and recheck calculations to ensure accuracy.