Find the centre of mass of laminae and solids

Centre of Mass on a Two-Dimensional Lamina

A lamina is a thin, flat object, such as a sheet of paper.
The centre of mass (or centroid) of a lamina is the point at which it balances perfectly.
To find the centre of mass of a simple regular shape (like a square, rectangle, triangle, or circle), it is sufficient to find the geometric centre.
For composite or irregular shaped laminae, the centroid can be found by splitting the shape into simpler “component” shapes, finding the centre of mass of each, and then combining these according to their areas.
If the lamina is split into n parts, the x-coordinate of the centre of mass (C.O.M) is calculated by: C.O.M (x) = ( ∑[Areas × their x coordinates] ) / Total area
And the y-coordinate of the centre of mass is calculated similarly: C.O.M (y) = ( ∑[Areas × their y coordinates] ) / Total area

Centre of Mass of Solids

The principles of finding the centre of mass in laminae extend naturally into three dimensions. We now consider solids of uniform density.
For simple regular solids (like a cube, sphere, cylinder etc.), the centre of mass is the geometric centre.
Just like laminae, irregular solids or composite solids can be split into simpler component solids, and the centroid can be found by looking at individual centres of mass of each of these components. The volumes of the components are taken into account.
If a solid volume is divided into n parts, the x, y, z coordinates for the centre of mass is: C.O.M (x) = ( ∑[Volumes × their x coordinates]) / Total volume C.O.M (y) = ( ∑[Volumes × their y coordinates]) / Total volume C.O.M (z) = ( ∑[Volumes × their z coordinates]) / Total volume

Centre of Mass on a Wire or Rod

A wire or rod is a one-dimensional object, the centre of mass can also be calculated using similar principles as the laminae and volumes.
Simple shapes like a straight wire or uniform density rod have their center of mass at the geometric centre.
Complex or composite structures must be divided into simpler components and their centres of mass found separately.
If a rod is split into n parts, the x, y coordinates for the centre of mass is: C.O.M (x) = ( ∑[lengths × their x coordinates]) / Total length C.O.M (y) = ( ∑[lengths × their y coordinates]) / Total length
Keep in mind that these equations assume the real-world effects such as wind resistance or bending flexibility are negligible for our model wire/rod.

Always remember to draw a diagram first if the situation is complex or has many parts. It helps to visualize the different components and their respective centres of mass. Also keep in mind real-world applications often have safety factors, so these calculations are idealized versions used for understanding the theoretical principles.