Centres of Mass

• Particles: x̄ = Σmx / Σm and ȳ = Σmy / Σm.
• Composite laminas: treat each simple shape as a particle at its own centre; subtract removed parts.
• A freely suspended body hangs with its centre of mass vertically below the point of suspension.

Weighted averages

Tabulate mass, x and y for every particle, then take the weighted averages.

Composite shapes

For a uniform lamina, mass is proportional to area. Standard results (triangle at the mean of its vertices, sector, arc) are in the formula book. A hole is a negative mass.

Hanging and tilting

Suspended from a point P, the line PG is vertical. The angle a given edge makes with the vertical comes from the right-angled triangle formed by G and that edge. A body on an inclined plane is about to topple when the line of action of its weight passes through the lowest corner.