For an extended body with uniform density, the centre of mass no longer depends on how discrete masses are distributed, but only on the geometric shape. In this case the CM coincides with the geometric centroid: the centre of the figure.
The practical rule exploits the symmetries of the figure:
- for a circle or a rectangle, the centroid is the geometric centre;
- for a triangle, it is the intersection of the medians (one third of the height from the base);
- more generally, every axis of symmetry passes through the centroid.
This greatly simplifies the calculation: instead of summing infinitely many mass contributions, it is enough to recognise the symmetry. For bodies made up of regular parts (a rectangle with a semicircle, for example), the centroid of each part is found and then a weighted average of the centroids is taken using their respective masses — exactly as for point masses.
Note
Uniform density is essential: if the material is denser on one side, the centre of mass shifts towards that part, even if the shape remains symmetric. The geometric centroid is the CM only when mass and volume are distributed proportionally.
Links
Topics: Centro di massa Concepts: Centro di massa Skills: Calcolo del centro di massa
Related exercises: Esercizio svolto — Il gatto sul carrello · Razzo che esplode a metà traiettoria · Disco con foro tangente