2.0 Position of centre of mass of continuous bodies

2.1 Rod
2.2 Semicircular ring
2.3 Semicircular disc
2.4 Solid sphere
2.5 Hemispherical Shell
2.6 Rectangular Plate
2.7 Square Plate
2.8 Circular Plate
2.9 Solid Cone
2.10 Hollow Cone
2.11 Questions
2.12 Centre of mass of a rigid complex bodies

If the bodies are not discrete and their distances are not specific, then the centre of mass can be found out by considering an infinitesimal small part of mass i.e. $dm$ at a distance $x$, $y$ & $z$ from the origin of the chosen co-ordinate system.

Mathematically, $${\overrightarrow r _{COM}} = \frac{{\int {\overrightarrow r dm} }}{{\int {dm} }}$$ Also, $${x_{COM}}\widehat i + {y_{COM}}\widehat j + {z_{COM}}\widehat k = \frac{{\int {\left( {x\widehat i + y\widehat j + z\widehat k} \right)dm} }}{{\int {dm} }}$$ or $${x_{COM}} = \frac{{\int {xdm} }}{{\int {dm} }},\quad {y_{COM}} = \frac{{\int {ydm} }}{{\int {dm} }},\quad {z_{COM}} = \frac{{\int {zdm} }}{{\int {dm} }}$$

Let us find the centre of mass of different uniform continuous bodies