Here is the center of mass formula list for different shapes. But, before we look at it, let’s examine the significance of center of mass.

So, center of mass of a group or system of particles is that (geometrical) point which behave as if

$\bullet$ all the mass of the system is located there &

$\bullet$ all the external forces are acting there

The above definition of center of mass is a result of the way, the position coordinates of center of mass of a group or system of particles is defined

$\overrightarrow{r}_{CM} = \cfrac{\sum_i m_i \overrightarrow{r}_i}{\sum_i m_i}$

Now, before we go any further, let’s quickly examine the difference between the terms center of mass and center of gravity.

For a rigid body, the gravitational force $\overrightarrow{F}_g$ effectively acts at a point, called the center of gravity. What this means is that, if the gravitational forces on the individual particles of the rigid body were somehow turned OFF and force $\overrightarrow{F}_g$ were to be turned ON at center of gravity, the net force and net torque (about any point) on the rigid body due to gravitational pull would not change and if the gravitational acceleration $g$ is same for all particles of the body (which will be true for any rigid body that we would encounter in our day to day lives), then the center of mass and center of gravity would coincide. If this didn’t make sense and you would like to understand more, sign up for free and go through the lecture video ‘Center of mass vs center of gravity’

With that, let’s review the center of mass of a two particle system, of a system of three particles, of a group of simple rigid bodies (for example a uniform $I$ shaped lamina), of a non-uniform rod of length $l$, of an arc, of a uniform circular arc, of a semicircular ring, of a semicircular disc, of an annular semicircular disc, of a triangle, of a solid hemisphere, of a hollow hemisphere, of a hollow cone, of a solid cone