Here is the list of center-of-mass formulas for different shapes. But before we look at it, let’s examine the significance of the center of mass.
So, the center of mass of a group or system of particles is the (geometrical) point that behaves as if
∙ all the mass of the system is located there &
∙ all the external forces are acting there
The above definition of center of mass is a result of the way the position coordinates of the center of mass of a group or system of particles is defined
→𝑟𝐶𝑀 =∑𝑖𝑚𝑖→𝑟𝑖∑𝑖𝑚𝑖
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 →𝐹𝑔 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 →𝐹𝑔 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.
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